Fast decoding of lifted interleaved linearized Reed–Solomon codes for multishot network coding

Abstract

Martínez-Peñas and Kschischang (IEEE Trans. Inf. Theory 65(8):4785–4803, 2019) proposed lifted linearized Reed–Solomon codes as suitable codes for error control in multishot network coding. We show how to construct and decode lifted interleaved linearized Reed–Solomon (LILRS) codes. Compared to the construction by Martínez-Peñas–Kschischang, interleaving allows to increase the decoding region significantly and decreases the overhead due to the lifting (i.e., increases the code rate), at the cost of an increased packet size. We propose two decoding schemes for LILRS that are both capable of correcting insertions and deletions beyond half the minimum distance of the code by either allowing a list or a small decoding failure probability. We propose a probabilistic unique Loidreau–Overbeck-like decoder for LILRS codes and an efficient interpolation-based decoding scheme that can be either used as a list decoder (with exponential worst-case list size) or as a probabilistic unique decoder. We derive upper bounds on the decoding failure probability of the probabilistic-unique decoders which show that the decoding failure probability is very small for most channel realizations up to the maximal decoding radius. The tightness of the bounds is verified by Monte Carlo simulations.

1 Introduction

Network coding [1] is a powerful approach to achieve the capacity of multicast networks. Unlike the classical routing schemes, network coding allows to mix (e.g. linearly combine) incoming packets at intermediate nodes. Kötter and Kschischang proposed codes in the subpsace metric as a suitable tool for error correction in (random) linear network coding [18] and defined the corresponding operator channel model. In an operator channel two type of errors, namely insertions and deletions, can occur. Insertions correspond to additional dimensions that are not contained in the transmitted space whereas deletions correspond to dimensions that are removed from the transmitted space.

The single-shot scenario from [18] was extended to the multishot case, i.e. the transmission over several instances of the operator channel, in [30]. It was shown in [27] that lifted linearized Reed–Solomon (LLRS) codes provide reliable and secure coding schemes for noncoherent multishot network coding, a scenario where the in-network linear combinations are not known or used at the receiver, under an adversarial model in the sum-subspace metric.

An $s$-interleaved code is a direct sum of $s$ codes of the same length (called constituent codes). This means that if the constituent codes are over $\mathbb F_{q}$, then the interleaved code can be viewed as a (not necessarily linear) code over $\mathbb F_{q^s}$. In the Hamming and rank metric, there are various decoders that can significantly increase the decoding radius of a constituent code by collaboratively decoding in an interleaved variant thereof. Such decoders are known in the Hamming metric for Reed–Solomon [8, 10, 13, 14, 19, 28, 34, 35, 37, 42, 43, 51, 52] and in general algebraic geometry codes [9, 17, 41], and in the rank metric for Gabidulin codes [7, 24, 39, 40, 44,45,46, 50]. All of these decoders have in common that they are either list decoders with exponential worst-case and small average-case list size, or probabilistic unique decoders that fail with a very small probability.

Interleaving was suggested in [48] as a method to decrease the overhead in lifted Gabidulin codes for error correction in noncoherent (single-shot) network coding, at the cost of a larger packet size while preserving a low decoding complexity. It was later shown [6, 46, 50] that it can also increase the error-correction capability of the code using suitable decoders for interleaved Gabidulin codes.

1.1 Our contributions

In this paper, we define and analyze lifted interleaved linearized Reed-Solomon (LILRS) codes that are obtained by lifting interleaved linearized Reed–Solomon (ILRS) codes as considered in [4]. We propose and analyze two decoding schemes that both allow for decoding insertions and deletions beyond the unique decoding region by allowing a (potentially exponential) list or a small decoding failure probability.

First, we propose a Loidreau–Overbeck-like probabilistic unique decoder and derive an upper bound on the decoding failure probability. The upper bound shows the the decoder succeeds with an overwhelming probability close to one for random realizations of the multishot operator channel that stay within the decoding region.

The second decoding approach is a novel interpolation-based list decoder that is based on the list decoder by Wachter-Zeh and Zeh [50] for interleaved Gabidulin codes. We derive a decoding region for the codes in the sum-subspace metric, analyze the complexity of the decoder and give an exponential upper bound on the list size. We derive an upper bound on the decoding failure probability of the interpretation as a probabilistic-unique decoder by relating the success conditions to the Loidreau–Overbeck-like decoder.

Compared to [27], we decrease the relative overhead introduced by lifting (or equivalently, increase the rate for the same code length and block size) and at the same time extend the decoding region, especially for insertions, significantly. These advantages come at the cost of a larger packet size of the packets within the network and a supposedly small failure probability. By considering the decoding problem in the complementary code, we show how the proposed LILRS coding schemes can be used to improve the decoding region w.r.t. deletions significantly.

Moreover, for the case $s=1$ (no interleaving), our algorithm does not require the assumption from [27, Sec. V.H] that $n_r\le n_t$, where $n_t$ and $n_r$ denotes the sum of the dimensions of the transmitted and received subspaces, respectively. Hence, the proposed decoder works in cases in which [27] does not work.

Compared to the conference version [3], which mainly considers interpolation-based decoding of LILRS codes, this work contains several new results. Apart from providing proofs that were omitted in [3] due to space restrictions, we propose and analyze a new Loidreau–Overbeck-like decoder for LILRS codes that allows for deriving a strict upper bound on the decoding failure probability. Compared to the heuristic upper bound provided in [3] the new upper bound takes into account the distribution of insertions in the multishot operator channel. Another novelty compared to the conference version is the definition and analysis of complementary LILRS codes which allows for an improved deletion-correction capability in the interleaved setting. We also provide tools that allow for an efficient implementation of the multishot operator channel that were also not considered in [3].

The main results of this paper, in particular the improvements upon the existing noninterleaved variants, are illustrated in Table 1.

Table 1 Overview of new decoding regions and computational complexities

1.2 Structure of the paper

In Sect. 2 the notation as well as basic definitions on vector spaces and skew polynomials are introduced. Section 3 is dedicated to a brief introduction to multishot network coding. In Sect. 4 we consider decoding of LILRS codes in the sum-subspace metric for error-control in noncoherent multishot network coding. In particular, we derive a Loidreau–Overbeck-like and an interpolation-based decoding scheme for LILRS codes, that can correct errors beyond the unique decoding radius in the sum-subspace metric. Section 5 concludes the paper.

2 Preliminaries

We now give some definitions and notation related to multishot network coding and LLRS codes. Since the notation for multishot network coding is quite involved, we tried to stick to common notation as much as possible such that readers familiar with the topic can skip parts of this section.

2.1 Notation

A set is a collection of distinct elements and is denoted by $\mathcal {S}=\{s_1,s_2,\dots ,s_{r}\}$. The cardinality of $\mathcal {S}$, i.e. the number of elements in $\mathcal {S}$, is denoted by $|\mathcal {S}|$. By $[i,j]$ with $i<j$ we denote the set of integers $\{i,i+1,\dots ,j\}$. We denote the set of nonnegative integers by $\mathbb {Z}_{\ge 0}=\{0,1,2,\dots \}$.

Let $\mathbb F_{q}$ be a finite field of order q and denote by $\mathbb F_{q^m}$ the extension field of $\mathbb F_{q}$ of degree m with primitive element $\alpha$. The multiplicative group $\mathbb F_{q^m}\setminus \{0\}$ of $\mathbb F_{q^m}$ is denoted by $\mathbb F_{q^m}^*$. Matrices and vectors are denoted by bold uppercase and lowercase letters like $\varvec{A}$ and $\varvec{a}$, respectively, and indexed starting from one. Under a fixed basis of $\mathbb F_{q^m}$ over $\mathbb F_{q}$ any element $a\in \mathbb F_{q^m}$ can be represented by a corresponding column vector $\varvec{a}\in \mathbb F_{q}^{m\times 1}$. For a matrix $\varvec{A}\in \mathbb F_{q^m}^{M\times N}$ the $\mathbb F_{q}$-rank ${{\,\textrm{rk}\,}}_q(\varvec{A})$ of $\varvec{A}$ is defined as the maximum number of $\mathbb F_{q}$-linearly independent columns of $\varvec{A}$. Let $\sigma :\mathbb F_{q^m}\rightarrow \mathbb F_{q^m}$ be a finite field automorphism given by $\sigma (a)=a^{q^r}$ for all $a\in \mathbb F_{q^m}$, where we assume that $1 \le r \le m$ and $\gcd (r,m)=1$. For a matrix $\varvec{A}$ and a vector ${\varvec{a}}$ we use the notation $\sigma (\varvec{A})$ and $\sigma ({\varvec{a}})$ to denote the element-wise application of the automorphism $\sigma$, respectively. For $\varvec{A}\in \mathbb F_{q^m}^{M\times N}$ we denote by ${\left\langle \varvec{A} \right\rangle }_{q}$ the $\mathbb F_{q}$-linear rowspace of the matrix $\varvec{A}_q\in \mathbb F_{q}^{M\times Nm}$ obtained by row-wise expanding the elements in $\varvec{A}$ over $\mathbb F_{q}$. The left and right kernel of a matrix $\varvec{A}\in \mathbb F_{q^m}^{M\times N}$ is denoted by $\ker _l(\varvec{A})$ and $\ker _r(\varvec{A})$, respectively.

For a set $\mathcal {I}\subset \mathbb {Z}_{>0}$ we denote by $[\varvec{A}]_{\mathcal {I}}$ (respectively $[{\varvec{a}}]_{\mathcal {I}}$) the matrix (vector) consisting of the columns (entries) of the matrix $\varvec{A}$ (vector ${\varvec{a}}$) indexed by $\mathcal {I}$.

2.2 Vector spaces and subspace metrics

Vector spaces are denoted by calligraphic letters such as e.g. $\mathcal {V}$. By $\mathcal {P}_q(N)$ we denote the set of all subspaces of $\mathbb F_{q}^{N}$. The Grassmannian is the set of all l-dimensional subspaces in $\mathcal {P}_q(N)$ and is denoted by $\mathcal {G}_q(N,l)$. The cardinality of the Grassmannian $|\mathcal {G}_q(N,l)|$, i.e. the number of l-dimensional subspace of $\mathbb F_{q}^N$ is given by the Gaussian binomial $\left[ \genfrac{}{}{0.0pt}{}{N}{l}\right] _{q}$ which is defined as

$$\begin{aligned} \left[ \genfrac{}{}{0.0pt}{}{N}{l}\right] _{q} = \prod _{i=1}^{l} \frac{q^{N-l+i}-1}{q^i-1}. \end{aligned}$$

The Gaussian binomial satisfies (see e.g. [18])

$$\begin{aligned} q^{(N-l)l} \le \textstyle \left[ \genfrac{}{}{0.0pt}{}{N}{l}\right] _{q} \le \kappa _qq^{(N-l)l}, \end{aligned}$$

where $\kappa _q:= \prod _{i=1}^{\infty } (1-q^{-i})^{-1}$. Note that $\kappa _q$ is monotonically decreasing in q with a limit of 1, and e.g. $\kappa _2 \approx 3.463$, $\kappa _3 \approx 1.785$, and $\kappa _4 \approx 1.452$.

The subspace distance between two subspaces $\mathcal {U},\mathcal {V}\in \mathcal {P}_q(N_i)$ is defined as (see [18])

$$\begin{aligned} d_S(\mathcal {U},\mathcal {V}):=\dim (\mathcal {U}+\mathcal {V})-\dim (\mathcal {U}\cap \mathcal {V}). \end{aligned}$$

For $\varvec{N}=(N_1,N_2,\dots ,N_\ell )$ and $\varvec{l}=(l_1, l_2, \dots , l_\ell )$ we define the $\ell$-fold Cartesian products

$$\begin{aligned} \mathcal {P}_q(\varvec{N}) :=\textstyle \prod _{i=1}^{\ell }\mathcal {P}_q(N_i)=\mathcal {P}_q(N_1)\times \dots \times \mathcal {P}_q(N_\ell ) \end{aligned}$$

and

$$\begin{aligned} \mathcal {G}_q(\varvec{N}, \varvec{l}) :=\textstyle \prod _{i=1}^{\ell }\mathcal {G}_q(N_i, l_i)=\mathcal {G}_q(N_1,l_1)\times \dots \times \mathcal {G}_q(N_\ell , l_\ell ). \end{aligned}$$

For any tuple of subspaces $\varvec{\mathcal {V}}\in \mathcal {P}_q(\varvec{N})$ we define its sum-dimension as

$$\begin{aligned} {{\,\mathrm{\dim _{\Sigma }}\,}}(\varvec{\mathcal {V}}):=\sum _{i=1}^{\ell }\dim ({\mathcal {V}}^{(i)}) \end{aligned}$$

and call the tuple

$$\begin{aligned} \left( \dim ({\mathcal {V}}^{(1)}),\dim ({\mathcal {V}}^{(2)}),\dots ,\dim ({\mathcal {V}}^{(\ell )})\right) \end{aligned}$$

the sum-dimension partition of $\mathcal {V}$.

We extend fundamental operators on subspaces to tuples of subspaces by considering their application in an element-wise manner, i.e. for $\varvec{\mathcal {V}},\varvec{\mathcal {U}}\in \mathcal {P}_q(\varvec{N}, \varvec{l})$ we define

$$\begin{aligned} \varvec{\mathcal {V}}^\perp&:=\left( {{\mathcal {V}}^{(1)}}^\perp , {{\mathcal {V}}^{(2)}}^\perp , \dots , {{\mathcal {V}}^{(\ell )}}^\perp \right) , \\ \varvec{\mathcal {V}}\cap \varvec{\mathcal {U}}&:=\left( {\mathcal {V}}^{(1)} \cap {\mathcal {U}}^{(1)}, {\mathcal {V}}^{(2)} \cap {\mathcal {U}}^{(2)}, \dots , {\mathcal {V}}^{(\ell )} \cap {\mathcal {U}}^{(\ell )} \right) , \\ \varvec{\mathcal {V}}\oplus \varvec{\mathcal {U}}&:=\left( {\mathcal {V}}^{(1)} \oplus {\mathcal {U}}^{(1)}, {\mathcal {V}}^{(2)} \oplus {\mathcal {U}}^{(2)}, \dots , {\mathcal {V}}^{(\ell )} \oplus {\mathcal {U}}^{(\ell )} \right) . \end{aligned}$$

We conclude this subsection considering the extension of the subspace distance to the multishot case [30].

Definition 1

(Sum-Subspace Distance [30]) Given $\varvec{\mathcal {U}}=({\mathcal {U}}^{(1)},{\mathcal {U}}^{(2)},\dots ,\mathcal {U}^{(\ell )})\in \mathcal {P}_q(\varvec{N})$ and $\varvec{\mathcal {V}}=({\mathcal {V}}^{(1)},{\mathcal {V}}^{(2)},\dots ,{\mathcal {V}}^{(\ell )})\in \mathcal {P}_q(\varvec{N})$ the sum-subspace distance between $\varvec{\mathcal {U}}$ and $\varvec{\mathcal {V}}$ is defined as

$$\begin{aligned} d_{\Sigma S}(\varvec{\mathcal {U}},\varvec{\mathcal {V}}):=&\sum _{i=1}^{\ell }d_S({\mathcal {U}}^{(i)},{\mathcal {V}}^{(i)}) =\sum _{i=1}^{\ell }\left( \dim ({\mathcal {U}}^{(i)}+{\mathcal {V}}^{(i)})-\dim ({\mathcal {U}}^{(i)}\cap {\mathcal {V}}^{(i)})\right) . \end{aligned}$$

(1)

In [29, 30] the sum-subspace distance is also called the extended subspace distance. Note, that there exists other metrics for multishot network coding like e.g. the sum-injection distance (see [27, Definition 7]) which are related to the sum-subspace distance and are not considered in this work.

2.3 Skew polynomials

Skew polynomials are a special class of non-commutative polynomials that were introduced by Ore [31]. A skew polynomial is a polynomial of the form

$$\begin{aligned} \textstyle f(x)=\sum _{i}f_i x^i \end{aligned}$$

with a finite number of coefficients $f_i\in \mathbb F_{q^m}$ being nonzero. The degree $\deg (f)$ of a skew polynomial f is defined as $\max \{i:f_i\ne 0\}$ if $f\ne 0$ and $-\infty$ otherwise.

The set of skew polynomials with coefficients in $\mathbb F_{q^m}$ together with ordinary polynomial addition and the multiplication rule

$$\begin{aligned} xa=\sigma (a)x,\qquad a\in \mathbb F_{q^m}\end{aligned}$$

forms a non-commutative ring denoted by $\mathbb F_{q^m}[x;\sigma ]$.

The set of skew polynomials in $\mathbb F_{q^m}[x;\sigma ]$ of degree less than k is denoted by $\mathbb F_{q^m}[x;\sigma ]_{<k}$. For any $a,b\in \mathbb F_{q^m}$ we define the operator

$$\begin{aligned} \mathcal {D}_{a}\left( b\right) :=\sigma (b)a. \end{aligned}$$

For an integer $i\ge 0$, we define (see [26, Proposition 32])

$$\begin{aligned} \mathcal {D}_{a}^{i+1}(b) =\mathcal {D}_{a}\left( \mathcal {D}_{a}^{i}(b)\right) =\sigma ^{i+1}(b)\mathcal {N}_{i+1}(a) \end{aligned}$$

where $\mathcal {D}_{a}^{0}(b)=b$ and $\mathcal {N}_{i}(a)=\sigma ^{i-1}(a)\sigma ^{i-2}(a)\dots \sigma (a)a$ is the generalized power function (see [21]). The generalized operator evaluation of a skew polynomial $f\in \mathbb F_{q^m}[x;\sigma ]$ at an element b w.r.t. a, where $a,b\in \mathbb F_{q^m}$, is defined as (see [22, 26])

$$\begin{aligned} {f}(b)_{a}=\sum _{i}f_i\mathcal {D}_{a}^{i}(b). \end{aligned}$$

For any fixed evaluation parameter $a \in \mathbb F_{q^m}$ the generalized operator evaluation forms an $\mathbb F_{q}$-linear map, i.e. for any $f\in \mathbb F_{q^m}[x;\sigma ]$, $\beta ,\gamma \in \mathbb F_{q}$ and $b,c\in \mathbb F_{q^m}$ we have that

$$\begin{aligned} {f}(\beta b+\gamma c)_{a}=\beta {f}(b)_{a}+\gamma {f}(c)_{a}. \end{aligned}$$

For two skew polynomials $f,g\in \mathbb F_{q^m}[x;\sigma ]$ and elements $a,b\in \mathbb F_{q^m}$ the generalized operator evaluation of the product $f\cdot g$ at b w.r.t a is given by (see [25])

$$\begin{aligned} {(f\cdot g)}(b)_{a} = {f}({g}(b)_{a})_{a}. \end{aligned}$$

An important notion for the generalized operator evaluation is the concept of conjugacy, which is defined as follows.

Definition 2

(Conjugacy [20]) For any two elements $a\in \mathbb F_{q^m}$ and $c\in \mathbb F_{q^m}^*$ define

$$\begin{aligned} a^c:=\sigma (c)ac^{-1}. \end{aligned}$$

• Two elements $a,b\in \mathbb F_{q^m}$ are called $\sigma$-conjugates, if there exists an element $c\in \mathbb F_{q^m}^*$ such that $b=a^c$.

• Two elements that are not $\sigma$-conjugates are called $\sigma$-distinct.

The notion of $\sigma$-conjugacy defines an equivalence relation on $\mathbb F_{q^m}$ and thus a partition of $\mathbb F_{q^m}$ into conjugacy classes [21]. The set

$$\begin{aligned} \mathcal {C}(a):=\left\{ a^c:c\in \mathbb F_{q^m}^*\right\} \end{aligned}$$

is called conjugacy class of a. A finite field $\mathbb F_{q^m}$ has at most $\ell \le q-1$ distinct conjugacy classes. For $\ell \le q-1$ the elements $1,\alpha ,\alpha ^2,\dots ,\alpha ^{\ell -2}$ are representatives of all (nontrivial) disjoint conjugacy classes of $\mathbb F_{q^m}$.

Let $a_1,\dots ,a_\ell$ be representatives from different conjugacy classes of $\mathbb F_{q^m}$ and let $b_1^{(i)},\dots ,b_{n_i}^{(i)}$ be elements from $\mathbb F_{q^m}$ for all $i=1,\dots ,\ell$. Then for any nonzero $f\in \mathbb F_{q^m}[x;\sigma ]$ satisfying

$$\begin{aligned} {f}(b_j^{(i)})_{a_i}=0, \qquad \forall i=1,\dots ,\ell ,j=1,\dots ,n_i \end{aligned}$$

we have that $\deg (f)\le \sum _{i=1}^{\ell }n_i$ where equality holds if and only if the $b_1^{(i)},\dots ,b_{n_i}^{(i)}$ are $\mathbb F_{q}$-linearly independent for each $i=1,\dots ,\ell$ (i.e. for each evaluation parameter $a_i$, see [12]).

The existence of a (generalized operator evaluation) interpolation polynomial is considered in Lemma 1 (see e.g. [12]).

Lemma 1

(Lagrange Interpolation (Generalized Operator Evaluation)) Let $b_1^{(i)},\dots ,b_{n_i}^{(i)}$ be $\mathbb F_{q}$-linearly independent elements from $\mathbb F_{q^m}$ for all $i=1,\dots ,\ell$. Let $c_1^{(i)},\dots ,c_{n_i}^{(i)}$ be elements from $\mathbb F_{q^m}$ and let $a_1,\dots ,a_\ell$ be representatives for different conjugacy classes of $\mathbb F_{q^m}$. Define the set of tuples $\mathcal {B} :=\{(b_j^{(i)}, c_j^{(i)}, a_i):i=1,\dots ,\ell , \, j=1,\dots ,n_i\}$. Then there exists a unique interpolation polynomial $\mathcal {I}_{\mathcal {B}}^{\textrm{op}}\in \mathbb F_{q^m}[x;\sigma ]$ such that

$$\begin{aligned} {\mathcal {I}_{\mathcal {B}}^{\textrm{op}}}(b_j^{(i)})_{a_i}=c_j^{(i)},\qquad \forall i=1,\dots ,\ell , \, \forall j=1,\dots , n_i, \end{aligned}$$

and $\deg (\mathcal {I}_{\mathcal {B}}^{\textrm{op}})<\sum _{i=1}^{\ell }n_i$.

For an element $a\in \mathbb F_{q^m}$, a vector $\varvec{b}\in \mathbb F_{q^m}^n$ and a skew polynomial $f\in \mathbb F_{q^m}[x;\sigma ]$ we define

$$\begin{aligned} {f}(\varvec{b})_{a}:=({f}(b_1)_{a},{f}(b_2)_{a},\dots ,{f}(b_n)_{a}). \end{aligned}$$

The set of all skew polynomials of the form

$$\begin{aligned} Q(x, y_1,\dots , y_s)=Q_0(x)+Q_1(x)y_1+\dots +Q_s(x)y_s, \end{aligned}$$

where $Q_j\in \mathbb F_{q^m}[x;\sigma ]$ for all $j=0,\dots ,s$ is denoted by $\mathbb F_{q^m}[x,y_1,\dots ,y_s;\sigma ]$.

