Almost Disjunctive List-Decoding Codes

TL;DR

This paper introduces a probabilistic generalization of disjunctive list-decoding codes, called almost disjunctive LD s_L-codes, and develops a random coding method to establish bounds on their capacity and error probability.

Contribution

It proposes a new probabilistic model for disjunctive list-decoding codes and derives asymptotically tight bounds using a random coding approach.

Findings

Lower bounds on capacity are established.

Error probability exponents are derived.

Bounds are asymptotically tight.

Abstract

A binary code is said to be a disjunctive list-decoding $s_L$-code, $s\ge1$, $L\ge1$, (briefly, LD $s_L$-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. In this paper, we introduce a natural {\em probabilistic} generalization of LD $s_L$-code when the code is said to be an almost disjunctive LD $s_L$-code if the unions of {\em almost all} $s$ sets satisfy the given condition. We develop a random coding method based on the ensemble of binary constant-weight codes to obtain lower bounds on the capacity and error probability exponent of such codes. For the considered ensemble our lower bounds are asymptotically tight.