Single-Exclusion Number and the Stopping Redundancy of MDS Codes

• Published in: IEEE transactions on information theory Vol. 55; no. 9; pp. 4155 - 4166
• Main Authors: Junsheng Han, Siegel, P.H., Roth, R.M.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01-09-2009; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Additive white noise; Applied sciences; Asymptotic properties; AWGN; Block codes; Codes; Coding, codes; Construction; Erasure channel; Exact sciences and technology; Graphs; Inequalities; Information theory; Information, signal and communications theory; Iterative decoding; Linear code; Magnetic recording; Maximum likelihood decoding; maximum-distance separable (MDS) code; Parity check codes; Probabilistic methods; Redundancy; Signal and communications theory; single-exclusion number; stopping redundancy; stopping set; Telecommunications and information theory; TurÁn number; Upper bound; Upper bounds; Block code; Probabilistic approach; Redundancy; Tanner graph; Erasure channel; Iterative decoding; maximum-distance separable (MDS) code; single-exclusion number; stopping set; Thrán number; Linear code; Upper bound; Minimal distance; stopping redundancy
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2009.2025578

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 a maximum-distance separable (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 i-subsets of an n-set, such that for each i-subset of the n-set, i =1,... ,t + 1, there exists a i-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 rarr infin , the stopping redundancy is asymptotic to S(n, d - 2), if d = o(radic(n)), or if k = o(radic(n)), k rarr infin, thus giving partial confirmation of the Schwartz-Vardy conjecture in the asymptotic sense.