Definition 3

($\varvec{w}$-weighted Degree) Given a vector $\varvec{w}\in \mathbb {Z}_{\ge 0}^{s+1}$, the $\varvec{w}$-weighted degree of a multivariate skew polynomial $Q\in \mathbb F_{q^m}[x,y_1,\dots ,y_s;\sigma ]$ is defined as

$$\begin{aligned} \deg _{\varvec{w}}(Q)=\max _j\{\deg (Q_j)+w_j\}. \end{aligned}$$

For an element $a\in \mathbb F_{q^m}$ and a vector $\varvec{b}=(b_1,b_2,\dots ,b_n)\in \mathbb F_{q^m}^n$ we define the vector

$$\begin{aligned} \mathcal {D}_{a}^{j}(\varvec{b}) :=\left( \mathcal {D}_{a}^{j}(b_1), \mathcal {D}_{a}^{j}(b_2), \dots , \mathcal {D}_{a}^{j}(b_n)\right) \end{aligned}$$

and the matrix

$$\begin{aligned} \varvec{V}_{d}(\varvec{b})_{a} :=\begin{pmatrix} \varvec{b}\\ \mathcal {D}_{a}^{1}(\varvec{b}) \\ \mathcal {D}_{a}^{2}(\varvec{b}) \\ \vdots \\ \mathcal {D}_{a}^{d-1}(\varvec{b}) \end{pmatrix} = \begin{pmatrix} b_1 &{} b_2 &{} \dots &{} b_n \\ \mathcal {D}_{a}^{1}(b_1) &{} \mathcal {D}_{a}^{1}(b_2) &{} \dots &{} \mathcal {D}_{a}^{1}(b_n) \\ \mathcal {D}_{a}^{2}(b_1) &{} \mathcal {D}_{a}^{2}(b_2) &{} \dots &{} \mathcal {D}_{a}^{2}(b_n) \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \mathcal {D}_{a}^{d-1}(b_1) &{} \mathcal {D}_{a}^{d-1}(b_2) &{} \dots &{} \mathcal {D}_{a}^{d-1}(b_n) \end{pmatrix} \in \mathbb F_{q^m}^{d\times n}. \end{aligned}$$

For a vector $\varvec{x}=\left( \varvec{x}^{(1)} \mid \varvec{x}^{(2)} \mid \dots \mid \varvec{x}^{(\ell )}\right) \in \mathbb F_{q^m}^n$ with $\varvec{x}^{(i)}\in \mathbb F_{q^m}^{n_i}$ for all $i=1,\dots ,\ell$, a length partition $\varvec{n}=(n_1,n_2,\dots ,n_\ell )\in \mathbb {Z}_{\ge 0}^\ell$ such that $\sum _{i=1}^{\ell }n_i=n$ and a vector ${\varvec{a}} = (a_1,a_2,\dots ,a_\ell )\in \mathbb F_{q^m}^\ell$ we define the vector¹

$$\begin{aligned} \mathcal {D}_{{\varvec{a}}}^{i}(\varvec{x}) := \begin{pmatrix}&\mathcal {D}_{a_1}^{i}(\varvec{x}^{(1)})&\big |&\mathcal {D}_{a_2}^{i}(\varvec{x}^{(2)})&\big |&\dots&\big |&\mathcal {D}_{a_\ell }^{i}(\varvec{x}^{(\ell )})&\end{pmatrix} \in \mathbb F_{q^m}^n. \end{aligned}$$

By the properties of the operator $\mathcal {D}_{a}^{i}(\cdot )$, we have that

$$\begin{aligned} \mathcal {D}_{{\varvec{a}}}^{i+j}(\varvec{x}) = \mathcal {D}_{{\varvec{a}}}^{j}(\mathcal {D}_{{\varvec{a}}}^{i}(\varvec{x})) \end{aligned}$$

and

$$\begin{aligned} \mathcal {D}_{{\varvec{a}}}^{i}(\xi \varvec{x}) = \sigma ^i(\xi ) \mathcal {D}_{{\varvec{a}}}^{i}(\varvec{x}) \quad \forall \, \xi \in \mathbb F_{q^m}. \end{aligned}$$

For a matrix

$$\begin{aligned} \varvec{X}= \begin{pmatrix} \varvec{x}_1 \\ \varvec{x}_2 \\ \vdots \\ \varvec{x}_d \end{pmatrix} \in \mathbb F_{q^m}^{d\times n}, \end{aligned}$$

an integer j and a vector ${\varvec{a}} = (a_1,a_2,\dots ,a_\ell )\in \mathbb F_{q^m}^\ell$ we define $\mathcal {D}_{{\varvec{a}}}^{j}(\cdot )$ applied to $\varvec{X}$ as

$$\begin{aligned} \mathcal {D}_{{\varvec{a}}}^{j}(\varvec{X}):=\begin{pmatrix} \mathcal {D}_{{\varvec{a}}}^{j}(\varvec{x}_1) \\ \mathcal {D}_{{\varvec{a}}}^{j}(\varvec{x}_2) \\ \vdots \\ \mathcal {D}_{{\varvec{a}}}^{j}(\varvec{x}_d) \end{pmatrix}. \end{aligned}$$

The element-wise application of the operator to matrices does not affect the rank, i.e. we have that ${{\,\textrm{rk}\,}}_{q^m}(\mathcal {D}_{{\varvec{a}}}^{j}(\varvec{X})) = {{\,\textrm{rk}\,}}_{q^m}(\varvec{X})$ (see [4, Lemma 3]).

Definition 4

($\sigma$-Generalized Moore Matrix) For an integer $d \in \mathbb {Z}_{>0}$, a length partition $\varvec{n}=(n_1,n_2,\dots ,n_\ell )\in \mathbb {Z}_{\ge 0}^\ell$ such that $\sum _{i=1}^{\ell }n_i=n$ and the vectors ${\varvec{a}}=(a_1,a_2,\dots ,a_\ell )\in \mathbb F_{q^m}^\ell$ and $\varvec{x}=\left( \varvec{x}^{(1)} \mid \varvec{x}^{(2)} \mid \dots \mid \varvec{x}^{(\ell )}\right) \in \mathbb F_{q^m}^n$ with $\varvec{x}^{(i)}\in \mathbb F_{q^m}^{n_i}$ for all $i=1,\dots ,\ell$, the $\sigma$-Generalized Moore matrix is defined as

$$\begin{aligned} \Lambda _d(\varvec{x})_{{\varvec{a}}} :=\begin{pmatrix} \mathcal {D}_{{\varvec{a}}}^{0}(\varvec{x}) \\ \mathcal {D}_{{\varvec{a}}}^{1}(\varvec{x}) \\ \vdots \\ \mathcal {D}_{{\varvec{a}}}^{d-1}(\varvec{x}) \end{pmatrix} = \left( \begin{array}{c|c|c|c} \varvec{V}_{d}(\varvec{x}^{(1)})_{a_1}&\varvec{V}_{d}(\varvec{x}^{(2)})_{a_2}&\cdots&\varvec{V}_{d}(\varvec{x}^{(\ell )})_{a_\ell } \end{array}\right) \in \mathbb F_{q^m}^{d \times n}. \end{aligned}$$

Similar as for ordinary polynomials and Vandermonde matrices, there is a relation between the generalized operator evaluation and the product with a $\sigma$-Generalized Moore matrix. In particular, for a skew polynomial $f(x)=\sum _{i=0}^{k-1} f_i x^i \in \mathbb F_{q^m}[x;\sigma ]_{<k}$ and vectors ${\varvec{a}}=(a_1,a_2,\dots ,a_\ell )\in \mathbb F_{q^m}^\ell$ and $\varvec{x}=\left( \varvec{x}^{(1)} \mid \varvec{x}^{(2)} \mid \dots \mid \varvec{x}^{(\ell )}\right) \in \mathbb F_{q^m}^n$ we have that

$$\begin{aligned} {f}({\varvec{a}})_{\varvec{x}} = (f_0, f_1, \dots , f_{k-1}) \cdot \Lambda _k(\varvec{x})_{{\varvec{a}}}. \end{aligned}$$

The rank of a $\sigma$-Generalized Moore matrix satisfies ${{\,\textrm{rk}\,}}_{q^m}(\Lambda _d(\varvec{x})_{{\varvec{a}}})=\min \{d,n\}$ if and only if ${{\,\textrm{wt}\,}}_{\Sigma R}(\varvec{x})=n$ (see e.g. [21, Theorem 4.5]).

Remark 1

To simplify the notation we omit the rank partition $\varvec{n}$ in $\Lambda _j(\cdot )_{{\varvec{a}}}$ since it will be always clear from the context (i.e. the length partition of the considered vector).

3 Multishot network coding

As a channel model we consider the multishot operator channel from [30] which consists of multiple independent channel uses of the operator channel from [18]. The operator channel is a discrete channel that relates the input $\mathcal {V}\in \mathcal {P}_q(N)$ with $n_t:=\dim (\mathcal {V})$ to the output $\mathcal {U}\in \mathcal {P}_q(N)$ by

$$\begin{aligned} \mathcal {U}= \mathcal {H}_{n_t-\delta }(\mathcal {V}) \oplus \mathcal {E}\end{aligned}$$

(2)

where $\mathcal {H}_{n_t-\delta }(\mathcal {V})$ is an erasure operator that returns an $(n_t-\delta )$-dimensional subspace of $\mathcal {V}$ and $\mathcal {E}\in \mathcal {G}_q(N,\gamma )$ is a $\gamma$-dimensional subspace with $\mathcal {V}\cap \mathcal {E}= \{\textbf{0}\}$. The dimension of the received space $n_r:=\dim (\mathcal {U})$ is then

$$\begin{aligned} n_r= n_t- \delta + \gamma \end{aligned}$$

where $\delta$ is called the number of deletions and $\gamma$ is called the number of insertions. Observe that the subspace distance between the input $\mathcal {V}$ and the output $\mathcal {U}$ is $d_S(\mathcal {V},\mathcal {U})=\gamma +\delta$.

3.1 Multishot operator channel

A multishot (or $\ell$-shot) operator channel [30] with overall $\gamma$ insertions and $\delta$ deletions is a discrete channel with input and output alphabet $\mathcal {P}_q(\varvec{N})$. Consider the partitions of insertions $\varvec{\gamma }=({\gamma }^{(1)},{\gamma }^{(2)},\dots ,{\gamma }^{(\ell )})$ and deletions $\varvec{\delta }=({\delta }^{(1)},{\delta }^{(2)},\dots ,{\delta }^{(\ell )})$ such that $\gamma = \sum _{i=1}^{\ell } {\gamma }^{(i)}$ and $\delta = \sum _{i=1}^{\ell } {\delta }^{(i)}$. The input is a tuple of subspaces $\varvec{\mathcal {V}}\in \mathcal {P}_q(\varvec{N})$ with sum-dimension ${{\,\mathrm{\dim _{\Sigma }}\,}}(\varvec{\mathcal {V}}) = n_t$ and sum-dimension partition $\varvec{n}_{t}$. The output $\varvec{\mathcal {U}}\in \mathcal {P}_q(\varvec{N})$ is related to the input $\varvec{\mathcal {V}}\in \mathcal {P}_q(\varvec{N})$ by

$$\begin{aligned} \varvec{\mathcal {U}}=\mathcal {H}_{n_t-\delta }(\varvec{\mathcal {V}})\oplus \varvec{\mathcal {E}}, \end{aligned}$$

(3)

where $\mathcal {H}_{n_t-\delta }(\varvec{\mathcal {V}})$ returns a tuple with sum-dimension $n_t-\delta$ and sum-dimension partition $\varvec{n}_{t}-\varvec{\delta }$ and $\varvec{\mathcal {E}}\in \mathcal {G}_q(\varvec{N},\varvec{\gamma })$ is a tuple of error spaces with ${{\,\mathrm{\dim _{\Sigma }}\,}}(\varvec{\mathcal {E}})=\gamma$ and $\varvec{\mathcal {V}}\cap \varvec{\mathcal {E}}=\{\textbf{0}\}$. The multishot operator channel can be considered as $\ell$ instances of the (single-shot) operator channel defined in (2) such that an overall number of $\gamma$ insertions and $\delta$ deletions occurs.

The output $\varvec{\mathcal {U}}$ of the multishot operator channel has sum-dimension

$$\begin{aligned} n_r=n_t-\delta +\gamma \end{aligned}$$

with sum-dimension partition

$$\begin{aligned} \varvec{n}_r= \varvec{n}_{t}- \varvec{\delta }+ \varvec{\gamma }. \end{aligned}$$

(4)

The multishot operator channel is illustrated in Figure 1.

Fig. 1

Illustration of the $\ell$-shot operator channel

In [27, 29] a sum-rank-metric representation of the multishot operator channel in the spirit of [47, 48] was considered. This equivalent channel representation is more suitable for decoding of LILRS codes in the sum-rank metric. In this work, we consider the interpretation as a multishot operator channel that is more closely related to the sum-subspace metric.

Remark 2

By the term “random instance of the $\ell$-shot operator channel with overall $\gamma$ insertions and $\delta$ deletions” we mean, that we draw uniformly at random an instance from all instances of the $\ell$-shot operator channel (3), i.e. we draw uniformly at random from all partitions of the insertions $\varvec{\gamma }$ and deletions $\varvec{\delta }$, and for fixed $\varvec{\mathcal {V}}$, $\varvec{\gamma }$ and $\varvec{\delta }$, the error space is chosen uniformly at random from the set

$$\begin{aligned} \mathcal {E}^{(\varvec{\gamma })}:=\left\{ \varvec{\mathcal {E}}\in \mathcal {G}_q(\varvec{N}, \varvec{\gamma }):{\mathcal {V}}^{(i)}\cap {\mathcal {E}}^{(i)}=\{\varvec{0}\},\forall i \wedge {{\,\mathrm{\dim _{\Sigma }}\,}}(\varvec{\mathcal {E}})=\gamma \right\} . \end{aligned}$$

(5)

In Appendix 1 we propose an efficient procedure to implement random instances of the multishot operator channel for parameters $\overline{\varvec{N}}=(\overline{N},\dots ,\overline{N})$ and $\overline{\varvec{n}}_{t}=(\overline{n}_t,\dots ,\overline{n}_t)$ by adapting the dynamic-programming routine in [36, Appendix A] for drawing an error of given sum-rank weight uniformly at random to the sum-subspace case (see Algorithm 5).

We now extend the definition of $(\gamma ,\delta )$ reachability for the operator channel [7] to the multishot operator channel (see also [3, Definition 2]).

Definition 5

($(\gamma ,\delta )$ Reachability) Given two tuples of subspaces $\varvec{\mathcal {U}},\varvec{\mathcal {V}}\in \mathcal {P}_q(\varvec{N})$ we say that $\varvec{\mathcal {V}}$ is $(\gamma ,\delta )$-reachable from $\varvec{\mathcal {U}}$ if there exists a realization of the multishot operator channel (3) with $\gamma$ insertions and $\delta$ deletions that transforms the input $\varvec{\mathcal {V}}$ to the output $\varvec{\mathcal {U}}$.

Next, we relate the $(\gamma ,\delta )$-reachability with the sum-subspace distance (see also [3, Proposition 2]).

Proposition 1

Consider $\varvec{\mathcal {U}}\in \mathcal {P}_q(\varvec{N})$ and $\varvec{\mathcal {V}}\in \mathcal {P}_q(\varvec{N})$. If $\varvec{\mathcal {V}}$ is $(\gamma ,\delta )$-reachable from $\varvec{\mathcal {U}}$, then we have that $d_{\Sigma S}(\varvec{\mathcal {U}},\varvec{\mathcal {V}})=\gamma +\delta$.

Similar as in [18] we now define normalized parameters for codes in the sum-subspace metric. The normalized weight $\lambda$, the code rate R and the normalized minimum distance $\eta$ of a sum-subspace code $\mathcal {C}$ with parameters $N=\sum _{i=1}^{\ell }N_i$ and $n_t=\sum _{i=1}^{\ell }n_t^{(i)}$ is defined as

$$\begin{aligned} \lambda :=\sum _{i=1}^{\ell }\frac{n_t^{(i)}}{N}=\frac{n_t}{N}, R:=\frac{\log _q(\mathcal {C})}{\sum _{i=1}^{\ell }n_t^{(i)}N_i} \quad \text {and}\quad \eta :=\frac{d_{\Sigma S}(\mathcal {C})}{2n_t}=\frac{d_{\Sigma S}(\mathcal {C})}{2\lambda N}, \end{aligned}$$

(6)

respectively. The normalized parameters $\lambda$, R and $\eta$ defined in (6) lie naturally within the interval [0, 1]. Define $\overline{n}_t:=n_t/\ell$. For $n_t^{(i)}=\overline{n}_t$ for all $i=1,\dots ,\ell$ we can write the code rate as

$$\begin{aligned} R=\frac{\log _q(|\mathcal {C}|)}{\sum _{i=1}^{\ell }\overline{n}_tN_i} =\frac{\ell \log _q(|\mathcal {C}|)}{n_tN} =\frac{\ell \log _q(|\mathcal {C}|)}{\lambda N^2}. \end{aligned}$$

A sum-subspace code $\mathcal {C}$ is a non-empty subset of $\mathcal {P}_q(\varvec{N})$, and has minimum subspace distance $d_{\Sigma S}(\mathcal {C})$ when all subspaces in the code have distance at least $d_{\Sigma S}(\mathcal {C})$ and there is at least one pair of subspaces with distance exactly $d_{\Sigma S}(\mathcal {C})$. In the following we consider constant-shot-dimension codes², i.e. codes that inject the same number of (linearly independent) packets $n_t^{(i)}$ in a given shot. In this setup, we transmit a tuple of subspaces

$$\begin{aligned} \varvec{\mathcal {V}}=\left( {\mathcal {V}}^{(1)},{\mathcal {V}}^{(2)},\dots ,{\mathcal {V}}^{(\ell )}\right) \in \mathcal {G}_q(\varvec{N}, \varvec{n}_{t}) \end{aligned}$$

and receive a tuple of subspaces

$$\begin{aligned} \varvec{\mathcal {U}}=\left( {\mathcal {U}}^{(1)},{\mathcal {U}}^{(2)},\dots ,{\mathcal {U}}^{(\ell )}\right) \in \mathcal {G}_q(\varvec{N}, \varvec{n}_r), \end{aligned}$$

where

(7)

for all $i=1,\dots ,\ell$.

Similar as for subspace codes in [18] we now define complementary sum-subspace codes. For any $\mathcal {V}\in \mathcal {P}_q(N)$ the dual space $\mathcal {V}^{\perp }$ is defined as

$$\begin{aligned} \mathcal {V}^{\perp }:=\{\varvec{u}\in \mathbb F_{q}^{N}:\varvec{u}\varvec{v}^\top =0,\,\forall \varvec{v}\in \mathcal {V}\} \end{aligned}$$

where $\dim (\mathcal {V}^{\perp })=N-\dim (\mathcal {V})$. For a tuple $\varvec{\mathcal {V}}=({\mathcal {V}}^{(1)},{\mathcal {V}}^{(2)},\dots ,{\mathcal {V}}^{(\ell )})\in \mathcal {P}_q(\varvec{N})$ we define the dual tuple as

$$\begin{aligned} \varvec{\mathcal {V}}^\perp :=\left( ({\mathcal {V}}^{(1)})^\perp ,({\mathcal {V}}^{(2)})^\perp ,\dots ,({\mathcal {V}}^{(\ell )})^\perp \right) \in \mathcal {P}_q(\varvec{N}). \end{aligned}$$

Note, that if $\varvec{\mathcal {V}}\in \mathcal {G}_q(\varvec{N},\varvec{n}_{t})$, then we have that $\varvec{\mathcal {V}}^\perp \in \mathcal {G}_q(\varvec{N},\varvec{N}-\varvec{n}_{t})$. By applying [18, Eq. 4] to the subspace distance between each component space of two tuples $\varvec{\mathcal {U}},\varvec{\mathcal {V}}\in \mathcal {P}_q(\varvec{N})$ in (1) we get that

$$\begin{aligned} d_{\Sigma S}(\varvec{\mathcal {V}}^\perp ,\varvec{\mathcal {U}}^\perp )=d_{\Sigma S}(\varvec{\mathcal {V}},\varvec{\mathcal {U}}). \end{aligned}$$

Consider a constant-shot-dimension sum-subspace code $\mathcal {C} \subseteq \mathcal {G}_q(\varvec{N},\varvec{n}_{t})$. Then the complementary constant-shot-dimension sum-subspace code $\mathcal {C}^\perp$ is defined as

$$\begin{aligned} \mathcal {C}^\perp :=\{\varvec{\mathcal {V}}^\perp :\varvec{\mathcal {V}}\in \mathcal {C}\} \subseteq \mathcal {G}_q(\varvec{N},\varvec{N}-\varvec{n}_{t}). \end{aligned}$$

The complementary code $\mathcal {C}^\perp$ has cardinality $|\mathcal {C}^\perp |=|\mathcal {C}|$, minimum sum-subspace distance $d_{\Sigma S}(\mathcal {C}^\perp )=d_{\Sigma S}(\mathcal {C})$ and code rate

$$\begin{aligned} R^\perp =\frac{\log _q(|\mathcal {C}^\perp |)}{\sum _{i=1}^{\ell }(N_i-n_t^{(i)})N_i}. \end{aligned}$$

3.2 Lifted linearized Reed–Solomon codes

Lifted linearized Reed–Solomon (LRS) codes [27] are constant-shot-dimension multishot network codes for error-control in noncoherent multishot network coding. The main idea behind the construction of LLRS codes is to lift codewords of an LRS code in a block-wise manner by augmenting each (transposed) codeword block by the corresponding $\mathbb F_{q}$-linearly independent code locators. For the special case of $\ell =1$ the construction coincides with the Kötter–Kschischang subspace codes [18].

