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Two pointsets in $ \mathrm{PG}(2,q^n) $ and the associated codes Adv. Math. Commun. (IF 0.9) Pub Date : 2022-02-16 Vito Napolitano, Olga Polverino, Paolo Santonastaso, Ferdinando Zullo
In this paper we consider two pointsets in \begin{document}$ \mathrm{PG}(2,q^n) $\end{document} arising from a linear set \begin{document}$ L $\end{document} of rank \begin{document}$ n $\end{document} contained in a line of \begin{document}$ \mathrm{PG}(2,q^n) $\end{document}: the first one is a linear blocking set of Rédei type, the second one extends the construction of translation KM-arcs. We point
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Correcting adversarial errors with generalized regenerating codes Adv. Math. Commun. (IF 0.9) Pub Date : 2022-02-16 Negin Karimi, Ahmad Yousefian Darani, Marcus Greferath
Traditional regenerating codes are efficient tools to optimize both storage and repair bandwidth in storing data across a distributed storage system, particularly in comparison to erasure codes and data replication. In traditional regenerating codes, the collection of any \begin{document}$ k $\end{document} nodes can reconstruct all stored information and is called the reconstruction set, \begin{document}$
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Constructions of optimal low hit zone frequency hopping sequence sets with large family size Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-12 Xiujie Zhang, Xianhua Niu, Xin Tan
Frequency hopping sequences with low hit zone is significant for application in quasi synchronous multiple-access systems. In this paper, we obtained two constructions of optimal frequency hopping sequence sets with low hit zone based on interleaving techniques. The presented low hit zone frequency hopping sequence sets are with new and flexible parameters and large family size which can meet the needs
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New quantum codes from metacirculant graphs via self-dual additive $\mathbb{F}_4$-codes Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-12 Padmapani Seneviratne, Martianus Frederic Ezerman
We use symplectic self-dual additive codes over \begin{document}$ \mathbb{F}_4 $\end{document} obtained from metacirculant graphs to construct, for the first time, \begin{document}$ \left[\kern-0.15em\left[ {\ell, 0, d} \right]\kern-0.15em\right] $\end{document} qubit codes with parameters \begin{document}$ (\ell,d) \in \{(78, 20), (90, 21), (91, 22), (93,21),(96,22)\} $\end{document}. Secondary constructions
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An algorithmic approach to entanglement-assisted quantum error-correcting codes from the Hermitian curve Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-12 René B. Christensen, Carlos Munuera, Francisco R. F. Pereira, Diego Ruano
We study entanglement-assisted quantum error-correcting codes (EAQECCs) arising from classical one-point algebraic geometry codes from the Hermitian curve with respect to the Hermitian inner product. Their only unknown parameter is \begin{document}$ c $\end{document}, the number of required maximally entangled quantum states since the Hermitian dual of an AG code is unknown. In this article, we present
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Upper bounds on the length function for covering codes with covering radius $ R $ and codimension $ tR+1 $ Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-12 Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco
The length function \begin{document}$ \ell_q(r,R) $\end{document} is the smallest length of a \begin{document}$ q $\end{document}-ary linear code with codimension (redundancy) \begin{document}$ r $\end{document} and covering radius \begin{document}$ R $\end{document}. In this work, new upper bounds on \begin{document}$ \ell_q(tR+1,R) $\end{document} are obtained in the following forms: \begin{document}$
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Rectangular, range, and restricted AONTs: Three generalizations of all-or-nothing transforms Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-11 Navid Nasr Esfahani, Douglas R. Stinson
All-or-nothing transforms (AONTs) were originally defined by Rivest [14] as bijections from \begin{document}$ s $\end{document} input blocks to \begin{document}$ s $\end{document} output blocks such that no information can be obtained about any input block in the absence of any output block. Numerous generalizations and extensions of all-or-nothing transforms have been discussed in recent years, many
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Construction for both self-dual codes and LCD codes Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-11 Keita Ishizuka, Ken Saito
