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On the Minimum/Stopping Distance of Array Low-Density Parity-Check Codes

2014; Institute of Electrical and Electronics Engineers; Volume: 60; Issue: 9 Linguagem: Inglês

10.1109/tit.2014.2333519

1557-9654

Eirik Rosnes, Marcel Ambroze, Martin Tomlinson,

DNA and Biological Computing

In this work, we study the minimum/stopping distance of array low-density parity-check (LDPC) codes.An array LDPC code is a quasi-cyclic LDPC code specified by two integers q and m, where q is an odd prime and m ≤ q.In the literature, the minimum/stopping distance of these codes (denoted by d(q, m) and h(q, m), respectively) has been thoroughly studied for m ≤ 5.Both exact results, for small values of q and m, and general (i.e., independent of q) bounds have been established.For m = 6, the best known minimum distance upper bound, derived by Mittelholzer (IEEE Int.Symp.Inf.Theory, Jun./Jul.2002), is d(q, 6) ≤ 32.In this work, we derive an improved upper bound of d(q, 6) ≤ 20 and a new upper bound d(q, 7) ≤ 24 by using the concept of a template support matrix of a codeword/stopping set.The bounds are tight with high probability in the sense that we have not been able to find codewords of strictly lower weight for several values of q using a minimum distance probabilistic algorithm.Finally, we provide new specific minimum/stopping distance results for m ≤ 7 and low-to-moderate values of q ≤ 79. Index Terms-Array codes, low-density parity-check (LDPC)codes, minimum distance, stopping distance, template support matrix. I. INTRODUCTIONIn this paper, we consider the array low-density parity-check (LDPC) codes, originally introduced by Fan in [1], and their minimum/stopping distance.Array LDPC codes are specified by two integers q and m, where q is an odd prime and m ≤ q.Furthermore, in this work, C(q, m) will denote the array LDPC code with parameters q and m, and d(q, m) (respectively h(q, m)) its minimum (respectively stopping) distance.Since the original work by Fan, several authors have considered the structural properties of these codes (see, e.g., [2][3][4][5][6][7][8]).For high rate and moderate length, these codes perform well under iterative decoding, and they are also well-suited for practical implementation due to their regular structure [9,10].The minimum distance of these codes was first analyzed by Mittelholzer in [2], where general (i.e., independent of q) minimum distance upper bounds for m ≤ 6 were provided.