Let ${\varvec{a}}=(a_1,a_2,\dots ,a_\ell )$ be a vector containing representatives from different conjugacy classes of $\mathbb F_{q^m}$. Let the vectors $\varvec{\beta }^{(i)}=(\beta _1^{(i)},\beta _2^{(i)},\dots ,\beta _{n_t^{(i)}}^{(i)})\in \mathbb F_{q^m}^{n_t^{(i)}}$ contain $\mathbb F_{q}$-linearly independent elements from $\mathbb F_{q^m}$ for all $i=1,\dots ,\ell$ and define $\varvec{\beta }=\left( \varvec{\beta }^{(1)}\mid \varvec{\beta }^{(2)}\mid \dots \mid \varvec{\beta }^{(\ell )}\right) \in \mathbb F_{q^m}^{n_t}$ and $\varvec{n}_{t}=(n_t^{(1)},n_t^{(2)},\dots ,n_t^{(\ell )})$. Then an LLRS code $\textrm{LLRS}[\varvec{\beta },{\varvec{a}},\ell ;\varvec{n}_{t},k]$ of sum-subspace dimension $n_t=n_t^{(1)}+n_t^{(2)}+\dots +n_t^{(\ell )}$, sum-dimension partition $\varvec{n}_{t}$ and dimension $k\le n_t$ is defined as

$$\begin{aligned} \Big \{\varvec{\mathcal {V}}(f) :=\left( {\mathcal {V}}^{(1)}(f), \dots , {\mathcal {V}}^{(\ell )}(f)\right) \, : \, f \in \mathbb F_{q^m}[x;\sigma ]_{<k}\Big \} \subseteq \mathcal {G}_q(\varvec{N},\varvec{n}_{t}) \end{aligned}$$

where $\varvec{N}=(N_1,\dots ,N_\ell )$ with $N_i=n_t^{(i)}+m$, and, for $f \in \mathbb F_{q^m}[x;\sigma ]_{<k}$, we have

$$\begin{aligned} {\mathcal {V}}^{(i)}(f) :={\left\langle \begin{pmatrix} \varvec{\beta }^{(i)\top }&{f}(\varvec{\beta }^{(i)})_{a_i}^\top \end{pmatrix} \right\rangle }_{q}\ \in \mathcal {G}_q(N_i,n_t^{(i)}), \qquad \forall i=1,\dots ,\ell . \end{aligned}$$

The lifting operation corresponds to augmenting each transposed codeword block of an LRS codeword by the corresponding (transposed) code locators and considering the $\mathbb F_{q}$-linear rowspace thereof (see [27, 29, 48]). The lifting operation causes a rate-loss since the code locators do not carry information since they are common for all codewords.

The minimum sum-subspace distance of $\textrm{LLRS}[\varvec{\beta },{\varvec{a}},\ell ;\varvec{n}_{t},k]$ equals (see [27])

$$\begin{aligned} d_{\Sigma S}(\textrm{LLRS}[\varvec{\beta },{\varvec{a}},\ell ;\varvec{n}_{t},k]) = 2(n_t- k + 1) \end{aligned}$$

(8)

and the code rate is

$$\begin{aligned} R = \frac{mk}{\sum _{i=1}^{\ell }n_t^{(i)}(n_t^{(i)} + m)}. \end{aligned}$$

In [27] an efficient interpolation-based decoding algorithm that can correct an overall number of $\gamma$ insertions and $\delta$ deletions up to

$$\begin{aligned} \gamma + \delta < n_t- k + 1 \end{aligned}$$

was presented. However, the decoder from [27] has the restriction that the dimension of the received spaces and the dimension of the transmitted spaces must be the same (c.f. [27, Sect. V.H]).

4 Decoding of lifted interleaved LRS codes for error-control in multishot network coding

In this section, we consider the application of lifted ILRS codes for error control in multishot network coding. In particular, we focus on noncoherent transmissions, where the network topology and/or the coefficients of the in-network linear combinations at the intermediate nodes are not known (or used) at the transmitter and the receiver. Therefore, we define and analyze lifted interleaved linearized Reed–Solomon (LILRS) codes. We derive a Loidreau–Overbeck-like decoder [24, 32, 33] for LILRS codes which is capable of correcting insertions and deletions beyond the unique decoding region at the cost of a (very) small decoding failure probability. Although the Loidreau–Overbeck-like decoder is not the most efficient decoder in terms of computational complexity, it allows to analyze the decoding failure probability and gives insights about the decoding procedure. We derive a tight upper bound on the decoding failure probability of the Loidreau–Overbeck-like decoder for LILRS codes, which, unlike simple heuristic bounds, considers the distribution of the error spaces caused by insertions.

We propose an efficient interpolation-based decoding scheme, which can correct insertions and deletions beyond the unique decoding region and which can be used as a list decoder or as a probabilistic unique decoder. We derive upper bounds on the worst-case list size and use the relation between the interpolation-based decoder and the Loidreau–Overbeck-like decoder to derive an upper bound on the decoding failure probability for the interpolation-based probabilistic unique decoding approach. Unlike the interpolation-based decoder in [27, Sect. V.H], the proposed decoding schemes do not have the restriction that the dimension of the received spaces and the dimension of the transmitted spaces must be the same.

4.1 Lifted interleaved linearized Reed–Solomon codes

In this section we consider LILRS codes for transmission over a multishot operator channel (3). We generalize the ideas from [27] to obtain multishot subspace codes by lifting the ILRS codes defined in [4] (see also [3]).

Definition 6

(Lifted Interleaved Linearized Reed–Solomon Code) Let ${\varvec{a}}=(a_1,a_2,\dots ,a_\ell )$ be a vector containing representatives from different conjugacy classes of $\mathbb F_{q^m}$. Let the vectors $\varvec{\beta }^{(i)}=(\beta _1^{(i)},\beta _2^{(i)},\dots ,\beta _{n_t^{(i)}}^{(i)})\in \mathbb F_{q^m}^{n_t^{(i)}}$ contain $\mathbb F_{q}$-linearly independent elements from $\mathbb F_{q^m}$ for all $i=1,\dots ,\ell$ and define $\varvec{\beta }=\left( \varvec{\beta }^{(1)}\mid \varvec{\beta }^{(2)}\mid \dots \mid \varvec{\beta }^{(\ell )}\right) \in \mathbb F_{q^m}^{n_t}$ and $\varvec{n}_{t}=(n_t^{(1)},n_t^{(2)},\dots ,n_t^{(\ell )})$. A homogeneous lifted $s$-interleaved linearized Reed–Solomon (LILRS) code $\textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k]$ of sum-subspace dimension $n_t=n_t^{(1)}+n_t^{(2)}+\dots +n_t^{(\ell )}$ and dimension $k\le n_t$ is defined as

$$\begin{aligned} \Big \{\varvec{\mathcal {V}}(\varvec{f}) :=\left( {\mathcal {V}}^{(1)}(\varvec{f}), \dots , {\mathcal {V}}^{(\ell )}(\varvec{f})\right) \, : \, \varvec{f}\in \mathbb F_{q^m}[x;\sigma ]_{<k}^s\Big \} \subseteq \mathcal {G}_q(\varvec{N},\varvec{n}_{t}) \end{aligned}$$

where $\varvec{N}=(N_1,\dots ,N_\ell )$ with $N_i=n_t^{(i)}+sm$, and, for $\varvec{f}= (f_1, \dots , f_s)$, we have

$$\begin{aligned} {\mathcal {V}}^{(i)}(\varvec{f}) :={\left\langle \begin{pmatrix} \varvec{\beta }^{(i)\top }&{f_1}(\varvec{\beta }^{(i)})_{a_i}^\top&\dots&{f_s}(\varvec{\beta }^{(i)})_{a_i}^\top \end{pmatrix} \right\rangle }_{q} \in \mathcal {G}_q(N_i,n_t^{(i)}). \end{aligned}$$

Remark 3

In order to not further complicate the notation, we consider homogeneous LILRS codes, i.e. codes where each component code has the same code dimension k, only. Note that the proposed results and concepts can be adapted to inhomogeneous codes with different code dimension $k_1,\dots ,k_s$ in a straight forward manner (analog to single-shot subspace codes in e.g. [5, 6, 49]).

Observe, that compared to LLRS codes the relative overhead due to lifting decreases in $s$ since the evaluations are performed at the same code locators and thus have to be appended only once. The reduction of the relative overhead comes at the cost of an increased packet size $N_i$ for each shot $i=1,\dots ,\ell$.

The definition of LILRS codes generalizes several code families. For $s=1$ we obtain the lifted linearized Reed–Solomon codes from [27, Sect. V.III]. For $\ell =1$ we obtain lifted interleaved Gabidulin codes as considered in e.g. [6, 50] with Kötter–Kschischang codes [18] as special case for $s=1$.

Proposition 2 shows that interleaving does not increase the minimum sum-subspace distance of the code.

Proposition 2

(Minimum Distance) The minimum sum-subspace distance of a LILRS code $\textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k]$ as in Definition 6 is

$$\begin{aligned} d_{\Sigma S}\left( \textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k]\right) =2\left( n_t-k+1\right) . \end{aligned}$$

Proof

Consider two distinct codewords

$$\begin{aligned} \varvec{\mathcal {V}}_1 :=\varvec{\mathcal {V}}(\varvec{f}_1), \, \varvec{\mathcal {V}}_2 :=\varvec{\mathcal {V}}(\varvec{f}_2) \in \textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k] \end{aligned}$$

constructed by the polynomial vectors $\varvec{f}_1=\left( f_1,0,\dots ,0\right) \text { and } \varvec{f}_2=\left( f_2,0,\dots ,0\right)$ with $f_1 \ne f_2$. Then the corresponding tuples of subspaces are $\varvec{\mathcal {V}}_j=\left( \mathcal {V}_j^{(1)}, \dots , \mathcal {V}_j^{(\ell )}\right)$ with

$$\begin{aligned} \mathcal {V}_j^{(i)} :={\left\langle \begin{pmatrix} \varvec{\beta }^{(i)\top }&{f_j}(\varvec{\beta }^{(i)})_{a_i}^\top&\textbf{0}&\dots&\textbf{0 }\end{pmatrix} \right\rangle }_{q} \end{aligned}$$

(9)

for $j=1,2$. Observe that the $\ell -1$ rightmost zero matrices in the component spaces in (9) do not contribute to the sum-subspace distance between $\varvec{\mathcal {V}}_1$ and $\varvec{\mathcal {V}}_2$. Therefore we have that

$$\begin{aligned} d_{\Sigma S}(\varvec{\mathcal {V}}_1, \varvec{\mathcal {V}}_2) = d_{\Sigma S}(\widetilde{\varvec{\mathcal {V}}}_1, \widetilde{\varvec{\mathcal {V}}}_2) \end{aligned}$$

where $\widetilde{\varvec{\mathcal {V}}}_j=\left( \widetilde{\mathcal {V}}_j^{(1)}, \dots , \widetilde{\mathcal {V}}_j^{(\ell )}\right)$ with

$$\begin{aligned} \widetilde{\mathcal {V}}_j^{(i)} :={\left\langle \begin{pmatrix} \varvec{\beta }^{(i)\top }&{f_j}(\varvec{\beta }^{(i)})_{a_i}^\top \end{pmatrix} \right\rangle }_{q} \end{aligned}$$

(10)

for $j=1,2$. Since $\widetilde{\varvec{\mathcal {V}}}_1, \widetilde{\varvec{\mathcal {V}}}_2 \in \textrm{LLRS}[\varvec{\beta },{\varvec{a}},\ell ;\varvec{n}_{t},k]$ we have that (see (8))

$$\begin{aligned} d_{\Sigma S}(\widetilde{\varvec{\mathcal {V}}}_1, \widetilde{\varvec{\mathcal {V}}}_2) \ge 2(n_t- k + 1). \end{aligned}$$

Now assume that there exist two codewords $\varvec{\mathcal {V}}_1',\varvec{\mathcal {V}}_2' \in \textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k]$ with component spaces of the form (9) such that $d_{\Sigma S}(\varvec{\mathcal {V}}_1',\varvec{\mathcal {V}}_2') < 2(n_t- k + 1)$. This would imply that the corresponding (truncated) spaces $\widetilde{\varvec{\mathcal {V}}}_1',\widetilde{\varvec{\mathcal {V}}}_2'$ of the form (10) which are codewords in $\textrm{LLRS}[\varvec{\beta },{\varvec{a}},\ell ;\varvec{n}_{t},k]$ also satisfy $d_{\Sigma S}(\widetilde{\varvec{\mathcal {V}}}_1',\widetilde{\varvec{\mathcal {V}}}_2') < 2(n_t- k + 1)$, which contradicts that $\textrm{LLRS}[\varvec{\beta },{\varvec{a}},\ell ;\varvec{n}_{t},k]$ has minimum sum-subspace distance $2(n_t- k + 1)$. Therefore, we conclude that $d_{\Sigma S}\left( \textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k]\right) =2\left( n_t-k+1\right)$. $\square$

For a LILRS code $\mathcal {C}=\textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k]$ we have that $N_i=n_t^{(i)}+sm$ for all $i=1,\dots ,\ell$ and therefore the code rate is

$$\begin{aligned} R =\frac{\log _q(|\mathcal {C}|)}{\sum _{i=1}^{\ell }n_t^{(i)}N_i} =\frac{smk}{\sum _{i=1}^{\ell }n_t^{(i)}(n_t^{(i)}+sm)}. \end{aligned}$$

(11)

Note, that there exist other definitions of the code rate for multishot codes that are not considered in this paper (see e.g. [30, Sect. IV.A]).

The normalized weight $\lambda$ and the normalized distance $\eta$ of an LILRS code $\textrm{LILRS}[\varvec{\beta },{\varvec{a}}, \ell ,s;\varvec{n}_{t},k]$ is

$$\begin{aligned} \lambda =\frac{n_t}{N} \quad \text {and}\quad \eta =\frac{n_t-k+1}{n_t}. \end{aligned}$$

For $n_t^{(i)}=\overline{n}_t$ for all $i=1,\dots ,\ell$ the code rate of an LILRS code $\textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k]$ in (11) becomes

$$\begin{aligned} R=\frac{\ell sm k}{n_tN}. \end{aligned}$$

For $n_t^{(i)}=\overline{n}_t$ for all $i=1,\dots ,\ell$ the Singleton-like upper bound for constant-shot-dimension sum-subspace codes [27, Theorem 7] evaluated for the parameters of LILRS codes becomes

$$\begin{aligned} |\mathcal {C}|\le \kappa _q^\ell q^{sm\left( n_t-\left( d_{\Sigma S}(\mathcal {C})/2-1\right) \right) } =\kappa _q^\ell q^{smk} \end{aligned}$$

which shows, that a Singleton-like bound achieving code can be at most $\kappa _q^\ell <3.5^\ell$ times larger than the corresponding LILRS code. Equivalently, the code rate is therefore upper-bounded by

$$\begin{aligned} R\le \frac{\ell (\log _q(\kappa _q)+smk)}{n_tN}. \end{aligned}$$

The benefit of the decreased relative overhead due to interleaving is illustrated in Figure 2. The figure shows, that the rate loss due to the overhead introduced by the lifting is reduced significantly, even for small interleaving orders. Further, we see that LILRS codes approach the Singleton-like bound for sum-subspace codes (see [27]) with increasing interleaving order while preserving the extension field degree m³.

Fig. 2

Normalized distance $\eta$ over the code rate R for an LILRS code $\textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell =2,s;\varvec{n}_{t}=(8,8),k=4]$ over $\mathbb F_{3^8}$ for interleaving orders $s\in \{1,3,10\}$ with the corresponding (average) normalized weight $\overline{\lambda }$. The case $s=1$ corresponds to the LLRS codes from [27]

4.2 Loidreau–Overbeck-like decoder for LILRS codes

Loidreau and Overbeck proposed the first efficient decoder for interleaved Gabidulin codes in the rank metric [24, 32, 33]. The main idea behind the Loidreau–Overbeck decoder is to compute an $\mathbb F_{q}$-linear transformation matrix from a decoding matrix (which depends on the code and the received word) that allows to transform the received word into a corrupted part and a noncorrupted part. The noncorrupted part is then used to recover the message polynomials e.g. via Lagrange interpolation.

The concept of the Loidreau–Overbeck decoder was generalized to decoding ILRS codes in the sum-rank metric [4]. In the sum-rank-metric case an $\mathbb F_{q}$-linear transformation matrix is obtained for each block.

Based on the previous decoders for the rank and sum-rank metric, we derive a Loidreau–Overbeck-like decoder for LILRS codes. Similar to the original decoder and its sum-rank-metric analogue we set up a decoding matrix that allows to compute $\mathbb F_{q}$-linear transformation matrices for each shot ${\mathcal {U}}^{(i)}$. The obtained transformation matrices allow to compute particular bases for the received subspaces ${\mathcal {U}}^{(i)}$ that can be split into a basis for the corrupted part (corresponding to the error space ${\mathcal {E}}^{(i)}$) and a noncurrupted part (i.e. a basis for ${\mathcal {V}}^{(i)} \cap {\mathcal {U}}^{(i)}$) for each shot. The noncorrupted part is then used to reconstruct the message polynomials via Lagrange interpolation. The qualitative structure of the tuple $\hat{\varvec{U}}=(\hat{\varvec{U}}^{(1)},\hat{\varvec{U}}^{(2)},\dots ,\hat{\varvec{U}}^{(\ell )})$ containing transformed basis matrices is illustrated in Figure 3.

Fig. 3

Qualitative illustration of the structure of the tuple $\hat{\varvec{U}}$ containing transformed basis matrices. The green parts form a basis for the non-corrupted spaces whereas the red parts indicate a basis for the erroneous spaces. The green part is used to reconstruct the message polynomials

The main motivation to derive the Loidreau–Overbeck-like decoder is to obtain an upper bound on the decoding failure probability that incorporates the distribution of the error spaces in $\varvec{\mathcal {E}}$. In Sect. 4.3 we will reduce the interpolation-based decoder for LILRS codes to the Loidreau–Overbeck-like decoder in order to obtain an upper bound on the decoding failure probability of the interpolation-based probabilistic unique decoder.

Up to our knowledge, this is the first Loidreau–Overbeck-like decoding scheme in the (sum-) subspace metric. It includes lifted interleaved Gabidulin (or interleaved Kötter–Kschischang) codes [5, 6] as special case for $\ell =1$. Hence, the results give a strict upper bound⁴ on the decoding failure probability of the decoders in [5, 6].

Suppose we transmit the tuple of subspaces

$$\begin{aligned} \varvec{\mathcal {V}}(\varvec{f}) = \left( {\mathcal {V}}^{(1)}(\varvec{f}), \dots , {\mathcal {V}}^{(\ell )}(\varvec{f})\right) \in \textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k] \end{aligned}$$

over an $\ell$-shot operator channel with overall $\gamma$ insertions and $\delta$ deletions and receive the tuple of subspaces

$$\begin{aligned} \varvec{\mathcal {U}}= \left( {\mathcal {U}}^{(1)}, \dots , {\mathcal {U}}^{(\ell )}\right) \in \mathcal {G}_q(\varvec{N}, \varvec{n}_r), \end{aligned}$$

where the received subspaces ${\mathcal {U}}^{(i)}$ are as defined in (7) and $\varvec{n}_r= \varvec{n}_{t}+ \varvec{\gamma }- \varvec{\delta }$ (see (4)). Define the vectors

$$\begin{aligned} \varvec{\xi }=\left( \varvec{\xi }^{(1)} \mid \varvec{\xi }^{(2)} \mid \dots \mid \varvec{\xi }^{(\ell )}\right) \in \mathbb F_{q^m}^{n_r} \quad \text {and}\quad \varvec{u}_j=\left( \varvec{u}_j^{(1)} \mid \varvec{u}_j^{(2)} \mid \dots \mid \varvec{u}_j^{(\ell )}\right) \in \mathbb F_{q^m}^{n_r}, \end{aligned}$$

for all $j=1,\dots ,s$ and consider the matrix

$$\begin{aligned} {\varvec{L}}:= \begin{pmatrix} \Lambda _{n_t-\delta -1}(\varvec{\xi })_{{\varvec{a}}} \\ \Lambda _{n_t-\delta -k}\left( \varvec{u}_1\right) _{{\varvec{a}}} \\ \vdots \\ \Lambda _{n_t-\delta -k}\left( \varvec{u}_s\right) _{{\varvec{a}}} \\ \end{pmatrix} \in \mathbb F_{q^m}^{((s+1)(n_t-\delta )-sk -1) \times n_r}. \end{aligned}$$

(12)

Lemma 2

(Transformed of Decoding Matrix) Consider the transmission of a tuple of subspaces $\varvec{\mathcal {V}}(\varvec{f}) \in \textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k]$ over an $\ell$-shot operator channel with overall $\gamma$ insertions and $\delta$ deletions and receive the tuple of subspaces $\varvec{\mathcal {U}}$. Let ${\varvec{L}}$ be as in (12). Then there exist invertible matrices $\varvec{W}^{(i)}\in \mathbb F_{q}^{n_r^{(i)}\times n_r^{(i)}}$ such that for $\varvec{W}={{\,\textrm{diag}\,}}(\varvec{W}^{(1)},\varvec{W}^{(2)},\dots ,\varvec{W}^{(\ell )})$ we have that

$$\begin{aligned} \overline{{\varvec{L}}}=\left( \overline{{\varvec{L}}}^{(1)},\dots ,\overline{{\varvec{L}}}^{(\ell )}\right) ={\varvec{L}}\cdot \varvec{W}\in \mathbb F_{q^m}^{((s+1)(n_t-\delta )-sk -1) \times n_r} \end{aligned}$$

(13)

consists of component matrices of the form

$$\begin{aligned} \overline{{\varvec{L}}}^{(i)}= \begin{pmatrix} \varvec{V}_{n_t-\delta -1}(\overline{\varvec{\xi }}_1^{(i)})_{a_i} &{} \varvec{V}_{n_t-\delta -1}(\overline{\varvec{\xi }}_2^{(i)})_{a_i} &{} \textbf{0 }\\ \textbf{0 }&{} \varvec{V}_{n_t-\delta -k}(\widetilde{\varvec{e}}_1^{(i)})_{a_i} &{} \varvec{V}_{n_t-\delta -k}(\hat{\varvec{e}}_1^{(i)})_{a_i} \\ \vdots &{} \vdots &{} \vdots \\ \textbf{0 }&{} \varvec{V}_{n_t-\delta -k}(\widetilde{\varvec{e}}_s^{(i)})_{a_i} &{} \varvec{V}_{n_t-\delta -k}(\hat{\varvec{e}}_s^{(i)})_{a_i} \end{pmatrix} \in \mathbb F_{q^m}^{((s+1)(n_t-\delta )-sk -1) \times n_r^{(i)}} \end{aligned}$$

where $\overline{\varvec{\xi }}_1^{(i)}\in \mathbb F_{q^m}^{n_t^{(i)}-{\delta }^{(i)}}$, $\overline{\varvec{\xi }}_2^{(i)},\widetilde{\varvec{e}}_l^{(i)}\in \mathbb F_{q^m}^{{t}^{(i)}},\hat{\varvec{e}}_l^{(i)}\in \mathbb F_{q^m}^{{\varkappa }^{(i)}}$ and

$$\begin{aligned} {{\,\textrm{rk}\,}}_{q}(\overline{\varvec{\xi }}_1^{(i)})&=n_t^{(i)}-{\delta }^{(i)}, \qquad{} & {} {{\,\textrm{rk}\,}}_{q}(\overline{\varvec{\xi }}_2^{(i)})={t}^{(i)} \\ {{\,\textrm{rk}\,}}_{q}(\widetilde{\varvec{e}}_l^{(i)})&={t}^{(i)},{} & {} {{\,\textrm{rk}\,}}_{q}(\hat{\varvec{e}}_l^{(i)})={\varkappa }^{(i)} \\ {{\,\textrm{rk}\,}}_{q}((\widetilde{\varvec{e}}_l^{(i)}\mid \hat{\varvec{e}}_l^{(i)}))&={t}^{(i)}+{\varkappa }^{(i)},{} & {} \end{aligned}$$

for all $i=1,\dots ,\ell$ such that ${{\,\textrm{rk}\,}}_{q^m}({\varvec{L}})={{\,\textrm{rk}\,}}_{q^m}(\overline{{\varvec{L}}})$.