From a given [n, k] code C, we give a method for constructing many [n, k] codes C' such that the hull dimensions of C and C' are identical. This method can be applied to constructions of both self-dual codes and linear complementary dual codes (LCD codes for short). Using the method, we construct 661 new inequivalent extremal doubly even [56, 28, 12] codes. Furthermore, constructing LCD codes by the
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The interplay of different metrics for the construction of constant dimension codes Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-11 Sascha Kurz
A basic problem for constant dimension codes is to determine the maximum possible size \begin{document}$ A_q(n,d;k) $\end{document} of a set of \begin{document}$ k $\end{document}-dimensional subspaces in \begin{document}$ \mathbb{F}_q^n $\end{document}, called codewords, such that the subspace distance satisfies \begin{document}$ d_S(U,W): = 2k-2\dim(U\cap W)\ge d $\end{document} for all pairs of
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Following Forrelation – quantum algorithms in exploring Boolean functions' spectra Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-11 Suman Dutta, Subhamoy Maitra, Chandra Sekhar Mukherjee
Here we revisit the quantum algorithms for obtaining Forrelation [Aaronson et al., 2015] values to evaluate some of the well-known cryptographically significant spectra of Boolean functions, namely the Walsh spectrum, the cross-correlation spectrum, and the autocorrelation spectrum. We introduce the existing 2-fold Forrelation formulation with bent duality-based promise problems as desirable instantiations
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Optimal data placements for triple replication in distributed storage systems Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Ruijing Liu,Junling Zhou
In distributed storage systems it is essential to store files (data) in replication to ensure reliability and fault-tolerance. Given a set \begin{document}$ V $\end{document} of \begin{document}$ v $\end{document} servers along with \begin{document}$ b $\end{document} files, each file is replicated (placed) on exactly \begin{document}$ k $\end{document} servers and thus a file can be represented by
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Differential faultt attack on DEFAULT Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Chandan Dey,Sumit Kumar Pandey,Tapabrata Roy,Santanu Sarkar
Block cipher DEFAULT has been proposed as a differential fault analysis immune cipher at Asiacrypt 2021. In this paper, we consider the initial version of DEFAULT with no permutation involved in the last round and show that one can find the key in this version with complexity \begin{document}$ 2^{16} $\end{document} by injecting 112 faults. However, our idea does not work for the modified version of
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Binary self-dual and LCD codes from generator matrices constructed from two group ring elements by a heuristic search scheme Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Steven Dougherty,Adrian Korban,Serap Șahinkaya,Deniz Ustun
We present a generator matrix of the form \begin{document}$ [ \sigma(v_1) \ | \ \sigma(v_2)] $\end{document}, where \begin{document}$ v_1 \in RG $\end{document} and \begin{document}$ v_2\in RH $\end{document}, for finite groups \begin{document}$ G $\end{document} and \begin{document}$ H $\end{document} of order \begin{document}$ n $\end{document} for constructing self-dual codes and linear complementary
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Low-density and high-density asymmetric CT-burst correcting integer codes Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Nabin Kumar Pokhrel,Pankaj Kumar Das
Two classes of integer codes correcting low-density and high-density asymmetric CT-bursts within a \begin{document}$ b $\end{document}-bit byte have been presented here. Unlike the previously studied integer codes correcting burst errors, here we study such codes with weight constraint on the burst. The proposed codes are compared with similar integer codes in terms of various properties, viz. memory
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Convolutional codes over finite chain rings, MDP codes and their characterization Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Gianira N. Alfarano,Anina Gruica,Julia Lieb,Joachim Rosenthal
In this paper, we develop the theory of convolutional codes over finite commutative chain rings. In particular, we focus on maximum distance profile (MDP) convolutional codes and we provide a characterization of these codes, generalizing the one known for fields. Moreover, we relate (reverse) MDP convolutional codes over a finite chain ring with (reverse) MDP convolutional codes over its residue field
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Quantum-safe identity-based broadcast encryption with provable security from multivariate cryptography Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Ramprasad Sarkar,Mriganka Mandal,Sourav Mukhopadhyay
