Single-Exclusion Number and the Stopping Redundancy of MDS Codes

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.

💡 Research Summary

The paper investigates the stopping redundancy of linear block codes, focusing on Maximum‑Distance‑Separable (MDS) codes. Stopping redundancy ρ(C) is defined as the smallest number of parity‑check nodes in a Tanner graph of a code C such that no stopping set of size smaller than the code’s minimum distance d exists. Schwartz and Vardy previously conjectured that for an MDS code the stopping redundancy depends only on the length n and the minimum distance d, but no tight bounds were known.

To address this gap the authors introduce a new combinatorial parameter, the (n, t)‑single‑exclusion number S(n, t). S(n, t) is the minimum size of a family of t‑subsets of an n‑element set with the property that for every i‑subset (1 ≤ i ≤ t + 1) there is a t‑subset in the family that contains all but one element of the i‑subset. This definition mirrors the requirement that a parity‑check matrix must “exclude” every small subset of variable nodes except possibly one, which is precisely the condition that eliminates stopping sets smaller than d. For an MDS code with parameters