The proof of Lemma 2 is based on particular bases for the received spaces in $\varvec{\mathcal {U}}$ and properties of the intersection and error spaces in $\varvec{\mathcal {V}}\cap \varvec{\mathcal {U}}$ and $\varvec{\mathcal {E}}$, respectively, and can be found in Appendix 1.

Lemma 3

(Properties of Decoding Matrix) Consider the notation and definitions as in Lemma 2 and define the vectors

$$\begin{aligned} \overline{\varvec{e}}_l:=\left( (\widetilde{\varvec{e}}_{l}^{(1)}\mid \hat{\varvec{e}}_{l}^{(1)})\mid \dots \mid (\widetilde{\varvec{e}}_{l}^{(\ell )}\mid \hat{\varvec{e}}_{l}^{(\ell )})\right) \in \mathbb F_{q^m}^{\gamma }, \quad \forall l=1,\dots s, \end{aligned}$$

and the matrix

$$\begin{aligned} \overline{\varvec{Z}}=\left( \overline{\varvec{Z}}^{(1)}\mid \dots \mid \overline{\varvec{Z}}^{(\ell )}\right) :=\left( \begin{array}{c} \Lambda _{n_t-\delta -k}(\overline{\varvec{e}}_1)_{\varvec{a}} \\ \vdots \\ \Lambda _{n_t-\delta -k}(\overline{\varvec{e}}_s)_{\varvec{a}} \end{array} \right) \in \mathbb F_{q^m}^{s(n_t-\delta -k)\times \gamma }. \end{aligned}$$

(14)

Let $\varvec{h}= (\varvec{h}^{(1)} \mid \varvec{h}^{(2)} \mid \dots \mid \varvec{h}^{(\ell )}) \in \mathbb F_{q^m}^{n_r}$ with $\varvec{h}^{(i)} \in \mathbb F_{q^m}^{n_r^{(i)}}$ for all $i=1,\dots ,\ell$ be a nonzero vector in the right kernel of the decoding matrix ${\varvec{L}}$ and suppose that $\overline{\varvec{Z}}$ has $\mathbb F_{q^m}$-rank $\gamma$. Then:

• 1. We have ${{\,\textrm{rk}\,}}_{q^m}({\varvec{L}}) = n_r-1$.

• 2. We have ${{\,\textrm{rk}\,}}_{q}(\varvec{h}^{(i)})=n_t^{(i)}-{\delta }^{(i)}$ for all $i=1,\dots ,\ell$, i.e., $\varvec{h}$ has sum-rank weight ${{\,\textrm{wt}\,}}_{\Sigma R}^{(\varvec{n}_r)}(\varvec{h}) = n_t-\delta$.

• 3. There are invertible matrices $\varvec{T}^{(i)} \in \mathbb F_{q}^{n_r^{(i)} \times n_r^{(i)}}$, for all $i=1,\dots ,\ell$, such that the last (rightmost) ${\gamma }^{(i)}$ positions of $\varvec{h}^{(i)} \varvec{T}^{(i)}$ are zero.

• 4. The first (upper) $n_t^{(i)}-{\delta }^{(i)}$ rows of $\hat{\varvec{U}}^{(i)} = \left( \varvec{T}^{(i)}\right) ^{-1}\varvec{U}^{(i)}$ form a basis for the non-corrupted received space ${\mathcal {U}}^{(i)} \cap {\mathcal {V}}^{(i)}$ for all $i=1,\dots ,\ell$.

• 5. The l-th message polynomial $f_l$ can be uniquely reconstructed from the transformed basis $\hat{\varvec{U}}^{(i)}$ for the received space ${\mathcal {U}}^{(i)}$ by Lagrange interpolation on the first $n_t^{(i)} - {\delta }^{(i)}$ rows of $\hat{\varvec{U}}^{(i)}$ for all $l=1,\dots ,s$ and $i=1,\dots ,\ell$.

We now provide a sketch of the proof. The full proof of Lemma 3 can be found in Appendix 1.

Sketch of the Proof

• Ad 1): The matrix $\overline{{\varvec{L}}}$ can be rearranged into an upper block-triangular matrix whose rank is determined by the two blocks on the diagonal, which have rank $n_t-\delta -1$ and $\gamma$ and thus imply that the $\mathbb F_{q^m}$-rank of the whole matrix equals $n_t-\delta -1+\gamma = n_r-1$. The statement follows since by Lemma 2 we have that ${{\,\textrm{rk}\,}}_{q^m}({\varvec{L}})={{\,\textrm{rk}\,}}_{q^m}(\overline{{\varvec{L}}})$.

• Ad 2): By assumption the $\mathbb F_{q^m}$-rank of $\overline{\varvec{Z}}$ equals $\gamma$ which implies that ${{\,\textrm{rk}\,}}_{q^m}({\overline{\varvec{Z}}}^{(i)})={\gamma }^{(i)}$ for all $i=1,\dots ,\ell$. Thus, for any $\overline{\varvec{h}} \in \ker _r(\overline{{\varvec{L}}}) \setminus \{\textbf{0}\}$ the ${\gamma }^{(i)}$ rightmost entries of ${\overline{\varvec{h}}}^{(i)}$ must be zero which implies that ${{\,\textrm{rk}\,}}_q({\overline{\varvec{h}}}^{(i)}) \le n_t^{(i)}-{\delta }^{(i)}$ for all $i=1,\dots ,\ell$. On the other hand $\overline{\varvec{h}}$ is contained in a code with minimum sum-rank distance $n_t-\delta$ which is the dual of the code spanned by the first $n_t-\delta -1$ rows of $\overline{{\varvec{L}}}$. The statement follows by combining these two facts.

• Ad 3): By 2) the $\mathbb F_{q}$-rank of ${\varvec{h}}^{(i)} \in \mathbb F_{q^m}^{n_r^{(i)}}$ equals $n_t^{(i)}-{\delta }^{(i)}$ for all $i=1,\dots ,\ell$. Hence there exist matrices ${\varvec{T}}^{(i)} \in \mathbb F_{q}^{n_r^{(i)} \times n_r^{(i)}}$ such that the $n_r^{(i)}-(n_t^{(i)}-{\delta }^{(i)})={\gamma }^{(i)}$ rightmost entries of ${\varvec{h}}^{(i)} {\varvec{T}}^{(i)}$ are equal to zero.

• Ad 4): Define the matrices $\varvec{D}^{(i)}=\left( \varvec{T}^{(i)-1}\right) ^\top$ for all $i=1,\dots ,\ell$ and observe that $\varvec{h}\varvec{T}\in \ker _r({\varvec{L}}\cdot {{\,\textrm{diag}\,}}(\varvec{D}^{(1)},\dots ,\varvec{D}^{(\ell )}))$. By using the $\mathbb F_{q^m}$-rank condition on $\overline{\varvec{Z}}$ one can show that the span of the ${\gamma }^{(i)}$ rightmost columns of the matrices ${\overline{{\varvec{L}}}}^{(i)}$ and ${{\varvec{L}}}^{(i)}{\varvec{D}}^{(i)}$ coincides. These columns correspond to the insertions which in turn implies that the last ${\gamma }^{(i)}$ rows of ${\hat{\varvec{U}}}^{(i)}=({\varvec{D}}^{(i)})^\top {\varvec{U}}^{(i)}$ form a basis for ${\mathcal {E}}^{(i)}$. The statement follows since by the definition of the operator channel we have that ${\mathcal {V}}^{(i)} \cap {\mathcal {E}}^{(i)} = \{\textbf{0}\}$ for all $i=1,\dots ,\ell$.

• Ad 5): By 4) the first $n_t^{(i)}-{\delta }^{(i)}$ rows of the transformed basis ${\hat{\varvec{U}}}^{(i)}$ form a basis for the noncorrupted intersection space ${\mathcal {V}}^{(i)} \cap {\mathcal {U}}^{(i)}$ for all $i=1,\dots ,\ell$. Due to the $\mathbb F_{q}$-linearity of the generalized operator evaluation for a fixed evaluation parameter (i.e. per shot), the message polynomials can be reconstructed by constructing the corresponding Lagrange interpolation polynomials (see Figure 4).

The complete procedure for the Loidreau–Overbeck-like decoder for LILRS codes is given in Algorithm 1. The structure of the transformed basis matrices $\hat{\varvec{U}}^{(i)}$ for all $i=1,\dots ,\ell$ is illustrated in Figure 4.

Algorithm 1

Loidreau−Overbeck-like Decoder for LILRS Codes

Fig. 4

Illustration of the structure of the transformed basis matrices $\hat{\varvec{U}}^{(i)}$. The green part forms a basis for the non-corrupted space ${\mathcal {U}}^{(i)}\cap {\mathcal {V}}^{(i)}$ whereas the red part forms a basis for the error space ${\mathcal {E}}^{(i)}$

An execution of the Loidreau–Overbeck-like decoder for LILRS codes is illustrated in Example 1.

Example 1

(Loidreau–Overbeck-like Decoder) Consider the finite field $\mathbb F_{3^2}$ with primitive element $\alpha$ defined by the primitive polynomial $x^2 + 2x + 2$. Consider the LILRS code $\textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k]$ over $\mathbb F_{3^2}$ with code locators $\varvec{\beta }= ((1,\alpha ) \mid (1,\alpha ))$, evaluation parameters ${\varvec{a}}=(1,\alpha )$, number of shots $\ell =2$, interleaving order $s=2$, sum-dimension partition $\varvec{n}_{t}= (2,2)$ and dimension $k=2$. Consider the transmission of a codeword with

$$\begin{aligned} {\mathcal {V}}^{(1)}(\varvec{f}) = {\left\langle \left( \begin{array}{ccc} 1 &{} \alpha + 2 &{} 2 \\ \alpha &{} 2 \alpha + 2 &{} 2 \end{array} \right) \right\rangle }_{q} \quad \text {and}\quad {\mathcal {V}}^{(2)}(\varvec{f})= {\left\langle \left( \begin{array}{ccc} 1 &{} \alpha &{} 1 \\ \alpha &{} \alpha &{} \alpha + 1 \end{array} \right) \right\rangle }_{q} \end{aligned}$$

(15)

that corresponds to the message polynomial vector $\varvec{f}=(f_1, f_2)$ with

$$\begin{aligned} f_1 = \alpha x + 2 \quad \text {and}\quad f_2 = 2\alpha x + \alpha + 2 \end{aligned}$$

over a multishot operator channel (3) with an overall number of $\gamma =2$ insertions and $\delta = 1$ deletion. Note, that a bounded minimum distance (BMD) decoder could only correct $\gamma =1$ insertions and $\delta =1$ deletion.

Suppose we receive the tuple of subspaces $\varvec{\mathcal {U}}=({\mathcal {U}}^{(1)},{\mathcal {U}}^{(2)})$ with

$$\begin{aligned} {\mathcal {U}}^{(1)}= {\left\langle \left( \begin{array}{ccc} \alpha + 2 &{} \alpha &{} 0 \\ 1 &{} 1 &{} 1 \\ 2 \alpha + 2 &{} 2 &{} 2 \end{array} \right) \right\rangle }_{q} \quad \text {and} \quad {\mathcal {U}}^{(2)}= {\left\langle \left( \begin{array}{ccc} 2 \alpha + 1 &{} 0 &{} 2 \\ 2 \alpha + 2 &{} \alpha &{} \alpha + 2 \end{array} \right) \right\rangle }_{q} \end{aligned}$$

(16)

where $\varvec{n}_r=(3,2)$ and $n_r= 5$.

According to (12) the Loidreau–Overbeck-like decoding matrix is

$$\begin{aligned} {\varvec{L}}= \left( \begin{array}{ccc|cc} \alpha + 2 &{} 1 &{} 2 \alpha + 2 &{} 2 \alpha + 1 &{} 2 \alpha + 2 \\ 2 \alpha &{} 1 &{} \alpha + 1 &{} \alpha + 1 &{} 2 \alpha + 1 \\ \alpha &{} 1 &{} 2 &{} 0 &{} \alpha \\ 0 &{} 1 &{} 2 &{} 2 &{} \alpha + 2 \end{array} \right) . \end{aligned}$$

The decoding matrix ${\varvec{L}}$ has $\mathbb F_{q^m}$-rank $n_r-1=4$ implying the right $\mathbb F_{3^2}$-kernel of ${\varvec{L}}$ has dimension one. We pick

$$\varvec{h}= (\varvec{h}^{(1)} \mid \varvec{h}^{(2)}) = \left( (\alpha , 0, 2) \mid (\alpha + 1, \alpha + 1)\right) \in \ker _r({\varvec{L}})$$

as nonzero element from the right $\mathbb F_{q^m}$-kernel of ${\varvec{L}}$ and recover the partition of insertions $\varvec{\gamma }$ as

$$\varvec{\gamma }= \left( n_r^{(1)}-{{\,\textrm{rk}\,}}_q(\varvec{h}^{(1)}) \mid n_r^{(2)}-{{\,\textrm{rk}\,}}_q(\varvec{h}^{(2)})\right) =(1 \mid 1).$$

Next we compute the transformation matrices

$$\begin{aligned} \varvec{T}^{(1)}= \begin{pmatrix} 0 &{} 1 &{} 0 \\ 2 &{} 1 &{} 1 \\ 2 &{} 1 &{} 0 \end{pmatrix} \quad \text {and} \quad \varvec{T}^{(2)}= \begin{pmatrix} 2 &{} 1 \\ 0 &{} 2 \end{pmatrix} \end{aligned}$$

such that the ${\gamma }^{(1)}={\gamma }^{(2)}=1$ rightmost entry of $\varvec{h}^{(1)}\varvec{T}^{(1)}$ and $\varvec{h}^{(2)}\varvec{T}^{(2)}$ is zero, i.e. we have

$$\begin{aligned} \varvec{h}^{(1)}\varvec{T}^{(1)} = (1, \alpha + 2, 0) \quad \text {and}\quad \varvec{h}^{(2)}\varvec{T}^{(2)} = (2\alpha + 2, 0). \end{aligned}$$

Now we can compute the transformed bases for the received spaces ${\mathcal {U}}^{(1)}$ and ${\mathcal {U}}^{(2)}$ as

$$\begin{aligned} \hat{\varvec{U}}^{(1)}&= \left( \varvec{T}^{(1)}\right) ^{-1}\varvec{U}^{(1)} = \begin{pmatrix} 2 \alpha &{} \alpha + 1 &{} 1 \\ \alpha + 2 &{} \alpha &{} 0 \\ \alpha + 2 &{} 2 &{} 2 \end{pmatrix}, \\ \hat{\varvec{U}}^{(2)}&= \left( \varvec{T}^{(2)}\right) ^{-1}\varvec{U}^{(2)} = \begin{pmatrix} 2 \alpha &{} 2 \alpha &{} 2 \alpha + 2 \\ \alpha + 1 &{} 2 \alpha &{} 2 \alpha + 1 \end{pmatrix}, \end{aligned}$$

where $\varvec{U}^{(1)}$ and $\varvec{U}^{(2)}$ are bases for ${\mathcal {U}}^{(1)}$ and ${\mathcal {U}}^{(2)}$, respectively. Then the upper $n_t^{(1)}-{\delta }^{(1)}=2$ rows of $\hat{\varvec{U}}^{(1)}$ form a basis for the noncorrupted space ${\mathcal {V}}^{(1)} \cap {\mathcal {U}}^{(1)}$ and the upper $n_t^{(2)}-{\delta }^{(2)}=1$ rows of $\hat{\varvec{U}}^{(2)}$ form a basis for the noncorrupted space ${\mathcal {V}}^{(2)} \cap {\mathcal {U}}^{(2)}$.

Applying Lagrange interpolation to the rows of $\hat{\varvec{U}}^{(1)}$ and $\hat{\varvec{U}}^{(2)}$ corresponding to the noncorrupted spaces we can recover the massage polynomials $f_1 = \alpha x + 2$ and $f_2 = 2\alpha x + \alpha + 2$.

Lemma 4

(Decoding Failure Probability) Suppose that a tuple of subspaces

$$\begin{aligned} \varvec{\mathcal {V}}(\varvec{f}) = \left( {\mathcal {V}}^{(1)}(\varvec{f}), \dots , {\mathcal {V}}^{(\ell )}(\varvec{f})\right) \in \textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k] \end{aligned}$$

is transmitted over a random instance of the $\ell$-shot operator channel (see Remark 2) with overall $\gamma$ insertions and $\delta$ deletions, where $\gamma$ and $\delta$ satisfy $\gamma \le \gamma _\textsf{max}:= s(n_t-\delta -k)$. Let $\overline{\varvec{Z}}$ be defined as in (14) (see Lemma 3). Then, we have

$$\begin{aligned} \Pr \!\left( {{\,\textrm{rk}\,}}_{q^m}(\Lambda _{n_t-\delta -k}(\overline{\varvec{Z}})_{{\varvec{a}}})<\gamma \right) \le \kappa _q^{\ell +1} q^{-m(\gamma _\textsf{max}-\gamma +1)}. \end{aligned}$$

Proof

Let $\varvec{\gamma }=({\gamma }^{(1)},\dots ,{\gamma }^{(\ell )})\in \mathbb {Z}_{\ge 0}^{\ell }$ and $\varvec{\delta }=({\delta }^{(1)},\dots ,{\delta }^{(\ell )})\in \mathbb {Z}_{\ge 0}^{\ell }$ be the partition of the insertions and deletions of the $\ell$-shot operator channel, respectively. By assumption, the tuple of error spaces $\varvec{\mathcal {E}}\in \mathcal {G}_q(\varvec{N}, \varvec{\gamma })$ is chosen uniformly at random from the set $\mathcal {E}^{(\varvec{\gamma })}$ as defined in (5). Each instance of the $\ell$-shot operator channel yields a decoding matrix $\overline{{\varvec{L}}}$ of the form (13) for some . Now let us fix $\varvec{\gamma }$ and $\varvec{\delta }$, which yields a particular instance of $\varvec{t}$ and $\varvec{\varkappa }$. By Lemma 3 the decoder succeeds if the $\mathbb F_{q^m}$-rank of the matrix

$$\begin{aligned} \overline{\varvec{Z}}:=\left( \begin{array}{c} \Lambda _{n_t-\delta -k}(\overline{\varvec{e}}_1)_{\varvec{a}} \\ \vdots \\ \Lambda _{n_t-\delta -k}(\overline{\varvec{e}}_s)_{\varvec{a}} \end{array} \right) \in \mathbb F_{q^m}^{s(n_t-\delta -k)\times \gamma }, \end{aligned}$$

equals $\gamma$ given that

$$\begin{aligned} {{\,\textrm{wt}\,}}_{\Sigma R}\left( {\widetilde{\varvec{E}}}^{(1)}{\hat{\varvec{E}}}^{(1)}\mid \dots \mid {\widetilde{\varvec{E}}}^{(\ell )}{\hat{\varvec{E}}}^{(\ell )}\right) =t+\varkappa =\gamma . \end{aligned}$$

(17)

Drawing $\varvec{\mathcal {E}}$ uniformly at random with a fixed rank partition $\varvec{\gamma }$ from $\mathcal {E}^{(\varvec{\gamma })}$ corresponds to drawing $\widetilde{\varvec{E}}$ and $\hat{\varvec{E}}$ uniformly at random from all matrices that satisfy (17). This implies that the probability

$$\begin{aligned} \Pr ({{\,\textrm{rk}\,}}_{q^m}(\Lambda _{n_t-\delta -1}(\overline{\varvec{Z}})_{\varvec{a}})<\gamma ) \end{aligned}$$

depends on $\varvec{\gamma }$ and not on $\varvec{t}$ and $\varvec{\varkappa }$. Hence, we can use [4, Lemma 7] with $t=\gamma$, $\varvec{t}=\varvec{\gamma }$, $n=n_r$ and get

$$\begin{aligned} \Pr ({{\,\textrm{rk}\,}}_{q^m}(\Lambda _{n_t-\delta -1}(\overline{\varvec{Z}})_{\varvec{a}})<\gamma ) \le \kappa _q^{\ell +1} q^{-m\left( \gamma _\textsf{max}-\gamma +1\right) }. \end{aligned}$$

Note that this expression is independent of the rank partition $\varvec{\gamma }$, so it is also an upper bound for $\Pr \!\left( {{\,\textrm{rk}\,}}_{q^m}(\Lambda _{n_t-\delta -k}(\overline{\varvec{Z}})_{{\varvec{a}}})<\gamma \right)$ with $\varvec{\mathcal {E}}$ drawn according to Remark 2. $\square$

Theorem 1

(Loidreau–Overbeck-like Decoder for LILRS Codes) Suppose we transmit the tuple of subspaces

$$\begin{aligned} \varvec{\mathcal {V}}(\varvec{f}) = \left( {\mathcal {V}}^{(1)}(\varvec{f}), \dots , {\mathcal {V}}^{(\ell )}(\varvec{f})\right) \in \textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k] \end{aligned}$$

over a random instance of the $\ell$-shot operator channel (see Remark 2)

$$\begin{aligned} \varvec{\mathcal {U}}=\mathcal {H}_{n_t-\delta }(\varvec{\mathcal {V}})\oplus \varvec{\mathcal {E}}, \end{aligned}$$

with overall $\gamma$ insertions and $\delta$ deletions, where

$$\begin{aligned} \gamma \le \gamma _\textsf{max}:= s(n_t-\delta -k). \end{aligned}$$

(18)

Then, Algorithm 1 with input $\varvec{\mathcal {V}}(\varvec{f})$ returns the correct message polynomial vector $\varvec{f}$ with success probability at least

$$\begin{aligned} \Pr (\text {success}) \ge 1 - \kappa _q^{\ell +1} q^{-m(\gamma _\textsf{max}-\gamma +1)}. \end{aligned}$$

Furthermore, the algorithm has complexity $O(sn_r^\omega )$ operations in $\mathbb F_{q^m}$ plus $O(m n_r^{\omega -1})$ operations in $\mathbb F_{q}$.

Proof

Due to Proposition 3, the algorithm returns the correct message polynomial vector $\varvec{f}$ if the $\mathbb F_{q^m}$-rank of $\Lambda _{n_t-\delta -k}(\overline{\varvec{Z}})_{\varvec{a}}$ is at least $\gamma$. Hence, the success probability is lower bounded by the probability that ${{\,\textrm{rk}\,}}_{q^m}(\Lambda _{n_t-\delta -k}(\overline{\varvec{Z}})_{{\varvec{a}}})=\gamma$, which is given in Lemma 4.

The lines of the algorithm have the following complexities:

• Lines 3 and 6: This can be done by solving the linear system of equations ${\varvec{L}}\varvec{h}^\top = \textbf{0}$. Since ${\varvec{L}}\in \mathbb F_{q^m}^{((s+1)(n_t-\delta )-sk -1) \times n_r}$, it costs $O\mathopen {}\left( sn_r^\omega \right) \mathclose {}$ operations in $\mathbb F_{q^m}$.

• Line 8 can be implemented by transforming the matrix representation of $\varvec{h}^{(i)}$, which is an $m\times n_r^{(i)}$ matrix over $\mathbb F_{q}$, into column echelon form. For each i, this costs $O\mathopen {}\left( m {n_r^{(i)}}^{\omega -1}\right) \mathclose {}$ operations in $\mathbb F_{q}$. In total, all $\ell$ calls of this line cost $O\mathopen {}\left( \ell m \sum _i {n_r^{(i)}}^{\omega -1}\right) \mathclose {} \subseteq O\mathopen {}\left( m n_r^{\omega -1}\right) \mathclose {}$ operations in $\mathbb F_{q}$.