Identity-Based Broadcast Encryption (\begin{document}$\textsf{IBBE}$\end{document}) is a novel concept that can efficiently and securely transmit confidential content to a group of authorized users without the traditional Public-Key Infrastructure (\begin{document}$\textsf{PKI}$\end{document}). After carefully exploring these areas, we have observed that none of the existing works have adopted the
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Constructions of optimal multiply constant-weight codes MCWC$ (3,n_1;1,n_2;1,n_3;8)s $ Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Liying Pan,R. Julian R. Abel,Jinhua Wang
A binary code \begin{document}$ {\mathcal{C}} $\end{document} of length \begin{document}$ n = \sum_{i = 1}^{m}n_i $\end{document} and minimum distance \begin{document}$ d $\end{document} is said to be of multiply constant-weight and denoted by MCWC\begin{document}$ (w_1, n_1 $\end{document}; \begin{document}$ w_2, n_2 $\end{document}; \begin{document}$ \ldots $\end{document}; \begin{document}$ w_m
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Some constructions of (almost) optimally extendable linear codes Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Xiaoshan Quan,Qin Yue,Liqin Hu
Let \begin{document}$ G $\end{document} be a generator matrix of a linear code \begin{document}$ \mathcal C $\end{document} and \begin{document}$ [G: I_k] $\end{document} be a generator matrix of its extendable linear code \begin{document}$ \mathcal {C}' $\end{document}, we call \begin{document}$ \mathcal C $\end{document} is optimally (almost optimally) extendable if \begin{document}$ d(\mathcal C^\perp)
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A novel genetic search scheme based on nature-inspired evolutionary algorithms for binary self-dual codes Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Adrian Korban,Serap Şahinkaya,Deniz Ustun
In this paper, a genetic algorithm, one of the evolutionary algorithm optimization methods, is used for the first time for the problem of computing extremal binary self-dual codes. We present a comparison of the computational times between the genetic algorithm and a linear search for different size search spaces and show that the genetic algorithm is capable of computing binary self-dual codes significantly
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Construction of three classes of strictly optimal frequency-hopping sequence sets Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Xianhong Xie,Yi Ouyang,Honggang Hu,Ming Mao
In this paper, we construct three classes of strictly optimal frequency-hopping sequence (FHS) sets with respect to partial Hamming correlation and family size. The first and second classes are based on the trace map, the third class is based on a generic construction.
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On the hardness of the Lee syndrome decoding problem Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Violetta Weger,Karan Khathuria,Anna-Lena Horlemann,Massimo Battaglioni,Paolo Santini,Edoardo Persichetti
In this paper we study the hardness of the syndrome decoding problem over finite rings endowed with the Lee metric. We first prove that the decisional version of the problem is NP-complete, by a reduction from the \begin{document}$ 3 $\end{document}-dimensional matching problem. Then, we study the complexity of solving the problem, by translating the best known solvers in the Hamming metric over finite
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Public key cryptography based on twisted dihedral group algebras Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Javier de la Cruz,Ricardo Villanueva-Polanco
In this paper, we propose to use a twisted dihedral group algebra for public-key cryptography. For this, we introduce a new \begin{document}$ 2 $\end{document}-cocycle \begin{document}$ \alpha_{\lambda} $\end{document} to twist the dihedral group algebra. Using the ambient space \begin{document}$ \mathbb{F}^{\alpha_{\lambda}} D_{2n} $\end{document}, we then introduce a key exchange protocol and present
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New type I binary $[72, 36, 12]$ self-dual codes from $M_6(\mathbb{F}_2)G$ - Group matrix rings by a hybrid search technique based on a neighbourhood-virus optimisation algorithm Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Adrian Korban,Serap Sahinkaya,Deniz Ustun
In this paper, a new search technique based on a virus optimisation algorithm is proposed for calculating the neighbours of binary self-dual codes. The aim of this new technique is to calculate neighbours of self-dual codes without reducing the search field in the search process (this technique is known in the literature due to the computational time constraint) but still obtaining results in a reasonable
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Three weight ternary linear codes from non-weakly regular bent functions Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Rumi Melih Pelen
