Improved Upper Bounds for the Rate of Separating and Completely Separating Codes

Abstract

A binary code is said to be an $(s,\ell)$-separating code if for any two disjoint sets of its codewords of cardinalities at most $s$ and $\ell$ respectively, there exists a coordinate in which all words of the first set have symbol 0 while all words of the second have 1. If, moreover, for any two sets there exists a second coordinate in which all words of the first set have 1 and all words of the second have 0, then such a code is called an $(s,\ell)$-completely separating code. We improve upper bounds on the rate of separating and completely separating codes.

Appendix: Lower Bounds On the Rate of Codes with a Separating Distance

Theorem 4.

The maximum cardinality $M_{u,v}(N,d)$ of a code having $(u,v)$-separating distance $d=\lfloor \tau N\rfloor$ satisfies the inequality

$$\log_2M_{u, v}(N, d)\ge N\frac{-h(\tau)-\tau\log_2\Delta(u,v)-(1-\tau)\log_2(1-\Delta(u,v))+o(1)}{u+v-1}$$

(25)

for $0\le\tau <\Delta(u,v)$ with $\Delta(u,v)$ as defined in (18).

Note that the right-hand side is positive for $\tau<\Delta(u,v)$ and tends to zero as $\tau\to\Delta(u,v)$. For $u=v=1$ we have $\Delta(1,1)=1/2$, and our lower bound turns into the Gilbert–Varshamov bound.

Proof.

Consider a random code of length $N$ and cardinality $M$ where each element of every codeword is chosen independently and equals $1$ with probability $p$.

The probability that a fixed coordinate separates two sets of codewords of sizes $u$ and $v$ is

$$q=p^u(1-p)^v+p^v(1-p)^u.$$

We choose the parameter $p$ so that to maximize $q$. Note that the maximum of the separation probability precisely equals the $\Delta(u,v)$ defined in (18). The number $\xi$ of coordinates separating two fixed sets of codewords has the binomial distribution with parameters $N$ and $\Delta=\Delta(u,v)$. Let us estimate the probability that $\xi<d$ given that $\Delta(u,v)>\tau$:

$$\begin{aligned} \mathcal{P}=\sum\limits_{k=0}^{d-1}\binom{N}{k}\Delta^k(1-\Delta)^{N-k}\le N2^{N(h(\tau)+\tau\log_2\Delta+(1-\tau)\log_2(1-\Delta)+o(1))}. \end{aligned}$$

Thus,

$$N^{-1}\log_2\mathcal{P}=h(\tau)+\tau\log_2\Delta+(1-\tau)\log_2(1-\Delta)+o(1).$$

The mathematical expectation of the number of sets of codewords with the separating distance between them being less than $d$ is at most $\mathcal{P}\cdot M^{u+v}$. Standard expurgation arguments result in the bound

$$\log_2M_{u, v}(N, d)\ge N \frac{-h(\tau)-\tau\log_2\Delta-(1-\tau)\log_2(1-\Delta)+o(1)} {u+v-1},$$

as required. △