• Line 9 can be implemented by transforming the matrix representation of $\varvec{h}^{(i)}$ into column echelon form, which was already accomplished in Line 8.

• Line 10 requires $O\mathopen {}\left( s{n_r^{(i)}}^2\right) \mathclose {}$ multiplications over $\mathbb F_{q^m}$ and thus $O\mathopen {}\left( s\sum _i{n_r^{(i)}}^2\right) \mathclose {}\subseteq O\mathopen {}\left( sn_r^2\right) \mathclose {}$ operations in $\mathbb F_{q^m}$ in total.

• Line 13 computes $s$ interpolation polynomials of degree less than $k\le n_t$ point tuples. This costs in total $\widetilde{O}\mathopen {}\left( s\mathcal {M}(n_t)\right) \mathclose {}$ operations in $\mathbb F_{q^m}$ (c.f. [11, 38]).

This proves the complexity statement.

The decoding region of the Loidreau–Overbeck-like decoder for LILRS codes is illustrated in Figure 5.

4.3 An interpolation-based decoding approach

We now derive an interpolation-based decoding approach for LILRS codes. The decoding principle consists of an interpolation step and a root-finding step. In [27], (lifted) linearized Reed–Solomon codes are decoded using the isometry between the sum-rank and the skew metric. In this work we consider an interpolation-based decoding scheme in the generalized operator evaluation domain. The new decoder is a generalization of [50] (interleaved Gabidulin codes in the rank metric) and [6] (lifted interleaved Gabidulin codes in the subspace metric). Compared to the Loidreau–Overbeck-like decoder from Sect. 4.2, which requires $O\mathopen {}\left( s^\omega n_r^2\right) \mathclose {}$ operations in $\mathbb F_{q^m}$, the proposed interpolation based decoder has a reduced computational complexity in the order of $\widetilde{O}\mathopen {}\left( s^\omega \mathcal {M}(n_r)\right) \mathclose {}$ operations in $\mathbb F_{q^m}$.

4.3.1 Interpolation step

Suppose we transmit the tuple of subspaces

$$\begin{aligned} \varvec{\mathcal {V}}(\varvec{f}) = \left( {\mathcal {V}}^{(1)}(\varvec{f}), \dots , {\mathcal {V}}^{(\ell )}(\varvec{f})\right) \in \textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k] \end{aligned}$$

over an $\ell$-shot operator channel (3) with $\gamma$ insertions and $\delta$ deletions and receive the tuple of subspaces

$$\begin{aligned} \varvec{\mathcal {U}}= \left( {\mathcal {U}}^{(1)},{\mathcal {U}}^{(2)}, \dots , {\mathcal {U}}^{(\ell )}\right) \in \mathcal {G}_q(\varvec{N},\varvec{n}_r) \end{aligned}$$

where the received subspaces ${\mathcal {U}}^{(i)}$ are represented as in (7). We describe $\varvec{\mathcal {U}}$ by the tuple containing the basis matrices of the received subspaces as

$$\begin{aligned} \varvec{U}:=\left( {\varvec{U}}^{(1)},{\varvec{U}}^{(2)},\dots ,{\varvec{U}}^{(\ell )}\right) \in \prod _{i=1}^{\ell }\mathbb F_{q^m}^{n_r^{(i)}\times (s+1)} \end{aligned}$$

where

$$\begin{aligned} {\varvec{U}}^{(i)}:=\begin{pmatrix} \varvec{\xi }^{(i)\top }&\varvec{u}_1^{(i)\top }&\varvec{u}_2^{(i)\top }&\dots&\varvec{u}_s^{(i)\top } \end{pmatrix} \in \mathbb F_{q^m}^{n_r^{(i)}\times (s+1)} \end{aligned}$$

has $\mathbb F_{q}$-rank ${{\,\textrm{rk}\,}}_q({\varvec{U}}^{(i)})=n_r^{(i)}$ and satisfies ${\mathcal {U}}^{(i)}={\left\langle {\varvec{U}}^{(i)} \right\rangle }_{q}$ for all $i=1,\dots ,\ell$.

Remark 4

In contrast to [27, Sect. V.H] we do not need the assumption that the $\mathbb F_{q}$-rank of $\varvec{\xi }^{(i)}$ equals $n_r^{(i)}$ for all $i=1,\dots ,\ell$, which is not the case in general (see also [47, Sect. 5.1.2]).

For a multivariate skew polynomial of the form

$$\begin{aligned} Q(x, y_1,\dots , y_s)=Q_0(x)+Q_1(x)y_1+\dots +Q_s(x)y_s\end{aligned}$$

where $Q_l(x)\in \mathbb F_{q^m}[x;\sigma ]$ for all $l\in [0,s]$ define the $n_r$ generalized operator evaluation maps

$$\begin{aligned} \mathbb F_{q^m}[x,y_1,\dots ,y_s;\sigma ]\times \mathbb F_{q^m}^{s+1}&\rightarrow \mathbb F_{q^m}\nonumber \\ \left( Q, (\xi _j^{(i)},u_{1,l}^{(i)},\dots ,u_{s,l}^{(i)})\right)&\mapsto \mathscr {E}_{j}^{(i)}(Q):={Q_0}(\xi _j^{(i)})_{a_i}+\textstyle \sum _{l=1}^{s}{Q_l}(u_{j,l}^{(i)})_{a_i}. \end{aligned}$$

(19)

for all $j=1,\dots ,n_r^{(i)}$ and $i=1,\dots ,\ell$. Now consider the following interpolation problem in $\mathbb F_{q^m}[x;\sigma ]$ (see also [3, Problem 1]).

Problem 1

(LILRS Interpolation Problem) Given the integers $D,s,\ell \in \mathbb {Z}_{\ge 0}$, a set

$$\begin{aligned} \mathscr {E}=\left\{ \mathscr {E}_{j}^{(i)}:i=1,\dots ,\ell , j=1,\dots ,n_i\right\} \end{aligned}$$

containing the generalized operator evaluation maps defined in (19) and a vector $\varvec{w}=(0,k-1,\dots ,k-1)\in \mathbb {Z}_{\ge 0}^{s+1}$, find a nonzero polynomial of the form

$$\begin{aligned} Q(x, y_1,\dots , y_s)=Q_0(x)+Q_1(x)y_1+\dots +Q_s(x)y_s\end{aligned}$$

with $Q_l(x)\in \mathbb F_{q^m}[x;\sigma ]$ for all $l\in [0,s]$ that satisfies:

• 1. $\mathscr {E}_{j}^{(i)}(Q)=0, \qquad \forall i=1,\dots ,\ell$, $j=1,\dots ,n_r^{(i)}$,

• 2. $\deg _{\varvec{w}}(Q(x,y_1,\dots ,y_s))<D$.

Define the skew polynomials

$$\begin{aligned} Q_0(x)=\sum _{i=0}^{D-1}q_{0,i}x^i \qquad \text {and}\qquad Q_j(x)=\sum _{i=0}^{D-k}q_{j,i}x^i, \end{aligned}$$

and the vectors

$$\begin{aligned} \varvec{\xi }=\left( \varvec{\xi }^{(1)} \mid \varvec{\xi }^{(2)} \mid \dots \mid \varvec{\xi }^{(\ell )}\right) \in \mathbb F_{q^m}^{n_r} \quad \text {and}\quad \varvec{u}_j=\left( \varvec{u}_j^{(1)} \mid \varvec{u}_j^{(2)} \mid \dots \mid \varvec{u}_j^{(\ell )}\right) \in \mathbb F_{q^m}^{n_r} \end{aligned}$$

for all $j=1,\dots ,s$. Then a solution of Problem 1 can be found by solving the $\mathbb F_{q^m}$-linear system

$$\begin{aligned} \varvec{R}_I\varvec{q}=\varvec{0} \end{aligned}$$

(20)

for

$$\begin{aligned} \varvec{q} = \left( q_{0,0},q_{0,1},\dots ,q_{0,D-1}\mid q_{1,0},q_{1,1},\dots ,q_{1,D-k}\mid \dots \mid q_{s,0},q_{s,1},\dots ,q_{s,D-k}\right) \end{aligned}$$

(21)

where the interpolation matrix $\varvec{R}_I\in \mathbb F_{q^m}^{n_r\times D(s+1)-s(k-1)}$ is given by

$$\begin{aligned} \varvec{R}_I= \begin{pmatrix} \Lambda _{D}(\varvec{\xi })_{\varvec{a}}^\top&\Lambda _{D-k+1}(\varvec{u}_1)_{\varvec{a}}^\top&\dots&\Lambda _{D-k+1}(\varvec{u}_s)_{\varvec{a}}^\top \end{pmatrix}. \end{aligned}$$

(22)

Problem 1 can be solved by the skew Kötter interpolation [23] with the generalized operator evaluation maps $\mathscr {E}_{j}^{(i)}$ as defined in (19) requiring $O\mathopen {}\left( s^2 n_r^2\right) \mathclose {}$ operations in $\mathbb F_{q^m}$. A solution of Problem 1 can be found efficiently requiring only $\widetilde{O}\mathopen {}\left( s^\omega \mathcal {M}(n_r)\right) \mathclose {}$ operations in $\mathbb F_{q^m}$ using a variant of the minimal approximant bases approach from [4]. Another approach yielding the same computational complexity of $\widetilde{O}\mathopen {}\left( s^\omega \mathcal {M}(n_r)\right) \mathclose {}$ operations in $\mathbb F_{q^m}$ is given by the fast divide-and-conquer Kötter interpolation from [2].

Lemma 5

(Existence of Solution [3, Lemma 2]) A nonzero solution of Problem 1 exists if .

Proof

Problem 1 corresponds to a system of $n_r$ $\mathbb F_{q^m}$-linear equations in $D(s+1)-s(k-1)$ unknowns (see (20)) which has a nonzero solution if the number of equations is less than the number of unknowns, i.e. if

$$\begin{aligned} n_r<D(s+1)-s(k-1) \quad \Longleftrightarrow \quad D\ge \tfrac{n_r+s(k-1)+1}{s+1}. \end{aligned}$$

(23)

The $\mathbb F_{q^m}$-linear solution space $\mathcal {Q}$ of Problem 1 is defined as

$$\begin{aligned} \mathcal {Q}:=\{Q\in \mathbb F_{q^m}[x,y_1,\dots ,y_s;\sigma ]:\varvec{q}(Q)\in \ker _r(\varvec{R}_I)\} \end{aligned}$$

where $\varvec{q}(Q)\in \mathbb F_{q^m}^{D(s+1)-s(k-1)}$ denotes the coefficient vector of Q as defined in (21). The dimension of the $\mathbb F_{q^m}$-linear solution space $\mathcal {Q}$ of Problem 1 (i.e. the dimension of the right kernel of $\varvec{R}_I$ in (22)) is denoted by

$$\begin{aligned} d_I:=\dim (\mathcal {Q})=\dim (\ker _r(\varvec{R}_I)). \end{aligned}$$

4.3.2 Root-finding step

The goal of the root-finding step is to recover the message polynomials $f_1,\dots ,f_s\in \mathbb F_{q^m}[x;\sigma ]_{<k}$ from the multivariate polynomial constructed in the interpolation step. We now derive a condition for the recovery of the message polynomials (see also [3, Lemma 3, Theorem 1]).

Lemma 6

(Roots of Polynomial) Let

$$\begin{aligned} P(x):=Q_0(x)+Q_1(x)f_1(x)+\dots +Q_s(x)f_s(x). \end{aligned}$$

Then there exist elements $\zeta _1^{(i)},\dots ,\zeta _{n_t^{(i)}-{\delta }^{(i)}}^{(i)}$ in $\mathbb F_{q^m}$ that are $\mathbb F_{q}$-linearly independent for each $i=1,\dots ,\ell$ such that

$$\begin{aligned} {P}(\zeta _j^{(i)})_{a_i}=0 \end{aligned}$$

for all $i=1,\dots ,\ell$ and $j=1,\dots ,n_t^{(i)}-{\delta }^{(i)}$.

Proof

In each shot the non-corrupted intersection space has dimension $\dim ({\mathcal {U}}^{(i)}\cap {\mathcal {V}}^{(i)})=n_t^{(i)}-{\delta }^{(i)}$ for all $i=1,\dots ,\ell$. A basis for each intersection space ${\mathcal {U}}^{(i)}\cap {\mathcal {V}}^{(i)}$ can be represented as

$$\begin{aligned} \left\{ \left( \zeta _j^{(i)},{f_1}(\zeta _j^{(i)})_{a_i},\!\dots ,{f_s}(\zeta _j^{(i)})_{a_i}\right) \!:\!j\!\in \![1,n_t^{(i)}\!-\!{\delta }^{(i)}]\right\} \end{aligned}$$

where $\zeta _1^{(i)},\dots ,\zeta _{n_t^{(i)}-{\delta }^{(i)}}^{(i)}$ are $(n_t^{(i)}-{\delta }^{(i)})$ $\mathbb F_{q}$-linearly independent elements from $\mathbb F_{q^m}$ for all $i=1,\dots ,\ell$. Since each intersection space ${\mathcal {U}}^{(i)}\cap {\mathcal {V}}^{(i)}$ is a subspace of the received space ${\mathcal {U}}^{(i)}$ we have that

$$\begin{aligned} {P}(\zeta _j^{(i)})_{a_i}:={Q_0}(\zeta _j^{(i)})_{a_i}+\sum _{l=1}^{s}{Q_l}({f_l}(\zeta _j^{(i)})_{a_i})_{a_i} =0 \end{aligned}$$

for all $i=1,\dots ,\ell , j=1,\dots ,n_t^{(i)}-{\delta }^{(i)}$.

Theorem 2

(Decoding Region) Let $\varvec{\mathcal {U}}\in \mathcal {G}_q(\varvec{N},\varvec{n}_r)$ be the tuple containing the received subspaces and let $Q(x,y_1,\dots ,y_s)\ne 0$ fulfill the constraints in Problem 1. Then for all codewords $\varvec{\mathcal {V}}(\varvec{f})\in \textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k]$ that are $(\gamma ,\delta )$-reachable from $\varvec{\mathcal {U}}$, where $\gamma$ and $\delta$ satisfy

$$\begin{aligned} \gamma +s\delta <s(n_t-k+1), \end{aligned}$$

(24)

we have that

$$\begin{aligned} P(x)\!=\!Q_0(x)+Q_1(x)f_1(x)+\!\dots \!+Q_s(x)f_s(x)\!=\!0. \end{aligned}$$

(25)

Proof

By Lemma 6 there exist elements $\zeta _1^{(i)},\dots ,\zeta _{n_t^{(i)}-{\delta }^{(i)}}$ in $\mathbb F_{q^m}$ that are $\mathbb F_{q}$-linearly independent for each $i=1,\dots ,\ell$ such that

$$\begin{aligned} {P}(\zeta _j^{(i)})_{a_i}=0 \end{aligned}$$

for all $i=1,\dots ,\ell$ and $j=1,\dots ,n_t^{(i)}-{\delta }^{(i)}$. By choosing

$$\begin{aligned} D\le n_t-\delta \end{aligned}$$

(26)

the degree of P(x) exceeds the degree bound from [12]) which is possible only if $P(x)=0$. Combining (23) and (26) we get

$$\begin{aligned} n_r+s(k-1)&<D(s+1)\le (s+1)(n_t-\delta ) \\\Longleftrightarrow \qquad \gamma +s\delta&<s(n_t-k+1). \end{aligned}$$

The decoding region in (24) shows and improved insertion-correction performance due to interleaving. The resulting improvement is illustrated in Figure 5.

In the root-finding step, all polynomials $f_1,\dots ,f_s\in \mathbb F_{q^m}[x;\sigma ]_{<k}$ that satisfy (25) need to be found. Instead of using only one solution of Problem 1 to set up the root-finding system we use a basis for the $d_I$-dimensional $\mathbb F_{q^m}$-linear solution space $\mathcal {Q}$ (see also [5, 50]. Alternatively, a degree-restricted subset of a Gröbner basis for the interpolation module of cardinality at most $s$ can be used to set up the root-finding system and find the minimal number of solutions (see [4]).

Fig. 5

Decoding region for Martínez-Peñas–Kschischang [27] codes $(s=1)$ and for decoding of lifted $(s=4)$-interleaved linearized Reed–Solomon codes. The decoding region for insertions increases with the interleaving order $s$

To set up the root-finding system set up with a basis for $\mathcal {Q}$ define the matrices

$$\begin{aligned} \sigma ^{i}({\varvec{Q}}_{j}):=\begin{pmatrix} \sigma ^i\left( q_{1,j}^{(1)}\right) &{} \sigma ^i\left( q_{2,j}^{(1)}\right) &{} \dots &{} \sigma ^i\left( q_{s,j}^{(1)}\right) \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \sigma ^i\left( q_{1,j}^{(d_I)}\right) &{} \sigma ^i\left( q_{2,j}^{(d_I)}\right) &{} \dots &{} \sigma ^i\left( q_{s,j}^{(d_I)}\right) \end{pmatrix}\in \mathbb F_{q^m}^{d_I \times s} \end{aligned}$$

and the vectors

$$\begin{aligned} \sigma ^{i}({\varvec{f}}_{j}) :=\left( \sigma ^i\left( f_j^{(1)}\right) ,\dots ,\sigma ^i\left( f_j^{(s)}\right) \right) \in \mathbb F_{q^m}^{s} \end{aligned}$$

and

$$\begin{aligned} \sigma ^{i}({\varvec{q}}_{0,j}) :=\left( \sigma ^i\left( q_{0,j}^{(1)}\right) ,\dots ,\sigma ^i\left( q_{0,j}^{(d_I)}\right) \right) \in \mathbb F_{q^m}^{d_I}. \end{aligned}$$

Defining the root-finding matrix

$$\begin{aligned} \varvec{Q}_R:=\begin{pmatrix} \varvec{Q}_0 &{} &{} &{} \\ \sigma ^{-1}({\varvec{Q}}_{1}) &{} \sigma ^{-1}({\varvec{Q}}_{0}) &{} &{} \\ \vdots &{} \sigma ^{-2}({\varvec{Q}}_{1}) &{} \ddots &{} \\ \sigma ^{-(D-k)}({\varvec{Q}}_{D-k}) &{} \vdots &{} \ddots &{} \sigma ^{-(k-1)}({\varvec{Q}}_{0}) \\ &{} \sigma ^{-(D-k-1)}({\varvec{Q}}_{D-k}) &{} \ddots &{} \sigma ^{-k}({\varvec{Q}}_{1}) \\ &{} &{} \ddots &{} \vdots \\ &{} &{} &{} \sigma ^{-(D-1)}({\varvec{Q}}_{D-k}) \end{pmatrix}\in \mathbb F_{q^m}^{Dd_I \times sk} \end{aligned}$$

(27)

and the vectors

$$\begin{aligned} \varvec{f}_R:=\left( \varvec{f}_0,\sigma ^{-1}({\varvec{f}}_{1}),\dots ,\sigma ^{-(k-1)}({\varvec{f}}_{k-1})\right) ^\top \in \mathbb F_{q^m}^{sk} \end{aligned}$$

(28)

and

$$\begin{aligned} \varvec{q}_{0}:=\left( \varvec{q}_{0,0},\sigma ^{-1}({\varvec{q}_{0,1}}_{}),\dots , \sigma ^{-(D-1)}({\varvec{q}_{{0,D-1}}}_{})\right) ^{\top }y\in \mathbb F_{q^m}^{Dy d_I} \end{aligned}$$

(29)

as in [4] we can write the root-finding system (25) as

$$\begin{aligned} \varvec{Q}_R\cdot \varvec{f}_R=-\varvec{q}_0. \end{aligned}$$

(30)

The root-finding system can be solved efficiently by the minimal approximant bases method in [7, Algorithm 7] (see also [4, Sect. IV.C]) requiring at most $\widetilde{O}\mathopen {}\left( s^\omega \mathcal {M}(n)\right) \mathclose {}$ operations in $\mathbb F_{q^m}$.

4.3.3 List decoding

In general, the root-finding matrix $\varvec{Q}_R$ in (30) can be rank deficient. In this case we obtain a list of potential message polynomials $f_1,\dots ,f_s$. By [4, Proposition 4] the root-finding system in (25) has at most $q^{m(k(s-1))}$ solutions $f_,\dots ,f_s\in \mathbb F_{q^m}[x;\sigma ]_{<k}$. In general, we have that $k\le n_t$, where $n_t\le \ell m$. Hence, for $m\approx n_t/\ell$ we get a worst-case list size of $q^{\frac{n_t}{\ell }(k(s-1))}$.

Algorithm 2

List Decoding of LILRS Codes

Theorem 3

(List Decoding of LILRS Codes) Let $\varvec{\mathcal {U}}\in \mathcal {G}_q(\varvec{N},\varvec{n}_r)$ be a tuple of received subspaces of a transmission of a codeword $\varvec{\mathcal {V}}\in \textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k]$ over an $\ell$-shot operator channel with overall $\gamma$ insertions and $\delta$ deletions. If the number of overall insertions $\gamma$ and deletions $\delta$ satisfy

$$\begin{aligned} \gamma +s\delta <s(n_t-k+1), \end{aligned}$$

then a list $\mathcal {L}$ of size

$$\begin{aligned} |\mathcal {L}|\le q^{m(k(s-1))} \end{aligned}$$

containing all message polynomial vectors $\varvec{f}\in \mathbb F_{q^m}[x;\sigma ]_{<k}^s$ corresponding to codewords $\varvec{\mathcal {V}}(\varvec{f})\in \textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k]$ that are $(\gamma ,\delta )$-reachable from $\varvec{\mathcal {U}}$ can be found requiring at most $\widetilde{O}\mathopen {}\left( s^\omega \mathcal {M}(n_r)\right) \mathclose {}$ operations in $\mathbb F_{q^m}$.

Proof

The proof follows directly from Lemma 6, Theorem 2 and the discussion above.

4.3.4 Probabilistic unique decoding

We now consider the interpolation-based decoder from Sect. 4.3 as a probabilistic unique decoder which either returns a unique solution (if the list size is equal to one) or a decoding failure. The main idea is to use a basis for the $d_I$-dimensional $\mathbb F_{q^m}$-linear solution space $\mathcal {Q}$ of the interpolation system (20) to set up the root-finding matrix (27) which in turn facilitates that the root-finding matrix $\varvec{Q}_R$ can have full rank.

Using similar arguments as in [4,5,6] we can lower bound the dimension $d_I$ of the $\mathbb F_{q^m}$-linear solution space $\mathcal {Q}$ of Problem 1.