This paper constructs several classes of three-weight ternary linear codes from non-weakly regular dual-bent functions based on a generic construction method. Instead of the whole space, we use the subspaces \begin{document}$ B_{\pm}(f) $\end{document} associated with a ternary non-weakly regular dual-bent function \begin{document}$ f $\end{document}. Unusually, we use the pre-image sets of the dual
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Reconstructing points of superelliptic curves over a prime finite field Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Jaime Gutierrez
Let \begin{document}$ p $\end{document} be a prime and \begin{document}$ \mathbb{F}_p $\end{document} the finite field with \begin{document}$ p $\end{document} elements. We show how, when given an superelliptic curve \begin{document}$ Y^n+f(X) \in \mathbb{F}_p[X,Y] $\end{document} and an approximation to \begin{document}$ (v_0,v_1) \in \mathbb{F}_p^2 $\end{document} such that \begin{document}$ v_1^n
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On Polynomial Modular Number Systems over $ \mathbb{Z}/{p}\mathbb{Z} $ Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Jean-Claude Bajard,Jérémy Marrez,Thomas Plantard,Pascal Véron
Since their introduction in 2004, Polynomial Modular Number Systems (PMNS) have become a very interesting tool for implementing cryptosystems relying on modular arithmetic in a secure and efficient way. However, while their implementation is simple, their parameterization is not trivial and relies on a suitable choice of the polynomial on which the PMNS operates. The initial proposals were based on
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Three constructions of Golay complementary array sets Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Bingsheng Shen,Yang Yang,Ruibin Ren
Recently, two-dimensional (2-D) arrays with good correlation have been used in MIMO systems. In this paper, we investigate new 2-D Golay complementary array sets (GCASs), whose 2-D aperiodic auto-correlation sums are zero for all 2-D nonzero shifts. Firstly, based on the 2-D generalized Boolean functions, we propose a direct construction of new GCASs. Secondly, using horizontal concatenation, we give
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On $ \mathbb{Z}_4\mathbb{Z}_4[u^3] $-additive constacyclic codes Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Om Prakash,Shikha Yadav,Habibul Islam,Patrick Solé
Let \begin{document}$ \mathbb{Z}_4 $\end{document} be the ring of integers modulo \begin{document}$ 4 $\end{document}. This paper studies mixed alphabets \begin{document}$ \mathbb{Z}_4\mathbb{Z}_4[u^3] $\end{document}-additive cyclic and \begin{document}$ \lambda $\end{document}-constacyclic codes for units \begin{document}$ \lambda = 1+2u^2,3+2u^2 $\end{document}. First, we obtain the generator polynomials
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Constructions of optimal rank-metric codes from automorphisms of rational function fields Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Rakhi Pratihar,Tovohery Hajatiana Randrianarisoa
We define a class of automorphisms of rational function fields of finite characteristic and employ these to construct different types of optimal linear rank-metric codes. The first construction is of generalized Gabidulin codes over rational function fields. Reducing these codes over finite fields, we obtain maximum rank distance (MRD) codes which are not equivalent to generalized twisted Gabidulin
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Binary self-dual codes of various lengths with new weight enumerators from a modified bordered construction and neighbours Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Joe Gildea,Adrian Korban,Adam M. Roberts,Alexander Tylyshchak
In this work, we define a modification of a bordered construction for self-dual codes which utilises \begin{document}$ \lambda $\end{document}-circulant matrices. We provide the necessary conditions for the construction to produce self-dual codes over finite commutative Frobenius rings of characteristic 2. Using the modified construction together with the neighbour construction, we construct many binary
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$\mathbb{F}_{p^{m}}\mathbb{F}_{p^{m}}{[u^2]}$-additive skew cyclic codes of length $2p^s $ Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Roghayeh Mohammadi Hesari,Mahboubeh Hosseinabadi,Rashid Rezaei,Karim Samei
In this paper, we first study the skew cyclic codes of length \begin{document}$ p^s $\end{document} over \begin{document}$ R_3 = \mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}, $\end{document} where \begin{document}$ p $\end{document} is a prime number and \begin{document}$ u^3 = 0. $\end{document} Then we characterize the algebraic structure of \begin{document}$ \mathbb{F}_{p^{m}}\mathbb{F}_{p^{m}}[u^2]
