Bounds on the rate of disjunctive codes (in Russian)

TL;DR

This paper investigates bounds on the rate of specific binary codes used for disjunctive and list-decoding purposes, providing new theoretical limits based on combinatorial and probabilistic methods.

Contribution

It introduces new lower and upper bounds on the rate of superimposed cover-free and list-decoding codes using ensemble analysis of constant weight codes.

Findings

Derived lower bounds on code rates using constant weight code ensembles.

Established an upper bound on the rate of superimposed list-decoding codes.

Provided theoretical limits for code performance in disjunctive coding scenarios.

Abstract

A binary code is called a superimposed cover-free $(s,\ell)$-code if the code is identified by the incidence matrix of a family of finite sets in which no intersection of $\ell$ sets is covered by the union of $s$ others. A binary code is called a superimposed list-decoding $s_L$-code if the code is identified by the incidence matrix of a family of finite sets in which the union of any $s$ sets can cover not more than $L-1$ other sets of the family. For $L=\ell=1$, both of the definitions coincide and the corresponding binary code is called a superimposed $s$-code. Our aim is to obtain new lower and upper bounds on the rate of the given codes. In particular, we derive lower bounds on the rates of a superimposed cover-free $(s,\ell)$-code and list-decoding $s_L$-code based on the ensemble of constant weight binary codes. Also, we establish an upper bound on the rate of superimposed list-decoding $s_L$-code.