Lemma 7

(Dimension of Solution Space) Let $\gamma$ and $\delta$ satisfy (24). Then the dimension $d_I=\dim (\mathcal {Q})$ of the $\mathbb F_{q^m}$-linear solution space $\mathcal {Q}$ of Problem 1 satisfies

$$\begin{aligned} d_I\ge s(D+1)-sk-\gamma . \end{aligned}$$

Proof

Let

$$\begin{aligned} \left\{ \left( \zeta _j^{(i)},{f_1}(\zeta _j^{(i)})_{a_i},\dots ,{f_s}(\zeta _j^{(i)})_{a_i}\right) :j\in [1,n_t^{(i)}-{\delta }^{(i)}]\right\} \end{aligned}$$

be a basis for each non-corrupted intersection space ${\mathcal {U}}^{(i)}\cap {\mathcal {V}}^{(i)}$ where $\zeta _1^{(i)},\dots ,\zeta _{n_t^{(i)}-{\delta }^{(i)}}^{(i)}$ are $(n_t^{(i)}-{\delta }^{(i)})$ $\mathbb F_{q}$-linearly independent elements from $\mathbb F_{q^m}$ for all $i=1,\dots ,\ell$. Define the vector $\varvec{\zeta }:=(\varvec{\zeta }^{(1)} \mid \varvec{\zeta }^{(2)} \mid \dots \mid \varvec{\zeta }^{(\ell )})\in \mathbb F_{q^m}^{n_t-\delta }$ where $\varvec{\zeta }^{(i)}=(\zeta _1^{(i)},\dots ,\zeta _{n_t^{(i)}-{\delta }^{(i)}}^{(i)})$ for all $i=1,\dots ,\ell$. Let

$$\begin{aligned} \left\{ \left( \epsilon _j^{(i)},\tilde{e}_{j,1}^{(i)},\dots ,\tilde{e}_{j,s}^{(i)},\right) :j\in [1,{\gamma }^{(i)}]\right\} \end{aligned}$$

be a basis for the error space ${\mathcal {E}}^{(i)}$ for all $i=1,\dots ,\ell$ and define

$$\begin{aligned} \varvec{\epsilon }&=(\varvec{\epsilon }^{(1)} \mid \varvec{\epsilon }^{(2)} \mid \dots \mid \varvec{\epsilon }^{(\ell )})\in \mathbb F_{q^m}^{{\gamma }^{(i)}} \quad \text {with}\quad \varvec{\epsilon }^{(i)}=(\epsilon _1^{(i)},\epsilon _2^{(i)},\dots ,\epsilon _{{\gamma }^{(i)}}^{(i)}), \\ \tilde{\varvec{e}}_j&=(\tilde{\varvec{e}}_j^{(1)} \mid \tilde{\varvec{e}}_j^{(2)} \mid \dots \mid \tilde{\varvec{e}}_j^{(\ell )})\in \mathbb F_{q^m}^{{\gamma }^{(i)}} \quad \text {with}\quad \tilde{\varvec{e}}_j^{(i)}=(\tilde{e}_{1,j}^{(i)},\tilde{e}_{2,j}^{(i)},\dots ,\tilde{e}_{{\gamma }^{(i)},j}^{(i)}). \end{aligned}$$

Then the matrix

$$\begin{aligned} \widetilde{\varvec{R}}_I= \begin{pmatrix} \Lambda _{D}(\varvec{\zeta })_{\varvec{a}}^\top &{} \textbf{0 }&{} \dots &{} \textbf{0 }\\ \Lambda _{D}(\varvec{\epsilon })_{\varvec{a}}^\top &{} \Lambda _{D-k+1}(\tilde{\varvec{e}}_1)_{\varvec{a}}^\top &{} \dots &{} \Lambda _{D-k+1}(\tilde{\varvec{e}}_s)_{\varvec{a}}^\top \end{pmatrix} \in \mathbb F_{q^m}^{n_r\times D(s+1)-s(k-1)} \end{aligned}$$

has the same column space as the matrix $\varvec{R}_I$ in (22). Since ${{\,\textrm{wt}\,}}_{\Sigma R}(\varvec{\zeta }) = n_t-\delta$ and $D\le n_t-\delta$ (see (26)) we have that the matrix $\Lambda _{D}(\varvec{\zeta })_{\varvec{a}}^\top$ has $\mathbb F_{q^m}$-rank $D$. The last $\gamma$ rows of $\widetilde{\varvec{R}}_I$ can increase the $\mathbb F_{q^m}$-rank of $\widetilde{\varvec{R}}_I$ by at most $\gamma$. Thus we have that ${{\,\textrm{rk}\,}}_{q^m}(\varvec{R}_I)={{\,\textrm{rk}\,}}_{q^m}(\widetilde{\varvec{R}}_I)\le D+\gamma$. Hence, the dimension $d_I$ of the $\mathbb F_{q^m}$-linear solution space $\mathcal {Q}$ of Problem 1 satisfies

$$\begin{aligned} d_I=\dim (\ker _r(\varvec{R}_I))&=D(s+1)-s(k-1)-{{\,\textrm{rk}\,}}_{q^m}(\varvec{R}_I) \\&\ge s(D+1)-sk -\gamma . \end{aligned}$$

The rank of the root-finding matrix $\varvec{Q}_R$ can be full if and only if the dimension of the solution space of the interpolation problem $d_I$ is at least $s$, i.e. if

$$\begin{aligned} d_I\ge s\qquad \Longleftrightarrow \qquad \gamma&\le sD-sk\nonumber \\ \Longleftrightarrow \qquad \gamma +s\delta&\le s(n_t-k). \end{aligned}$$

(31)

The probabilistic unique decoding region in (31) is only sightly smaller than the list decoding region in (24). The improved decoding region for LILRS codes is illustrated in Figure 5. Recall from Remark 4 that unlike the proposed decoder, the decoder in [27] has the restriction that $n_r=n_t$, which corresponds to the case that $\gamma = \delta$.

Combining (26) and (31) we get the degree constraint for the probabilistic unique decoder (see [5])

In order to get an estimate of the probability of successful decoding, we use similar assumptions as in [5, 50] to derive a heuristic upper bound on the decoding failure probability $P_f$.

The root-finding matrix $\varvec{Q}_R$ in (27) contains a lower block-diagonal matrix with $\varvec{Q}_0, \sigma ^{-1}({\varvec{Q}}_{0}),\dots ,\sigma ^{-(k-1)}({\varvec{Q}}_{0})$ on the diagonal. Since ${{\,\textrm{rk}\,}}_{q^m}(\sigma ^{i}({\varvec{Q}}_{0}))={{\,\textrm{rk}\,}}_{q^m}(\varvec{Q}_0)$ for all i, this implies that ${{\,\textrm{rk}\,}}_{q^m}(\varvec{Q}_R) = sk$ if ${{\,\textrm{rk}\,}}_{q^m}(\varvec{Q}_0) = s$.

Under the assumption that the coefficients $q_{i,j}^{(r)}$ are uniformly distributed over $\mathbb F_{q^m}$ (see [5, 50, Lemma 9]), we can upper bound the decoding failure probability $P_f :=\Pr ({{\,\textrm{rk}\,}}_{q^m}(\varvec{Q}_R) < sk)$ by the probability that the $(d_I \times s)$ matrix $\varvec{Q}_0$ with uniformly distributed elements from $\mathbb F_{q^m}$ has $\mathbb F_{q^m}$-rank less than $s$ and get

(32)

Note, that for $D=n_t-\delta$ (see (26)) we get

$$\begin{aligned} P_f \le \kappa _qq^{-m(\gamma _\textsf{max}-\gamma +1)}. \end{aligned}$$

Note, that the assumption that the coefficients $q_{i,j}^{(r)}$ are uniformly distributed over $\mathbb F_{q^m}$ does not reflect the distribution of the error space tuple $\varvec{\mathcal {E}}$. Although there is evidence that this assumption is reasonable (see e.g. [16] for folded LRS codes), it does not reflect the actual error model of the multishot operator channel.

Similar as in [50, Lemma 8] for interleaved Gabidulin codes and [4, Theorem 4] for ILRS codes, the conditions of successful decoding of the interpolation-based decoder can be reduced to the conditions of the Loidreau–Overbeck-like decoder from Sect. 4.2. This reduction allows to obtain an upper bound on the decoding failure probability since the distribution of the error space tuple $\varvec{\mathcal {E}}$ is considered in the derivation. The results of the interpolation-based probabilistic unique decoder are summarized in Algorithm 3 and Theorem 4.

Algorithm 3

Probabilistic Unique Decoding of LILRS Codes

Theorem 4

(Probabilistic Unique Decoding of LILRS Codes) Let $\varvec{\mathcal {U}}\in \mathcal {G}_q(\varvec{N},\varvec{n}_r)$ be a tuple of received subspaces of a transmission of a codeword $\varvec{\mathcal {V}}\in \textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k]$ over random instance of the $\ell$-shot operator channel (see Remark 2) with overall $\gamma$ insertions and $\delta$ deletions. If the number of overall insertions $\gamma$ and deletions $\delta$ satisfy

$$\begin{aligned} \gamma +s\delta \le s(n_t-k), \end{aligned}$$

then a the unique message polynomial vector $\varvec{f}\in \mathbb F_{q^m}[x;\sigma ]_{<k}^s$ corresponding to the codeword $\varvec{\mathcal {V}}(\varvec{f})\in \textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k]$ satisfying $d_{\Sigma S}(\varvec{\mathcal {V}}(\varvec{f}),\varvec{\mathcal {U}})=\gamma +\delta$ can be found with probability at least

$$\begin{aligned} \Pr (\text {success}) \ge 1 - \kappa _q^{\ell +1} q^{-m(\gamma _\textsf{max}-\gamma +1)} \end{aligned}$$

(33)

requiring at most $\widetilde{O}\mathopen {}\left( s^\omega \mathcal {M}(n_r)\right) \mathclose {}$ operations in $\mathbb F_{q^m}$.

Proof

For the purpose of the proof (but not algorithmically), we consider the root-finding problem set up with an $\mathbb F_{q^m}$-basis $Q^{(1)},\dots ,Q^{(d_I)}$ of $\mathcal {Q}$. The unique decoder fails if there are at least two distinct roots $\varvec{f}$ and $\varvec{f}'$. In this case, the $\mathbb F_{q^m}$-linear system $\varvec{Q}_R\cdot \varvec{f}_R=-\varvec{q}_0$ in (30) set up with the $\mathbb F_{q^m}$-basis $\widetilde{Q}^{(r)}\in \mathcal {Q}$ for $r=1,\dots ,d_I$ has at least two solutions. This means that $\varvec{Q}_R\in \mathbb F_{q}^{Dd_I\times sk}$ must have rank $<sk$.

The matrix $\varvec{Q}_R$ contains a lower block triangular matrix with matrices $\varvec{Q}_0,\sigma ^{-1}(\varvec{Q}_0), \dots ,\sigma ^{-(k-1)}(\varvec{Q}_0)$ on the upper diagonal, which have all $\mathbb F_{q^m}$-rank ${{\,\textrm{rk}\,}}_{q^m}(\varvec{Q}_0)$ (see [4]). Thus, if ${{\,\textrm{rk}\,}}_{q^m}(\varvec{Q}_0)=s$ the matrix $\varvec{Q}_R$ has full $\mathbb F_{q^m}$-rank $sk$. Therefore, ${{\,\textrm{rk}\,}}_{q^m}(\varvec{Q}_R)<sk$ implies that $\varvec{Q}_0$ has rank $<s$.

Since the root-finding system (30) has at least one solution $\varvec{f}_R$, there is a vector $\varvec{f}_0 \in \mathbb F_{q^m}^s$ such that

$$\begin{aligned} \varvec{Q}_0 \varvec{f}_0 = - \varvec{q}_{0,0}^\top . \end{aligned}$$

Thus, the matrix

$$\begin{aligned} \overline{\varvec{Q}}_0 := \begin{pmatrix} \varvec{Q}_0&\varvec{q}_{0,0}^\top \end{pmatrix} \in \mathbb F_{q^m}^{d_I\times (s+1)} \end{aligned}$$

has rank ${{\,\textrm{rk}\,}}_{q^m}(\overline{\varvec{Q}}_0) = {{\,\textrm{rk}\,}}_{q^m}(\varvec{Q}_0) < s$. Hence, there are at least $d_I-s+1$ $\mathbb F_{q^m}$-linearly independent polynomials $\widetilde{Q}^{(1)},\dots ,\widetilde{Q}^{(d_I-s+1)}\in \mathcal {Q}$ such that their zeroth coefficients $\widetilde{q}_{l,0}^{\,(1)},\dots ,\widetilde{q}_{l,0}^{\,(d_I-s+1)}$ are zero for all $l=0,\dots ,s$ (obtained by suitable) $\mathbb F_{q^m}$-linear combinations of the original basis polynomials $Q^{(1)},\dots ,Q^{(d_I)}$, such that the corresponding $\mathbb F_{q^m}$-linear row operations on $\overline{\varvec{Q}}_0$ give a $(d_I-s+1)\times (s+1)$ zero matrix (recall that $\overline{\varvec{Q}}_0$ has $d_I$ rows, but rank at most $s-1$).

The $d_I-s+1$ $\mathbb F_{q^m}$-linearly independent coefficient vectors of $\widetilde{Q}^{(1)},\dots ,\widetilde{Q}^{(d_I-s+1)}$ of the form (21) are in the left kernel of the matrix

$$\begin{aligned} \varvec{R}_I^\top = \begin{bmatrix} \Lambda _{D}(\varvec{\xi })_{\varvec{a}} \\ \Lambda _{D-k+1}(\varvec{u}_1)_{\varvec{a}} \\ \vdots \\ \Lambda _{D-k+1}(\varvec{u}_s)_{\varvec{a}} \end{bmatrix}\in \mathbb F_{q^m}^{D(s+1)-s(k-1)\times n_r}. \end{aligned}$$

Since the zeroth components $\widetilde{q}_{l,0}^{\,(r)}$ of all $\widetilde{Q}^{(r)}$ are zero for all $l=0,\dots ,s$ and $r=1,\dots ,d_I-s+1$, this means that the left kernel of the matrix

$$\begin{aligned} \widetilde{\varvec{R}}_I^\top = \begin{bmatrix} \mathcal {D}_{{\varvec{a}}}\left( \Lambda _{D-1}(\varvec{\xi })_{\varvec{a}}\right) \\ \mathcal {D}_{{\varvec{a}}}\left( \Lambda _{D-k}(\varvec{u}_1)_{\varvec{a}}\right) \\ \vdots \\ \mathcal {D}_{{\varvec{a}}}\left( \Lambda _{D-k}(\varvec{u}_s)_{\varvec{a}}\right) \end{bmatrix}\mathbb F_{q^m}^{D(s+1)-sk -1 \times n} \end{aligned}$$

has dimension at least $d_I-s+1$. The maximum decoding region corresponds to the degree constraint $D=n_r-\gamma _\textsf{max}=n_t-\delta$ (see (26)) and thus

$$\begin{aligned} \dim (\ker _l(\widetilde{\varvec{R}}_I^\top ))&\ge s(n_t-\delta +1)-sk-\gamma _\textsf{max}-s+1 \ge 1. \end{aligned}$$

Therefore, we have that

$$\begin{aligned} {{\,\textrm{rk}\,}}_{q^m}(\widetilde{\varvec{R}}_I^\top )&\le D(s+1)-sk -1 -\dim (\ker _l(\widetilde{\varvec{R}}_I^\top )) \\&<(n_t-\delta )(s+1)-sk-1 \\&=n_r-1. \end{aligned}$$

Observe, that for $D=n_t-\delta$ we have that

$$\begin{aligned} \widetilde{\varvec{R}}_I^\top =\mathcal {D}_{{\varvec{a}}}\left( {\varvec{L}}\right) \end{aligned}$$

where ${\varvec{L}}$ is the Loidreau–Overbeck decoding matrix from (12). By [4, Lemma 3] the $\mathbb F_{q^m}$-rank of ${\varvec{L}}$ and $\mathcal {D}_{{\varvec{a}}}\left( {\varvec{L}}\right)$ is the same and thus we have that

$$\begin{aligned} {{\,\textrm{rk}\,}}_{q^m}({\varvec{L}})={{\,\textrm{rk}\,}}_{q^m}(\widetilde{\varvec{R}}_I)< n_r-1 \end{aligned}$$

which shows that in this case the Loidreau–Overbeck-like decoder fails as well. Therefore, we conclude that

$$\begin{aligned} \Pr ({{\,\textrm{rk}\,}}_{q^m}(\varvec{Q}_R)<sk)\le \Pr ({{\,\textrm{rk}\,}}_{q^m}(\varvec{Q}_0)<s)\le \Pr ({{\,\textrm{rk}\,}}_{q^m}({\varvec{L}})<n_r-1) \end{aligned}$$

and thus the lower bound on the probability of successful decoding follows from Theorem 1. The complexity statement follows from [4, Corollary 1] and [4, Corollary 2].

The lower bound on the probability of successful decoding in (33) yields an upper bound on the decoding failure probability $P_f$, i.e. we have that

$$\begin{aligned} P_f\le \kappa _q^{\ell +1} q^{-m(\gamma _\textsf{max}-\gamma +1)}. \end{aligned}$$

(34)

The simulations results in Sect. 4.5 show that the upper bound on the decoding failure probability in (34) gives a good estimate of the performance of the probabilistic unique decoder.

The interpolation-based probabilistic unique decoding scheme for LILRS codes is illustrated in Example 2.

Example 2

(Interpolation-Based Decoding) Consider again the code $\textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k]$, the codeword $\varvec{\mathcal {V}}(\varvec{f})$ from (15) and received spaces in $\varvec{\mathcal {U}}$ from (16) considered in Example 1.

First, we compute $\mathbb F_{q^m}[x;\sigma ]$-linearly independent polynomials of minimal $\varvec{w}=(0,1,1)$-weighted degree that span the solution space of Problem 1 as

$$\begin{aligned} Q^{(1)}&= \left( \alpha + 2\right) x^{2} + \left( 2 \alpha + 1\right) x + 1 + (x + 1)y_1 \\ Q^{(2)}&= \left( \alpha + 1\right) x^{2} + \left( \alpha + 2\right) x + \alpha + 2 + (\left( 2 \alpha + 2\right) x + 2 \alpha )y_1 + (x + 2 \alpha + 1)y_2 \end{aligned}$$

using e.g. the skew Kötter interpolation from [23].

Using the coefficients of $Q^{(1)}$ and $Q^{(2)}$ we set up the root-finding matrix $\varvec{Q}_R$ as (see (27))

$$\begin{aligned} \varvec{Q}_R= \left( \begin{array}{rrrr} 1 &{} 0 &{} 0 &{} 0 \\ 2 \alpha &{} 2 \alpha + 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 \\ \alpha + 1 &{} 1 &{} \alpha + 2 &{} \alpha \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 2 \alpha + 2 &{} 1 \end{array}\right) \end{aligned}$$

and the vector $\varvec{q}_0$ as (see (29))

$$\begin{aligned} \varvec{q}_0 = \left( 1, \alpha + 2, \alpha , 2\alpha , \alpha + 2, \alpha + 1\right) ^\top . \end{aligned}$$

The unique solution of the $\mathbb F_{q^m}$-linear root-finding system $\varvec{Q}_R\cdot \varvec{f}_R=-\varvec{q}_0$ in (30) is

$$\begin{aligned} \varvec{f}_R:=&\left( f_0^{(1)}, f_0^{(2)}, \sigma ^{-1}(f_1^{(1)}), \sigma ^{-1}(f_1^{(2)})\right) \\ =&\left( 2, \alpha + 2, 2\alpha + 1, \alpha + 2)\right) ^\top . \end{aligned}$$

Considering the structure of $\varvec{f}_R$ (cf. (28)) we can recover the message polynomials

$$\begin{aligned} f_1(x)&= \alpha x + 2,\\ f_2(x)&= 2 \alpha x + \alpha + 2 \end{aligned}$$

which correspond to the transmitted codeword $\varvec{C}(\varvec{f})$.

4.4 Insertion/deletion-correction with the complementary code

In [6, Sect. 4.4] it was shown that the complementary of an interleaved (single-shot) subspace code is capable of correcting more deletions than insertions. We will now briefly describe how to extend the concept from [6] to the multishot scenario. In particular, we show that the complementary code of a LILRS code is more resilient against deletions than insertions. By using the arguments from [6, Lemma 14] and [6, Theorem 4] on each of the components of $\varvec{\mathcal {U}}^\perp$ (and $\varvec{\mathcal {U}})$ we obtain the following result.

Proposition 3

Consider a LILRS code $\mathcal {C}=\textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell ,s;\varvec{n}_{t},k]$ and the corresponding complementary code $\mathcal {C}^\perp$. Suppose we transmit a tuple $\varvec{\mathcal {V}}^\perp \in \mathcal {C}^\perp$ over a multishot operator channel (3) with overall $\gamma$ insertions and $\delta$ deletions and receive

$$\begin{aligned} \varvec{\mathcal {U}}^\perp =\mathcal {H}_{N-n_t-\delta }(\varvec{\mathcal {V}}^\perp )\oplus \varvec{\mathcal {E}}_\gamma \end{aligned}$$

where ${{\,\mathrm{\dim _{\Sigma }}\,}}(\varvec{\mathcal {E}}_\gamma )=\gamma$. Then we have that

$$\begin{aligned} \varvec{\mathcal {U}}&=\mathcal {H}_{n_t-\gamma }(\varvec{\mathcal {V}})\oplus \varvec{\mathcal {E}}_\delta \end{aligned}$$

where ${{\,\mathrm{\dim _{\Sigma }}\,}}(\varvec{\mathcal {E}}_\delta )=\delta$.

The proof of Proposition 3 can be found in Appendix 1.

Proposition 3 shows, that the dual of the received tuple $\varvec{\mathcal {U}}^\perp$, which is $\varvec{\mathcal {U}}$, is a codeword $\varvec{\mathcal {V}}\in \mathcal {C}$ that is corrupted by $\delta$ insertions and $\gamma$ deletions. Therefore we can use the decoder from Sect. 4.3.3 on $\varvec{\mathcal {U}}$ to perform list decoding of $\gamma$ insertions and $\delta$ deletions up to

$$\begin{aligned} \delta +s\gamma <s(n_t-k+1) \end{aligned}$$

or the decoders from Sects. 4.2 and 4.3.4 to perform probabilistic unique decoding up to

$$\begin{aligned} \delta +s\gamma \le s(n_t-k). \end{aligned}$$

The decoding steps can be summarized as follows:

• 1. Transmit a tuple $\varvec{\mathcal {V}}^\perp \in \mathcal {C}^\perp$ over a multishot operator channel with overall $\gamma$ insertions and $\delta$ deletions.

• 2. Compute the dual of the received tuple $(\varvec{\mathcal {U}}^\perp )^\perp =\varvec{\mathcal {U}}$.

• 3. Use a decoder from Sects. 4.3.3, 4.2 and 4.3.4 to recover the tuple $\varvec{\mathcal {V}}\in \mathcal {C}$.

• 4. Compute the dual tuple of $\varvec{\mathcal {V}}$ to obtain $\varvec{\mathcal {V}}^\perp$.

4.5 Simulation results

In order to verify the upper bound on the decoding failure probability in (33) we performed a Monte Carlo simulation (100 errors) of a code $\textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell =2,s=3;\varvec{n}_{t}=(3,3),k=3]$ over $\mathbb F_{3^3}$ over a multishot operator channel with overall $\delta =1$ deletion and $\gamma \in \{4,5,6\}$ insertions.