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On some codes from rank 3 primitive actions of the simple Chevalley group $ G_2(q) $ Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Tung Le,Bernardo G. Rodrigues
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On the polycyclic codes over $ \mathbb{F}_q+u\mathbb{F}_q $ Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Wei Qi
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Self-orthogonal codes from equitable partitions of distance-regular graphs Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Dean Crnković,Sanja Rukavina,Andrea Švob
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Parameterization of Boolean functions by vectorial functions and associated constructions Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Claude Carlet
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Finding small roots for bivariate polynomials over the ring of integers Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Jiseung Kim,Changmin Lee
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Infinite families of 2-designs from a class of affine-invariant codes Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Yan Liu,Xiwang Cao
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On the generalised rank weights of quasi-cyclic codes Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Enhui Lim,Frédérique Oggier
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The zero-error capacity of binary channels with 2-memories Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Guofen Zhang,Ping Li,Jianfeng Hou,Bo Bai
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Balanced ($\mathbb{Z} _{2u}\times \mathbb{Z}_{38v}$, {3, 4, 5}, 1) difference packings and related codes Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Hengming Zhao,Rongcun Qin,Dianhua Wu
Let \begin{document}$ m $\end{document}, \begin{document}$ n $\end{document} be positive integers, and \begin{document}$ K $\end{document} a set of positive integers with size greater than 2. An \begin{document}$ (m,n,K,1) $\end{document} optical orthogonal signature pattern code, \begin{document}$ (m,n,K,1) $\end{document}-OOSPC, was introduced by Kwong and Yang for 2-D image transmission in multicore-fiber
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Computing square roots faster than the Tonelli-Shanks/Bernstein algorithm Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Palash Sarkar
Let \begin{document}$ p $\end{document} be a prime such that \begin{document}$ p = 1+2^nm $\end{document}, where \begin{document}$ n\geq 1 $\end{document} and \begin{document}$ m $\end{document} is odd. Given a square \begin{document}$ u $\end{document} in \begin{document}$ \mathbb{Z}_p $\end{document} and a non-square \begin{document}$ z $\end{document} in \begin{document}$ \mathbb{Z}_p $\end{document}
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Additive polycyclic codes over $ \mathbb{F}_{4} $ induced by binary vectors and some optimal codes Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Arezoo Soufi Karbaski,Taher Abualrub,Nuh Aydin,Peihan Liu
In this paper, we study the structure and properties of additive right and left polycyclic codes induced by a binary vector \begin{document}$ a $\end{document} in \begin{document}$ \mathbb{F}_{2}^{n}. $\end{document} We find the generator polynomials and the cardinality of these codes. We also study different duals for these codes. In particular, we show that if \begin{document}$ C $\end{document}
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Two classes of cyclic extended double-error-correcting Goppa codes Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Yanyan Gao,Qin Yue,Xinmei Huang,Yun Yang
Let \begin{document}$ \Bbb F_{2^m} $\end{document} be a finite extension of the field \begin{document}$ \Bbb F_2 $\end{document} and \begin{document}$ g(x) = x^2+\alpha x+1 $\end{document} a quadratic polynomial over \begin{document}$ \Bbb F_{2^m} $\end{document}. In this paper, two classes of cyclic extended double-error-correcting Goppa codes are proposed. We obtain the following two classes of Goppa
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Galois LCD codes over rings Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Zihui Liu
We define the generalized Galois inner product for codes over Frobenius rings, and then present the Galois linear complementary dual (LCD) codes over such rings which generalize the Galois LCD codes over finite fields defined in a recent reference. We describe the judging criterions for a generalized Galois LCD code over a Frobenius ring by introducing the rank of a matrix over such a ring. We also
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An algorithm for solving over-determined multivariate quadratic systems over finite fields Adv. Math. Commun. (IF 0.9) Pub Date : 2022-01-01 Lih-Chung Wang,Tzer-jen Wei,Jian-Ming Shih,Yuh-Hua Hu,Chih-Cheng Hsieh
