Single-Exclusion Number and the Stopping Redundancy of MDS Codes

TL;DR

This paper investigates the stopping redundancy of MDS codes by introducing the (n,t)-single-exclusion number, providing new bounds and partial asymptotic confirmation of the Schwartz-Vardy conjecture.

Contribution

It defines the (n,t)-single-exclusion number and derives new bounds using probabilistic and explicit methods, advancing understanding of MDS code stopping redundancy.

Findings

New upper bounds on the single-exclusion number

Asymptotic equivalence of stopping redundancy and S(n,d-2) under certain conditions

Partial confirmation of the Schwartz-Vardy conjecture asymptotically

Abstract

For a linear block code C, its stopping redundancy is defined as the smallest number of check nodes in a Tanner graph for C, such that there exist no stopping sets of size smaller than the minimum distance of C. Schwartz and Vardy conjectured that the stopping redundancy of an MDS code should only depend on its length and minimum distance. We define the (n,t)-single-exclusion number, S(n,t) as the smallest number of t-subsets of an n-set, such that for each i-subset of the n-set, i=1,...,t+1, there exists a t-subset that contains all but one element of the i-subset. New upper bounds on the single-exclusion number are obtained via probabilistic methods, recurrent inequalities, as well as explicit constructions. The new bounds are used to better understand the stopping redundancy of MDS codes. In particular, it is shown that for [n,k=n-d+1,d] MDS codes, as n goes to infinity, the stopping redundancy is asymptotic to S(n,d-2), if d=o(\sqrt{n}), or if k=o(\sqrt{n}) and k goes to infinity, thus giving partial confirmation of the Schwartz-Vardy conjecture in the asymptotic sense.