The channel realization is chosen uniformly at random from all possible realizations of the multishot operator channel with exactly this number of deletions and insertions (see Remark 2). The drawing procedure was implemented using the adapted dynamic-programming routine in Appendix 1.

The results in Figure 6 show, that the upper bound in (33) gives a good estimate of the decoding failure probability. Although the heuristic upper bound from (32) looks tighter for the considered parameters, it is not a strict upper bound for the considered multishot operator channel.

Fig. 6

Result of a Monte Carlo simulation of the code $\textrm{LILRS}[\varvec{\beta },{\varvec{a}},\ell =2,s=3;\varvec{n}_{t}=(3,3),k=3]$ over $\mathbb F_{3^3}$ transmitted over a multishot operator channel with overall $\delta =1$ deletions and $\gamma =2,3,4,5,6$ insertions

For the same parameters a (non-interleaved) lifted linearized Reed–Solomon code [27] (i.e. $s=1$) can only correct $\gamma$ insertions and $\delta$ deletions up to $\gamma +\delta <4$.

5 Conclusion

5.1 Summary

We considered lifted $s$-interleaved linearized Reed–Solomon (LILRS) codes for error-control in noncoherent multishot network coding and showed that the relative overhead due to lifting can be reduced significantly compared to the construction by Martínez-Peñas–Kschischang. We proposed two decoding schemes for the multishot operator channel that are capable of correcting insertions and deletions beyond the unique decoding region in the sum-subspace metric.

We proposed an efficient interpolation-based decoding scheme for LILRS codes, which can be used as a list decoder or as a probabilistic unique decoder and can correct a total number of $\gamma$ insertions and $\delta$ deletions up to $\gamma + s\delta < s(n_t-k+1)$ and $\gamma + s\delta \le s(n_t-k)$, respectively, where $s$ is the interleaving order, $n_t$ the sum of the dimensions of the transmitted spaces and k the dimension of the code. We derived a Loidreau–Overbeck-like decoder for LILRS codes, which provides arguments to upper bound on the decoding failure probability for the interpolation-based probabilistic unique decoder.

We showed how to construct and decode lifted $s$-interleaved linearized Reed–Solomon codes for error control in random linear multishot network coding. Compared to the construction by Martínez-Peñas–Kschischang, interleaving allows to increase the decoding region significantly (especially w.r.t. the number of insertions) and decreases the overhead due to the lifting (i.e., increases the code rate), at the cost of an increased packet size.

Up to our knowledge, the proposed decoding schemes are the first being able to correct errors beyond the unique decoding region in the sum-subspace metric efficiently. The tightness of the upper bounds on the decoding failure probability of the proposed decoding schemes for LILRS codes were validated via Monte Carlo simulations.

5.2 Remarks on generality

In this paper, we considered codes constructed by skew polynomials with zero derivations, i.e. polynomials from $\mathbb F_{q^m}[x;\sigma ]$, only. The main reason for this is that for operations in $\mathbb F_{q^m}[x;\sigma ]$ we can give complexity bounds, which are of interest in the implementation point of view. However, the complexity analysis has to be performed w.r.t. this setup (computational complexity may be larger).

We considered decoding of homogeneous LILRS codes, respectively, i.e. interleaved codes where the component codes have the same code dimension. We consider these simpler code classes in order to not further complicate the quite involved notation. The decoding schemes proposed in this paper can be generalized to heterogeneous interleaved codes, where each component code may have a different dimension, in a straight-forward manner like e.g. in [5, 49]. Denote by $k_1,\dots ,k_s$ the dimensions of the component codes and define $\overline{k}:=\frac{1}{s}\sum _{l=1}^{s}k_l$. The resulting decoding regions are then $\gamma +s\delta <s(n_t-\overline{k}+1)$ for list decoding and $\gamma +s\delta \le s(n_t-\overline{k})$ for probabilistic unique decoding.

5.3 Outlook and future work

For future work it would be interesting to see the considered concepts applied to interleaved LILRS codes that are based on the construction of LRS codes over smaller fields. Further, the investigation of lifted interleaved variants of other maximum sum-rank distance (MSRD) codes such as e.g. one-weight codes in the sum-rank metric and the application of subspace designs to sum-subspace-metric codes, is subject to future work.

So far, no results on the list-decodability of random sum-subspace-metric codes, like e.g. for single-shot subspace codes [15], are available. Once such results are available it would be interesting to compare the list-decodability of random sum-subspace-metric codes with constructive results proposed in this paper.

Appendices

Appendix A: Proofs from Sect. 4.2

1.1 A.1 Proof of Lemma 2

Proof

First observe, that every received space ${\mathcal {U}}^{(i)}$ can be represented by a matrix $\widetilde{\varvec{U}}^{(i)}$ of the form

$$\begin{aligned} \widetilde{\varvec{U}}^{(i)}= \left( \begin{array}{c|ccc} \varvec{\xi }_1^{(i)\top } &{} \varvec{x}_1^{(i)\top } &{} \dots &{} \varvec{x}_s^{(i)\top } \\ \hline \varvec{\xi }_2^{(i)\top } &{} \widetilde{\varvec{u}}_1^{(i)\top } &{} \dots &{} \widetilde{\varvec{u}}_s^{(i)\top } \\ \hline \varvec{0} &{} \hat{\varvec{e}}_1^{(i)\top } &{} \dots &{} \hat{\varvec{e}}_s^{(i)\top } \end{array} \right) \in \mathbb F_{q^m}^{n_r^{(i)}\times (s+1)} \end{aligned}$$

such that first $n_t^{(i)}-{\delta }^{(i)}$ rows of $\widetilde{\varvec{U}}^{(i)}$ form a basis for the non-corrupted received space ${\mathcal {U}}^{(i)}\cap {\mathcal {V}}^{(i)}$ whereas the last ${t}^{(i)}+{\varkappa }^{(i)}={\gamma }^{(i)}$ rows correspond to the insertions for all $i=1,\dots ,\ell$, i.e. we have that

$$\begin{aligned} {\left\langle \left( \begin{array}{c|ccc} \varvec{\xi }_1^{(i)\top }&\varvec{x}_1^{(i)\top }&\dots&\varvec{x}_s^{(i)\top } \end{array} \right) \right\rangle }_{q}&= {\mathcal {U}}^{(i)}\cap {\mathcal {V}}^{(i)}, \nonumber \\ {\left\langle \left( \begin{array}{c|ccc} \varvec{\xi }_2^{(i)\top } &{} \widetilde{\varvec{u}}_1^{(i)\top } &{} \dots &{} \widetilde{\varvec{u}}_s^{(i)\top } \\ \hline \varvec{0} &{} \hat{\varvec{e}}_1^{(i)\top } &{} \dots &{} \hat{\varvec{e}}_s^{(i)\top } \end{array} \right) \right\rangle }_{q}&= {\mathcal {E}}^{(i)} \end{aligned}$$

(A1)

for all $i=1,\dots ,\ell$ where $\varvec{\xi }_1^{(i)},\varvec{x}_j^{(i)}\in \mathbb F_{q^m}^{n_t^{(i)}-{\delta }^{(i)}}$, $\varvec{\xi }_2^{(i)},\widetilde{\varvec{u}}_j^{(i)}\in \mathbb F_{q^m}^{{t}^{(i)}}$ and $\hat{\varvec{E}}^{(i)}= \begin{pmatrix} \hat{\varvec{e}}_1^{(i)\top }&\dots&\hat{\varvec{e}}_s^{(i)\top } \end{pmatrix} \in \mathbb F_{q^m}^{{\varkappa }^{(i)}\times s}$.

Since the rows of the matrices $\varvec{U}^{(i)}$ also form a basis of ${\mathcal {U}}^{(i)}$, there exist invertible matrices $\varvec{W}^{(i)}\in \mathbb F_{q}^{n_r^{(i)}}$ such that

$$\begin{aligned} \left( \varvec{W}^{(i)}\right) ^\top \varvec{U}^{(i)}=\widetilde{\varvec{U}}^{(i)},\quad \forall i=1,\dots ,\ell . \end{aligned}$$

Since ${\mathcal {U}}^{(i)}\cap {\mathcal {V}}^{(i)}$ is an $n_t^{(i)}-{\delta }^{(i)}$-dimensional subspace ${\mathcal {V}}^{(i)}$, we have by the definition of LILRS codes that ${{\,\textrm{rk}\,}}_{q}(\varvec{\xi }_1^{(i)})=n_t^{(i)}-{\delta }^{(i)}$ for all $i=1,\dots ,\ell$. We also have that ${{\,\textrm{rk}\,}}_{q}(\hat{\varvec{E}}^{(i)})={\varkappa }^{(i)}$ for all $i=1,\dots ,\ell$ since otherwise the matrix $\widetilde{\varvec{U}}^{(i)}$ would not have full $\mathbb F_{q}$-rank $n_r^{(i)}$, which contradicts that it is a basis for ${\mathcal {U}}^{(i)}$ that is obtained from $\varvec{U}^{(i)}$ via $\mathbb F_{q}$-elementary row operations. Since ${\mathcal {V}}^{(i)}\cap {\mathcal {E}}^{(i)}=\{\textbf{0}\}$, we have that $\widetilde{\varvec{u}}_l^{(i)}\ne {f_1}(\varvec{\xi }_2^{(i)})_{a_i}$ for all $i=1,\dots ,\ell$ and $l=1,\dots ,s$, since otherwise there would exist a vector such that, for some j, we have

$$\begin{aligned} {\left\langle (\xi _{2,j}^{(i)}\mid {f_1}(\xi _{2,j}^{(i)})_{a_i}\dots {f_1}(\xi _{2,j}^{(i)})_{a_i}) \right\rangle }_{q}\in {\mathcal {V}}^{(i)} \end{aligned}$$

which contradicts that ${\mathcal {V}}^{(i)}\cap {\mathcal {E}}^{(i)}=\{\textbf{0}\}$. We now show that

$$\begin{aligned} {{\,\textrm{rk}\,}}_{q}(\varvec{\xi }_2^{(i)})={t}^{(i)}, \quad \text {and}\quad {{\,\textrm{rk}\,}}_{q}((\varvec{\xi }_1^{(i)},\varvec{\xi }_2^{(i)}))=n_t^{(i)}-{\delta }^{(i)}+{t}^{(i)}, \quad \forall i=1,\dots ,\ell . \end{aligned}$$

Suppose that ${{\,\textrm{rk}\,}}_{q}(\varvec{\xi }_2^{(i)})<{t}^{(i)}$. Then there exists an $\mathbb F_{q}$-linear combination of the rows of the matrix in (A1) such that the first entry of the vector becomes zero. In this case it would belong to the lower part of the matrix having a leading zero. If ${{\,\textrm{rk}\,}}_{q}((\varvec{\xi }_1^{(i)},\varvec{\xi }_2^{(i)}))<n_t^{(i)}-{\delta }^{(i)}+{t}^{(i)}$, then there exist an element $\xi _{2,j}$ such that ${\left\langle \xi _{2,j} \right\rangle }_{q}\subseteq {\left\langle \varvec{\xi }_{1}^\top \right\rangle }_{q}$. In this case we can subtract the corresponding $\mathbb F_{q}$-linear combination from the row having $\xi _{2,j}$ as first entry, which again yields a row with leading zero element.

Using the matrices $\widetilde{\varvec{U}}^{(i)}$ for all $i=1,\dots ,\ell$ and performing $\mathbb F_{q^m}$-linear row operations, we can set up a matrix of the form

$$\begin{aligned} \overline{{\varvec{L}}}=\left( {\varvec{L}}^{(1)},\dots ,{\varvec{L}}^{(\ell )}\right) :=\left( \begin{array}{c} \Lambda _{n_t-\delta -1}(\overline{\varvec{\xi }})_{\varvec{a}} \\ \Lambda _{n_t-\delta -k}(\overline{\varvec{u}}_{1})_{\varvec{a}} \\ \vdots \\ \Lambda _{n_t-\delta -k}(\overline{\varvec{u}}_{s})_{\varvec{a}} \end{array} \right) \in \mathbb F_{q^m}^{((s+1)(n_t-\delta )-sk -1) \times n_r} \end{aligned}$$

where

$$\begin{aligned} \overline{\varvec{\xi }}&=\left( (\varvec{\xi }_1^{(1)}\mid \varvec{\xi }_2^{(1)}\mid \textbf{0})\mid \dots \mid (\varvec{\xi }_1^{(\ell )}\mid \varvec{\xi }_2^{(\ell )}\mid \textbf{0})\right) , \\ \overline{\varvec{u}}_{l}&=\left( (\textbf{0}\mid \widetilde{\varvec{e}}_l^{(1)}\mid \hat{\varvec{e}}_l^{(1)})\mid \dots \mid (\textbf{0}\mid \widetilde{\varvec{e}}_l^{(\ell )}\mid \hat{\varvec{e}}_l^{(\ell )})\right) , \quad \forall l=1,\dots ,s. \end{aligned}$$

The component matrices are then of the form

$$\begin{aligned} \overline{{\varvec{L}}}^{(i)}= \begin{pmatrix} \varvec{V}_{n_t-\delta -1}(\overline{\varvec{\xi }}_1^{(i)})_{a_i} &{} \varvec{V}_{n_t-\delta -1}(\overline{\varvec{\xi }}_2^{(i)})_{a_i} &{} \textbf{0 }\\ \textbf{0 }&{} \varvec{V}_{n_t-\delta -k}(\widetilde{\varvec{e}}_1^{(i)})_{a_i} &{} \varvec{V}_{n_t-\delta -k}(\hat{\varvec{e}}_1^{(i)})_{a_i} \\ \vdots &{} \vdots &{} \vdots \\ \textbf{0 }&{} \varvec{V}_{n_t-\delta -k}(\widetilde{\varvec{e}}_s^{(i)})_{a_i} &{} \varvec{V}_{n_t-\delta -k}(\hat{\varvec{e}}_s^{(i)})_{a_i} \end{pmatrix} \in \mathbb F_{q^m}^{((s+1)(n_t-\delta )-sk -1) \times n_r^{(i)}} \end{aligned}$$

where $\varvec{x}^{(i)}_l-{f_l}(\varvec{\xi }_1^{(i)})_{a_i}=\textbf{0}$ (yielding the zero matrices in the leftmost block) and $\widetilde{\varvec{e}}_l^{(i)}=\widetilde{\varvec{u}}_l^{(i)}-{f_l}(\varvec{\xi }_2^{(i)})_{a_i}$ for all $l=1,\dots ,s$ and $i=1,\dots ,\ell$. Define the matrices

$$\begin{aligned} \widetilde{\varvec{E}}^{(i)}= \begin{pmatrix} \widetilde{\varvec{e}}_1^{(i)} \\ \vdots \\ \widetilde{\varvec{e}}_s^{(i)} \end{pmatrix}\in \mathbb F_{q^m}^{s\times t^{(i)}}, \qquad \hat{\varvec{E}}^{(i)}= \begin{pmatrix} \hat{\varvec{e}}_1^{(i)} \\ \vdots \\ \hat{\varvec{e}}_s^{(i)} \end{pmatrix}\in \mathbb F_{q^m}^{s\times {\varkappa }^{(i)}} \end{aligned}$$

for all $i=1,\dots ,\ell$. Suppose that the $\mathbb F_{q}$-rank of $\widetilde{\varvec{E}}^{(i)}$ is less than $t^{(i)}$. Then there exist a nontrivial $\mathbb F_{q}$-linear combination of the columns of the matrix

$$\begin{aligned} \begin{pmatrix} \varvec{V}_{n_t-\delta -1}(\overline{\varvec{\xi }}_2^{(i)})_{a_i} \\ \varvec{V}_{n_t-\delta -k}(\widetilde{\varvec{e}}_1^{(i)})_{a_i} \\ \vdots \\ \varvec{V}_{n_t-\delta -k}(\widetilde{\varvec{e}}_s^{(i)})_{a_i} \end{pmatrix} \in \mathbb F_{q^m}^{((s+1)(n_t-\delta )-sk -1) \times t^{(i)}} \end{aligned}$$

such that the first (upper) $(n_t-\delta -1)$ rows are nonzero and the last (lower) $s(n_t-\delta -k)$ rows are all zero. This contradicts that ${\mathcal {V}}^{(i)}\cap {\mathcal {E}}^{(i)}=\{\textbf{0}\}$ since each $\overline{{\varvec{L}}}^{(i)}$ is obtained by subtracting the evaluations of each $f_l$ at the corresponding values in $\varvec{\xi }_1^{(i)}$ and $\varvec{\xi }_2^{(i)}$ (and the corresponding row-operator powers thereof). Now assume that ${{\,\textrm{rk}\,}}_q(\widetilde{\varvec{E}}^{(i)}\mid \hat{\varvec{E}}^{(i)})<{\gamma }^{(i)}$ for some $i\in \{1,\dots ,\ell \}$. Then there exist nontrivial $\mathbb F_{q}$-linear combinations of the $t^{(i)}+{\varkappa }^{(i)}={\gamma }^{(i)}$ rightmost columns of $\overline{{\varvec{L}}}^{(i)}$ such that the $s(n_t-\delta -k)$ rows are all zero and the upper $n_t-\delta -1$ rows are nonzero. This contradicts that ${\mathcal {V}}^{(i)}\cap {\mathcal {E}}^{(i)}=\{\textbf{0}\}$ by the same argument as above.

Since $\overline{{\varvec{L}}}$ is obtained from ${\varvec{L}}$ via $\mathbb F_{q^m}$-elementary row operations and $\mathbb F_{q}$-elementary column operations, we have that

$$\begin{aligned} {{\,\textrm{rk}\,}}_{q^m}({\varvec{L}})={{\,\textrm{rk}\,}}_{q^m}(\overline{{\varvec{L}}}). \end{aligned}$$

1.2 A.2 Proof of Lemma 3

Proof

• Ad 1): By column permutations, i.e. by rearranging the columns of $\overline{{\varvec{L}}}$ according to the vectors

  $$\begin{aligned} \varvec{\xi }_1&:=\left( \varvec{\xi }_1^{(1)} \mid \varvec{\xi }_1^{(2)} \mid \dots \mid \varvec{\xi }_1^{(\ell )}\right) \in \mathbb F_{q^m}^{n_t-\delta }, \quad&\varvec{\xi }_2&:=\left( \varvec{\xi }_2^{(1)} \mid \varvec{\xi }_2^{(2)} \mid \dots \mid \varvec{\xi }_2^{(\ell )}\right) \in \mathbb F_{q^m}^{t}, \\ \widetilde{\varvec{e}}_l&:=\left( \widetilde{\varvec{e}}_l^{(1)} \mid \widetilde{\varvec{e}}_l^{(2)} \mid \dots \mid \widetilde{\varvec{e}}_l^{(\ell )}\right) \in \mathbb F_{q^m}^{t},&\hat{\varvec{e}}_l&:=\left( \hat{\varvec{e}}_l^{(1)} \mid \hat{\varvec{e}}_l^{(2)} \mid \dots \mid \hat{\varvec{e}}_l^{(\ell )}\right) \in \mathbb F_{q^m}^{\varkappa }, \end{aligned}$$

  we obtain another (equivalent) matrix

  $$\begin{aligned} \widetilde{{\varvec{L}}} :=\left( \begin{array}{c|c|c} \Lambda _{n_t-\delta -1}(\varvec{\xi }_1)_{\varvec{a}} &{} \Lambda _{n_t-\delta -1}(\varvec{\xi }_2)_{\varvec{a}} &{} \varvec{0} \\ \varvec{0} &{} \Lambda _{n_t-\delta -k}(\widetilde{\varvec{E}})_{\varvec{a}} &{} \Lambda _{n_t-\delta -k}(\hat{\varvec{E}})_{\varvec{a}} \end{array} \right) \end{aligned}$$

  where

  $$\begin{aligned} \widetilde{\varvec{E}} = \left( \widetilde{\varvec{E}}^{(1)}\mid \dots \mid \widetilde{\varvec{E}}^{(\ell )}\right) \in \mathbb F_{q^m}^{s\times t} \quad \text {and}\quad \hat{\varvec{E}}= \left( \hat{\varvec{E}}^{(1)}\mid \dots \mid \hat{\varvec{E}}^{(\ell )}\right) \in \mathbb F_{q^m}^{s\times \varkappa } \end{aligned}$$

  such that ${{\,\textrm{rk}\,}}_{q^m}({\varvec{L}})={{\,\textrm{rk}\,}}_{q^m}(\overline{{\varvec{L}}})={{\,\textrm{rk}\,}}_{q^m}(\widetilde{{\varvec{L}}})$. By Lemma 2 we have that

  $$\begin{aligned} {{\,\textrm{wt}\,}}_{\Sigma R}(\widetilde{\varvec{E}})=t,\quad {{\,\textrm{wt}\,}}_{\Sigma R}(\hat{\varvec{E}})=\varkappa \end{aligned}$$

  and

  $$\begin{aligned} {{\,\textrm{wt}\,}}_{\Sigma R}\left( ({\widetilde{\varvec{E}}}^{(1)}\mid {\hat{\varvec{E}}}^{(1)})\mid \dots \mid ({\widetilde{\varvec{E}}}^{(\ell )}\mid {\hat{\varvec{E}}}^{(\ell )})\right) =t+\varkappa =\gamma . \end{aligned}$$

  Defining the matrix

  $$\begin{aligned} \varvec{Z}:=\begin{pmatrix} \Lambda _{n_t-\delta -k}(\widetilde{\varvec{E}})_{\varvec{a}}&\Lambda _{n_t-\delta -k}(\hat{\varvec{E}})_{\varvec{a}} \end{pmatrix} \in \mathbb F_{q^m}^{s(n_t-\delta -k)\times \gamma } \end{aligned}$$

  we can write $\widetilde{{\varvec{L}}}$ as a block matrix of the form

  $$\begin{aligned} \widetilde{{\varvec{L}}}= \begin{pmatrix} \Lambda _{n_t-\delta -1}(\varvec{\xi }_1)_{\varvec{a}} &{} \star \\ \textbf{0 }&{} \varvec{Z}\end{pmatrix}, \end{aligned}$$

  where $\star$ denotes an arbitrary matrix, and thus we have that ${{\,\textrm{rk}\,}}_{q^m}(\widetilde{{\varvec{L}}})={{\,\textrm{rk}\,}}_{q^m}(\Lambda _{n_t-\delta -1} (\varvec{\xi }_1)_{\varvec{a}})+{{\,\textrm{rk}\,}}_{q^m}(\varvec{Z})$. By Lemma 2 the vector $\varvec{\xi }_1\in \mathbb F_{q^m}^{n_t-\delta }$ has sum-rank $n_t-\delta$. Recall that we have

  $$\begin{aligned} {{\,\textrm{rk}\,}}_{q^m}\left( \Lambda _{n_t-\delta -1}(\varvec{\xi }_1)_{\varvec{a}}\right) =\min \{n_t-\delta -1,n_t-\delta \}=n_t-\delta -1. \end{aligned}$$