An algorithm for solving over-determined multivariate quadratic systems over finite fields is given. It is more efficient than other known algorithms over finite fields of relatively large size in terms of both performance and memory comsumption. It is also simpler for computer programming. The complexity estimate of our algorithm can be used to estimate the security level of multivariate cryptosystems
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Classification of $ \mathbf{(3 \!\mod 5)} $ arcs in $ \mathbf{ \operatorname{PG}(3,5)} $ Adv. Math. Commun. (IF 0.9) Pub Date : 2021-12-27 Sascha Kurz, Ivan Landjev, Assia Rousseva
The proof of the non-existence of Griesmer \begin{document}$ [104, 4, 82]_5 $\end{document}-codes is just one of many examples where extendability results are used. In a series of papers Landjev and Rousseva have introduced the concept of \begin{document}$ (t\mod q) $\end{document}-arcs as a general framework for extendability results for codes and arcs. Here we complete the known partial classification
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Constructions of asymptotically optimal codebooks with respect to Welch bound and Levenshtein bound Adv. Math. Commun. (IF 0.9) Pub Date : 2021-12-27 Gang Wang, Deng-Ming Xu, Fang-Wei Fu
Codebooks with small maximum cross-correlation amplitudes are used to distinguish the signals from different users in code division multiple access communication systems. In this paper, several classes of codebooks are introduced, whose maximum cross-correlation amplitudes asymptotically achieve the corresponding Welch bound and Levenshtein bound. Specially, a class of optimal codebooks with respect
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Niederreiter cryptosystems using quasi-cyclic codes that resist quantum Fourier sampling Adv. Math. Commun. (IF 0.9) Pub Date : 2021-12-17 Upendra Kapshikar, Ayan Mahalanobis
McEliece and Niederreiter cryptosystems are robust and versatile cryptosystems. These cryptosystems work with many linear error-correcting codes. They are popular these days because they can be quantum-secure. In this paper, we study the Niederreiter cryptosystem using non-binary quasi-cyclic codes. We prove, if these quasi-cyclic codes satisfy certain conditions, the corresponding Niederreiter cryptosystem
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Aperiodic/periodic complementary sequence pairs over quaternions Adv. Math. Commun. (IF 0.9) Pub Date : 2021-12-17 Zhen Li, Cuiling Fan, Wei Su, Yanfeng Qi
Aperodic (or called Golay)/Periodic complementary pairs (GCPs/ PCPs) are pairs of sequences whose aperiodic/periodic autocorrelation sums are zero everywhere, except at the zero shift. In this paper, we introduce GCPs/PCPs over the quaternion group \begin{document}$ Q_8 $\end{document}, which is a generalization of quaternary GCPs/PCPs. Some basic properties of autocorrelations of \begin{document}$
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On the exponents of APN power functions and Sidon sets, sum-free sets, and Dickson polynomials Adv. Math. Commun. (IF 0.9) Pub Date : 2021-12-17 Claude Carlet, Stjepan Picek
We derive necessary conditions related to the notions, in additive combinatorics, of Sidon sets and sum-free sets, on those exponents \begin{document}$ d\in {\mathbb Z}/(2^n-1){\mathbb Z} $\end{document}, which are such that \begin{document}$ F(x) = x^d $\end{document} is an APN function over \begin{document}$ {\mathbb F}_{2^n} $\end{document} (which is an important cryptographic property). We study
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Rotational analysis of ChaCha permutation Adv. Math. Commun. (IF 0.9) Pub Date : 2021-12-09 Stefano Barbero, Emanuele Bellini, Rusydi H. Makarim
We show that the underlying permutation of ChaCha20 stream cipher does not behave as a random permutation for up to 17 rounds with respect to rotational cryptanalysis. In particular, we derive a lower and an upper bound for the rotational probability through ChaCha quarter round, we show how to extend the bound to a full round and then to the full permutation. The obtained bounds show that the probability
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Reversible $ G $-codes over the ring $ {\mathcal{F}}_{j,k} $ with applications to DNA codes Adv. Math. Commun. (IF 0.9) Pub Date : 2021-12-09 Yasemin Cengellenmis, Abdullah Dertli, Steven T. Dougherty, Adrian Korban, Serap Şahinkaya, Deniz Ustun
In this paper, we show that one can construct a \begin{document}$ G $\end{document}-code from group rings that is reversible. Specifically, we show that given a group with a subgroup of order half the order of the ambient group with an element that is its own inverse outside the subgroup, we can give an ordering of the group elements for which \begin{document}$ G $\end{document}-codes are reversible
