1Single-Exclusion Number and the Stopping Redundancy of MDS Codes
Junsheng Han, Student Member, Paul H. Siegel, Ron M. Roth · 2007
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 → ∞, the stopping redundancy is asymptotic to S(n, d − 2), if d = o(√n), or if k = o( n), k → ∞, thus giving partial confirmation of the Schwartz-Vardy conjecture in the asymptotic sense. Index Terms — erasure channel, iterative decoding, MDS code, single-exclusion number, stopping redundancy, stopping set, Turán number. I.