  The matrix $\varvec{Z}$ can be obtained from $\overline{\varvec{Z}}$ via column permutations, which implies that ${{\,\textrm{rk}\,}}_{q^m}(\varvec{Z})={{\,\textrm{rk}\,}}_{q^m}(\overline{\varvec{Z}})$. Since the $\mathbb F_{q^m}$-rank of $\overline{\varvec{Z}}$ (and thus also of $\varvec{Z}$) equals $\gamma$, we conclude by the upper block triangular structure of $\widetilde{{\varvec{L}}}$ that

  $$\begin{aligned} {{\,\textrm{rk}\,}}_{q^m}(\widetilde{{\varvec{L}}})={{\,\textrm{rk}\,}}_{q^m}({\varvec{L}})=n_t-\delta -1+\gamma =n_r-1. \end{aligned}$$

• Ad 2): Let $\overline{\varvec{h}}=\left( \varvec{h}^{(1)}\mid \varvec{h}^{(2)}\mid \dots \mid \varvec{h}^{(\ell )}\right) \in \ker _r(\overline{{\varvec{L}}})\setminus \{\textbf{0}\}$ be a nonzero element in the right kernel of $\overline{{\varvec{L}}}$. If the $\mathbb F_{q^m}$-rank of $\overline{\varvec{Z}}$ equals $\gamma$, we have that

  $$\begin{aligned} {{\,\textrm{rk}\,}}_{q^m}(\overline{\varvec{Z}}^{(i)})={\gamma }^{(i)} \end{aligned}$$

  for all $i=1,\dots ,\ell$. Thus, for any $\overline{\varvec{h}}\in \ker _r(\overline{{\varvec{L}}})\setminus \{\textbf{0}\}$ we must have that the last (rightmost) ${\gamma }^{(i)}$ entries of each $\overline{\varvec{h}}^{(i)}$ must be zero, which implies that

  $$\begin{aligned} {{\,\textrm{rk}\,}}_{q}(\overline{\varvec{h}}^{(i)})\le n_t^{(i)}-{\delta }^{(i)},\quad \forall i=1,\dots ,\ell . \end{aligned}$$

  (A2)

  Let $\mathcal {C}$ be the $[n_t-\delta +t,n_t-\delta -1]$ code generated by $(\Lambda _{n_t-\delta -1}(\varvec{\xi }_1)_{\varvec{a}} \mid \Lambda _{n_t-\delta -1}(\varvec{\xi }_2)_{\varvec{a}})$ and let $\mathcal {C}'$ be the $[n_r,n_t-\delta -1]$ code generated by $(\Lambda _{n_t-\delta -1}(\varvec{\xi }_1)_{\varvec{a}} \mid \Lambda _{n_t-\delta -1}(\varvec{\xi }_2)_{\varvec{a}} \mid \varvec{0})$. Since $(\Lambda _{n_t-\delta -1}(\varvec{\xi }_1)_{\varvec{a}} \mid \Lambda _{n_t-\delta -1}(\varvec{\xi }_2)_{\varvec{a}})$ is a generator matrix of an $[n_t-\delta +t,n_t-\delta -1]$ LRS code (up to column permutations), we have that the dual code $\mathcal {C}^\perp$ is equivalent to an LRS code with minimum sum-rank distance $n_t-\delta$. The dual code of $\mathcal {C}'$ is an $[n_r, \gamma +1]$ code $\mathcal {C}'^\perp$ that can be decomposed into

  $$\begin{aligned} \mathcal {C}_1'^\perp&=\{(\varvec{c}\mid \varvec{0}):\varvec{c}\in \mathcal {C}^\perp \} \\ \mathcal {C}_2'^\perp&=\{(\varvec{0}\mid \varvec{\widetilde{\varvec{c}}}):\varvec{\widetilde{\varvec{c}}}\in \mathbb F_{q^m}^{\varkappa }\} \end{aligned}$$

  such that $\mathcal {C}'^\perp =\mathcal {C}_1'^\perp +\mathcal {C}_2'^\perp$ and $\mathcal {C}_1'^\perp \cap \mathcal {C}_2'^\perp =\{\textbf{0}\}$. Therefore, we have that the minimum distance of the dual code $\mathcal {C}'^\perp$ is $n_t-\delta$. The first $n_t-\delta -1$ rows of $\overline{{\varvec{L}}}$ span code $\overline{\mathcal {C}}$ which is equivalent to $\mathcal {C}'$ and thus we have that the minimum sum-rank distance of the dual code $\overline{\mathcal {C}}^\perp$ equals $n_t-\delta$. Since any nonzero element $\overline{\varvec{h}}$ in the right kernel of $\overline{{\varvec{L}}}$ is contained in $\overline{\mathcal {C}}^\perp$, we have that

  $$\begin{aligned} {{\,\textrm{wt}\,}}_{\Sigma R}(\overline{\varvec{h}})\ge n_t-\delta . \end{aligned}$$

  (A3)

  Therefore, (A2) and (A3) are satisfied only if

  $$\begin{aligned} {{\,\textrm{rk}\,}}_{q}(\overline{\varvec{h}}^{(i)})=n_t^{(i)}-{\delta }^{(i)},\quad \forall i=1,\dots ,\ell . \end{aligned}$$

  This proves the statement, since by Lemma 2 we have that $\ker _r({\varvec{L}})=\ker _r(\overline{{\varvec{L}}}\varvec{W}^{-1})$ and thus

  $$\begin{aligned} \overline{\varvec{h}}\in \ker _r(\overline{{\varvec{L}}}) \quad \Longleftrightarrow \quad \overline{\varvec{h}}\left( \varvec{W}^{-1}\right) ^\top \in \ker _r({\varvec{L}}). \end{aligned}$$

• Ad 3): Expand $\varvec{h}^{(i)}$ into an $m \times n_r^{(i)}$ matrix over $\mathbb F_{q}$, which has $\mathbb F_{q}$-rank $n_t^{(i)}-{\delta }^{(i)}$ by 2). Then, we can perform elementary column operations on this matrix to bring it into reduced column echelon form, where the $n_t^{(i)}-{\delta }^{(i)}$ nonzero columns are the leftmost ones. The matrix $\varvec{T}^{(i)}\in \mathbb F_{q}^{n_r^{(i)}\times n_r^{(i)}}$ is then chosen to be the matrix that, by multiplication from the right, performs the used sequence of elementary column operations. Note that the $n_t^{(i)}-{\delta }^{(i)}$ nonzero entries of $\varvec{h}^{(i)}\varvec{T}^{(i)}$ are linearly independent over $\mathbb F_{q}$.

• Ad 4): The matrices $\varvec{T}^{(i)} \in \mathbb F_{q}^{n_r^{(i)} \times n_r^{(i)}}$ are chosen such that the rightmost ${\gamma }^{(i)}$ positions of $\varvec{h}^{(i)} \varvec{T}^{(i)}$ are zero. Recall from 2), that if ${{\,\textrm{rk}\,}}_{q^m}(\overline{\varvec{Z}})=\gamma$, we have that for any nonzero element $\overline{\varvec{h}}=(\overline{\varvec{h}}^{(1)}\mid \dots \mid \overline{\varvec{h}}^{(\ell )})$ in the right kernel of $\overline{{\varvec{L}}}$ the ${\gamma }^{(i)}$ rightmost positions of each $\overline{\varvec{h}}^{(i)}$ are zero. Define the matrices $\varvec{D}^{(i)}=\left( \varvec{T}^{(i)-1}\right) ^\top$ for all $i=1,\dots ,\ell$. Then we have that

  $$\begin{aligned} \varvec{h}\varvec{T}\in \ker _r({\varvec{L}}\cdot {{\,\textrm{diag}\,}}(\varvec{D}^{(1)},\dots ,\varvec{D}^{(\ell )})). \end{aligned}$$

  We will now show that the column span of the matrices

  $$\begin{aligned} \left[ \overline{{\varvec{L}}}^{(i)}\right] _{\{n_t^{(i)}-{\delta }^{(i)}+1,\dots ,n_r^{(i)}\}} \quad \text {and}\quad \left[ {\varvec{L}}^{(i)}\varvec{D}^{(i)}\right] _{\{n_t^{(i)}-{\delta }^{(i)}+1,\dots ,n_r^{(i)}\}} \end{aligned}$$

  must be the same. Since ${{\,\textrm{rk}\,}}_{q^m}(\overline{\varvec{Z}})=\gamma$ implies ${{\,\textrm{rk}\,}}_{q^m}(\overline{\varvec{Z}}^{(i)})={\gamma }^{(i)}$ for all $i=1,\dots ,\ell$, the $s(n_t-\delta -k)$ last (lower) rows of ${\varvec{L}}^{(i)}\varvec{D}^{(i)}$ have $\mathbb F_{q^m}$-rank ${\gamma }^{(i)}$ for all $i=1,\dots ,\ell$. Now assume that the column span of the matrices above are not the same for some $i\in \{1,\dots ,\ell \}$. Then there exist $\mathbb F_{q^m}$-linearly independent columns for some index $j\in \{1,\dots ,n_t^{(i)}-{\delta }^{(i)}\}$, which contradicts that $\varvec{h}\varvec{T}$ (which has the ${\gamma }^{(i)}$ rightmost entries per block equal to zero) is in the right kernel of ${\varvec{L}}\cdot {{\,\textrm{diag}\,}}(\varvec{D}^{(1)},\dots ,\varvec{D}^{(\ell )})$. Therefore, we must have that the column space is the same. Since the ${\gamma }^{(i)}$ rightmost columns of each $\overline{{\varvec{L}}}^{(i)}$ correspond to the insertions, we have that the rightmost ${\gamma }^{(i)}$ columns of ${\varvec{L}}^{(i)}\varvec{D}^{(i)}$ also correspond to the insertions. Since ${\varvec{L}}$ is set up using the matrices $\varvec{U}^{(i)}$ for all $i=1,\dots ,\ell$, we have that the last (lower) ${\gamma }^{(i)}$ rows of $\left( \varvec{T}^{(i)}\right) ^{-1}\varvec{U}^{(i)}$ span the error space ${\mathcal {E}}^{(i)}$. This implies that the first $n_t^{(i)}-{\delta }^{(i)}$ rows of $\left( \varvec{T}^{(i)}\right) ^{-1}\varvec{U}^{(i)}$ form a basis of the noncorrupted space ${\mathcal {U}}^{(i)}\cap {\mathcal {V}}^{(i)}$, since by the definition of the operator channel we have that ${\mathcal {V}}^{(i)}\cap {\mathcal {E}}^{(i)}=\{\textbf{0}\}$ for all $i=1,\dots ,\ell$.

• Ad 5): By 4) the first $n_t^{(i)}-{\delta }^{(i)}$ rows of

  $$\begin{aligned} \hat{\varvec{U}}^{(i)}=\left( \varvec{T}^{(i)}\right) ^{-1}\varvec{U}^{(i)} = \begin{pmatrix} \hat{\varvec{\xi }}^{(i)\top }&\hat{\varvec{u}}_1^{(i)\top }&\dots&\hat{\varvec{u}}_s^{(i)\top } \end{pmatrix} \in \mathbb F_{q^m}^{n_r^{(i)}\times (s+1)}. \end{aligned}$$

  form a basis for the noncorrupted subspace ${\mathcal {U}}^{(i)}\cap {\mathcal {V}}^{(i)}$, i.e. we have that

  $$\begin{aligned} \hat{u}^{(i)}_{l,\mu }={f_l}(\hat{\xi }^{(i)})_{a_i}, \quad \forall \mu =1,\dots ,n_t^{(i)}-{\delta }^{(i)},\,\forall l=1,\dots ,s,\,\forall i=1,\dots ,\ell . \end{aligned}$$

  Therefore, the message polynomials $f_1,\dots ,f_s\in \mathbb F_{q^m}[x;\sigma ]_{<k}$ can be reconstructed via Lagrange interpolation (cf. Lemma 1).

Appendix B: Proofs from Sect. 4.4

1.1 B.1 Proof of Proposition 3

Proof

The proof proceed similar as the proof of [6, Theorem 4]. At the output of the operator channel we have

$$\begin{aligned} \varvec{\mathcal {U}}^\perp =\mathcal {H}_{N-n_t-\delta }(\varvec{\mathcal {V}}^\perp )\oplus \varvec{\mathcal {E}}_\gamma \end{aligned}$$

where ${{\,\mathrm{\dim _{\Sigma }}\,}}(\varvec{\mathcal {E}}_\gamma )=\gamma$. Let $\varvec{\gamma }$ and $\varvec{\delta }$ denote the partition of the insertions and deletions, respectively.

For all $i=1,\dots ,\ell$ we have that

$$\begin{aligned} {\mathcal {U}}^{(i)}=\left( ({\mathcal {U}}^{(i)})^\perp \right) ^\perp&=\left( \mathcal {H}_{N_i-n_t^{(i)}-{\delta }^{(i)}}\left( {({\mathcal {V}}^{(i)})}^\perp \right) \oplus {\mathcal {E}}^{(i)}_{{\gamma }^{(i)}}\right) ^\perp \\&=\left( \mathcal {H}_{N_i-n_t^{(i)}-{\delta }^{(i)}}\left( {({\mathcal {V}}^{(i)})}^\perp \right) \right) ^\perp \cap ({\mathcal {E}}^{(i)}_{{\gamma }^{(i)}})^\perp . \end{aligned}$$

By [6, Lemma 14] we have that

$$\begin{aligned} \left( \mathcal {H}_{N_i-n_t^{(i)}-{\delta }^{(i)}}\left( {({\mathcal {V}}^{(i)})}^\perp \right) \right) ^\perp ={\mathcal {V}}^{(i)}\oplus {\mathcal {E}}^{(i)}_{{\delta }^{(i)}} \end{aligned}$$

(B4)

where $\dim ({\mathcal {E}}^{(i)}_{{\delta }^{(i)}})={\delta }^{(i)}$ and therefore

$$\begin{aligned} {\mathcal {U}}^{(i)}&=\left( {\mathcal {V}}^{(i)}\oplus {\mathcal {E}}^{(i)}_{{\delta }^{(i)}}\right) \cap ({\mathcal {E}}^{(i)}_{{\gamma }^{(i)}})^\perp \nonumber \\&=\left( {\mathcal {V}}^{(i)}\cap ({\mathcal {E}}^{(i)}_{{\gamma }^{(i)}})^\perp \right) \oplus {\mathcal {E}}^{(i)}_{{\delta }^{(i)}} \end{aligned}$$

(B5)

for all $i=1,\dots ,\ell$ since ${\mathcal {E}}^{(i)}_{{\delta }^{(i)}}\cap ({\mathcal {E}}^{(i)}_{{\gamma }^{(i)}})^\perp ={\mathcal {E}}^{(i)}_{{\delta }^{(i)}}$. From the operator channel we have that

$$\begin{aligned} ({\mathcal {V}}^{(i)})^\perp \cap {\mathcal {E}}^{(i)}_{{\gamma }^{(i)}}=0 \quad \Longrightarrow \quad ({\mathcal {V}}^{(i)})^\perp \subseteq ({\mathcal {E}}^{(i)}_{{\gamma }^{(i)}})^\perp \end{aligned}$$

since $({\mathcal {V}}^{(i)})^\perp \oplus {\mathcal {V}}^{(i)}=\mathbb F_{q}^{N_i}$ and $\dim (({\mathcal {V}}^{(i)})^\perp )\le \dim (({\mathcal {E}}^{(i)}_{{\gamma }^{(i)}})^\perp )$. From (B4) we get ${\mathcal {V}}^{(i)}\cap {\mathcal {E}}^{(i)}_{{\delta }^{(i)}}=0$ and therefore

$$\begin{aligned} {\mathcal {E}}^{(i)}_{{\delta }^{(i)}}\subseteq ({\mathcal {V}}^{(i)})^\perp \subseteq ({\mathcal {E}}^{(i)}_{{\gamma }^{(i)}})^\perp \end{aligned}$$

for all $i=1,\dots ,\ell$. We have that $\dim (({\mathcal {E}}^{(i)}_{{\gamma }^{(i)}})^\perp )=N_i-{\gamma }^{(i)}$ which therefore deletes exactly ${\gamma }^{(i)}$ dimensions from ${\mathcal {V}}^{(i)}$ since $\dim ({\mathcal {U}}^{(i)})=N_i-\dim (({\mathcal {U}}^{(i)})^\perp )=\dim ({\mathcal {V}}^{(i)})-{\gamma }^{(i)}+{\delta }^{(i)}$ for all $i=1,\dots ,\ell$. Hence, we can write (B5) as

$$\begin{aligned} {\mathcal {U}}^{(i)}=\mathcal {H}_{n_t^{(i)}-{\gamma }^{(i)}}({\mathcal {V}}^{(i)})\oplus {\mathcal {E}}^{(i)}_{{\delta }^{(i)}},\quad \forall i=1,\dots ,\ell \end{aligned}$$

which proves the statement.

Appendix C: Efficient implementation of the multishot operator channel

For implementing the multishot operator channel we need to draw tuples that contain subspaces and have a particular constant sum-dimension uniformly at random from the set of all such tuples efficiently. In the following we assume that $\ell \mid N$. In order to accomplish this subspace-tuple drawing algorithm efficiently we adapted the enumerative coding approach from [36] for the case where the dimension of the ambient space as well as the dimensions of the transmitted spaces are equal in each shot, i.e. for $\overline{\varvec{N}}=(\overline{N}, \overline{N},\dots ,\overline{N})$ with $\overline{N}:=\frac{N}{\ell }$ and $\overline{\varvec{n}}_{t}= (\overline{n}_t, \overline{n}_t, \dots , \overline{n}_t)$.

Suppose that we want to draw tuples with fixed sum-dimension $\gamma$ uniformly at random from the set

$$\begin{aligned} \{\varvec{\mathcal {E}}\in \mathcal {P}_q(\overline{\varvec{N}}) : {{\,\mathrm{\dim _{\Sigma }}\,}}(\varvec{\mathcal {E}}) = \gamma \}. \end{aligned}$$

Define the set $\mathcal {T}_{\overline{N}, \gamma , \ell } :=\left\{ \varvec{\gamma }\in \{0,\dots , \min \{\overline{N}, \gamma \}\}^\ell : \sum _{i=1}^{\ell } {\gamma }^{(i)} = \gamma \right\}$.

Then the number $\textsf {N}_{\Sigma S}(\overline{N},t,\ell )$ of tuples in $\mathcal {P}_q(\overline{\varvec{N}})$ with sum-dimension $\gamma$ is given by

$$\begin{aligned} \textsf {N}_{\Sigma S}(\overline{N},\gamma ,\ell ) = \sum _{\varvec{\gamma }\in \mathcal {T}_{\overline{N}, \gamma , \ell }} \prod _{i=1}^{\ell } \left[ \genfrac{}{}{0.0pt}{}{\overline{N}}{{\gamma }^{(i)}}\right] _{q}. \end{aligned}$$

Since the cardinality of the set $\mathcal {T}_{\overline{N}, \gamma , \ell }$ and thus the number of terms in the sum above is large for most parameters (see [36, Sect. III]) we may compute $\textsf {N}_{\Sigma S}(\overline{N},\gamma ,\ell )$ recursively as

$$\begin{aligned} \textsf {N}_{\Sigma S}(\overline{N},\gamma ,\ell ) = \left\{ \begin{array}{ll} \displaystyle \left[ \genfrac{}{}{0.0pt}{}{\overline{N}}{\gamma }\right] _{q}, &{} \text {if } \ell =1, \\ \displaystyle \sum _{\gamma '=\max \{0, \gamma -(\ell -1)\overline{N}\}}^{\min \{\overline{N},\gamma \}} \left[ \genfrac{}{}{0.0pt}{}{\overline{N}}{\gamma '}\right] _{q} \textsf {N}_{\Sigma S}(\overline{N},\gamma - \gamma ',\ell -1) &{} \text {if } \ell > 1, \end{array} \right. \end{aligned}$$

with $\textsf {N}_{\Sigma S}(\overline{N},\gamma ,\ell ) = 0$ for $\gamma > \overline{N}\ell$.

For $\ell =1$ the number coincides with the number of $\gamma$-dimensional vector spaces of $\mathbb F_{q}^{\overline{N}}$ which is $\left[ \genfrac{}{}{0.0pt}{}{\overline{N}}{\gamma }\right] _{q}$. The recursion for $\ell > 1$ follows by summing over the product of all possible $\gamma '$-dimensional subspaces in the first block and the number of remaining $(\ell -1)$-tuples of sum-dimension $\gamma -\gamma '$.

As in [36] this recursion provides an efficient method to draw dimension partitions of $\ell$-tuples with fixed sum-dimension that are chosen uniformly at random from $\mathcal {P}_q(\overline{\varvec{N}})$ which is given in Algorithm 4.

Algorithm 4

DrawDimensionPartition

Equipped with the efficient method to draw dimension partitions for sampling $\ell$-tuples with constant sum-dimension from $\mathcal {P}_q(\overline{\varvec{N}})$ uniformly at random (see Algorithm 4) we can implement the multishot operator channel (see (3))

$$\begin{aligned} \varvec{\mathcal {U}}=\mathcal {H}_{n_t-\delta }(\varvec{\mathcal {V}}) \oplus \varvec{\mathcal {E}}\end{aligned}$$

with input alphabet $\mathcal {G}_q(\overline{\varvec{N}}, \overline{\varvec{n}}_{t})$ and an overall number of $\gamma$ insertions and $\delta$ deletions as follows.

First, we require the partition $\varvec{\delta }$ of the $\delta$ deletions. For each shot i there are $\left[ \genfrac{}{}{0.0pt}{}{\overline{n}_t}{{\delta }^{(i)}}\right] _{q}$ ways to chose an ${\delta }^{(i)}$-dimensional subspace of the transmitted space ${\mathcal {V}}^{(i)} \in \mathcal {G}_q(\overline{N}, \overline{n}_t)$. Hence, we can use the routine ${ \textsc {DrawDimensionPartition}}(q, \overline{n}_t, \delta , \ell )$ in Algorithm 4 to draw the dimension partition $\varvec{\delta }$ for the deletions operator $\mathcal {H}_{n_t-\delta }(\varvec{\mathcal {V}})$.

Recall that for the error space $\varvec{\mathcal {E}}$ we have the restriction that $\varvec{\mathcal {V}}\cap \varvec{\mathcal {E}}= \textbf{0}$. Hence, we have to sample $\varvec{\mathcal {E}}$ uniformly at random from $\mathcal {G}_q(\overline{\varvec{N}}-\overline{\varvec{n}}_{t}, \varvec{\gamma })$( rather than from $\mathcal {G}_q(\overline{\varvec{N}}, \varvec{\gamma })$) and thus have to call ${ \textsc {DrawDimensionPartition}}(q, \overline{N}-\overline{n}_t, \gamma , \ell )$ in order to get the corresponding insertion partition $\varvec{\gamma }$.

Once the partitions of insertions $\varvec{\gamma }$ and deletions $\varvec{\delta }$ are obtained we can apply the single-shot operator channel from [18] to each shot with the corresponding number of ${\gamma }^{(i)}$ insertions and ${\delta }^{(i)}$ deletions. The whole procedure is provided in Algorithm 5.

Algorithm 5

Random Instance of Multishot Operator Channel