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New nonexistence results on perfect permutation codes under the hamming metric Adv. Math. Commun. (IF 0.9) Pub Date : 2021-12-09 Xiang Wang, Wenjuan Yin
Permutation codes under the Hamming metric are interesting topics due to their applications in power line communications and block ciphers. In this paper, we study perfect permutation codes in \begin{document}$ S_n $\end{document}, the set of all permutations on \begin{document}$ n $\end{document} elements, under the Hamming metric. We prove the nonexistence of perfect \begin{document}$ t $\end{do
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Generic constructions of MDS Euclidean self-dual codes via GRS codes Adv. Math. Commun. (IF 0.9) Pub Date : 2021-12-09 Ziteng Huang, Weijun Fang, Fang-Wei Fu, Fengting Li
Recently, the construction of new MDS Euclidean self-dual codes has been widely investigated. In this paper, for square \begin{document}$ q $\end{document}, we utilize generalized Reed-Solomon (GRS) codes and their extended codes to provide four generic families of \begin{document}$ q $\end{document}-ary MDS Euclidean self-dual codes of lengths in the form \begin{document}$ s\frac{q-1}{a}+t\frac{q-1}{b}
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Differential spectra of a class of power permutations with Niho exponents Adv. Math. Commun. (IF 0.9) Pub Date : 2021-12-09 Zhen Li, Haode Yan
Let \begin{document}$ m\geq3 $\end{document} be a positive integer and \begin{document}$ n = 2m $\end{document}. Let \begin{document}$ f(x) = x^{2^m+3} $\end{document} be a power permutation over \begin{document}$ {\mathrm {GF}}(2^n) $\end{document}, which is a monomial with a Niho exponent. In this paper, the differential spectrum of \begin{document}$ f $\end{document} is investigated. It is shown
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Minimal codewords arising from the incidence of points and hyperplanes in projective spaces Adv. Math. Commun. (IF 0.9) Pub Date : 2021-12-09 Daniele Bartoli, Lins Denaux
Over the past few years, the codes \begin{document}$ {\mathcal{C}}_{n-1}(n,q) $\end{document} arising from the incidence of points and hyperplanes in the projective space \begin{document}$ {\rm{PG}}(n,q) $\end{document} attracted a lot of attention. In particular, small weight codewords of \begin{document}$ {\mathcal{C}}_{n-1}(n,q) $\end{document} are a topic of investigation. The main result of this
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A proof of the conjectured run time of the Hafner-McCurley class group algorithm Adv. Math. Commun. (IF 0.9) Pub Date : 2021-11-22 Jean-François Biasse, Muhammed Rashad Erukulangara
We present a proof under a generalization of the Riemann Hypothesis that the class group algorithm of Hafner and McCurley runs in expected time \begin{document}$ e^{\left(3/\sqrt{8}+o(1)\right)\sqrt{\log d\log\log d}} $\end{document} where \begin{document}$ -d $\end{document} is the discriminant of the input imaginary quadratic order. In the original paper, an expected run time of \begin{document}$
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Security analysis of public key encryption with filtered equality test Adv. Math. Commun. (IF 0.9) Pub Date : 2021-11-18 Yu-Chi Chen
Public key encryption with equality test can provide a very simple add-on in which any one can directly perform testing over a pair of ciphertexts to check whether the underlying messages are identical or not without decryption. To restrict the such test power for different scenarios, that of delegated equality test is introduced to allow only the authenticated party to perform the test. In this paper
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Splitting authentication codes with perfect secrecy: New results, constructions and connections with algebraic manipulation detection codes Adv. Math. Commun. (IF 0.9) Pub Date : 2021-11-18 Maura B. Paterson, Douglas R. Stinson
A splitting BIBD is a type of combinatorial design that can be used to construct splitting authentication codes with good properties. In this paper we show that a design-theoretic approach is useful in the analysis of more general splitting authentication codes. Motivated by the study of algebraic manipulation detection (AMD) codes, we define the concept of a group generated splitting authentication
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Nonexistence of some ternary linear codes with minimum weight -2 modulo 9 Adv. Math. Commun. (IF 0.9) Pub Date : 2021-11-10 Toshiharu Sawashima, Tatsuya Maruta
One of the fundamental problems in coding theory is to find \begin{document}$ n_q(k,d) $\end{document}, the minimum length \begin{document}$ n $\end{document} for which a linear code of length \begin{document}$ n $\end{document}, dimension \begin{document}$ k $\end{document}, and the minimum weight \begin{document}$ d $\end{document} over the field of order \begin{document}$ q $\end{document} exists