Abstract-For a linear block code with minimum distance d, its stopping redundancy is the minimum number of check nodes in a Tanner graph representation of the code, such that all nonempty stopping sets have size d or larger. We derive new upper bounds on stopping redundancy for all linear codes in general, and for maximum distance separable (MDS) codes specifically, and show how they improve upon previous results. For MDS codes, the new bounds are found by upper-bounding the stopping redundancy by a combinatorial quantity closely related to Turán numbers. (The Turán number, T (v; k; t), is the smallest number of t-subsets of a v-set, such that every k-subset of the v-set contains at least one of the t-subsets.) Asymptotically, we show that the stopping redundancy of MDS codes with length n and minimum distance d > 1 is T (n; d 0 1; d 0 2)(1 + O(n 01 )) for fixed d, and is at most T (n; d 0 1; d 0 2)(3 + O(n 01 )) for fixed code dimension k = n 0 d + 1. For d = 3; 4, we prove that the stopping redundancy of MDS codes is equal to T (n; d 0 1; d 0 2), for which exact formulas are known. For d = 5, we show that the stopping redundancy of MDS codes is either T (n; 4; 3) or T (n; 4; 3) + 1.
We study memories protected with error control codes, in which the memory's contents are organized in lines which are read and written to in isolation from other lines. In these memories the available redundancy is structured so as to protect individual lines rather than the entire memory as a whole. Often designers wish to read and write only parts of the memory line, as in some instances this leads to various favorable system design tradeoffs, including better power consumption, increased data access concurrency, etc. (alternatively one may say that designers sometimes would prefer smaller line sizes). Nevertheless when designing systems with such subline accesses it is often found that in order to mantain a given level of reliability, the total amount of redundancy allocated in the memory needs to be increased beyond desirable levels. In this work, we initiate a study of the problem of structuring error control codes to allow subline accesses with good tradeoffs between reliability and redundancy. We motivate and explore a setting in which a "double-lookup" protocol is used in conjunction with certain types of two-level codes, whereby error detection is attained in a first level and error correction using the second level is performed whenever errors are detected in the first level. We obtain lower bounds on redundancy for a given level of reliability and offer a code construction that attains this bound for a certain important class of parameters. We also introduce an alternate construction which allows us to find longer codes under restrictions of the Galois Field size used in the 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.Comment: 12 pages, 1 figure. Submitted to IEEE Transactions on Information Theor
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