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arXiv:1504.06416 · IEEE International Symposium on Information Theory (ISIT)

On the minimum distance of array-based spatially-coupled low-density parity-check codes

Eirik Rosnes

Published Jun 1, 2015• 10 citations
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Abstract

An array low-density parity-check (LDPC) code is a quasi-cyclic LDPC code specified by two integers q and m, where q is an odd prime and m ≤ q. The exact minimum distance, for small q and m, has been calculated, and tight upper bounds on it for m ≤ 7 have been derived. In this work, we study the minimum distance of the spatially-coupled version of these codes. In particular, several tight upper bounds on the optimal minimum distance for coupling length at least two and m = 3; 4; 5, that are independent of q and that are valid for all values of q ≥ q0where q0depends on m, are presented. Furthermore, we show by exhaustive search that by carefully selecting the edge spreading or unwrapping procedure, the minimum distance (when q is not very large) can be significantly increased, especially for m = 5.

DOI
10.1109/isit.2015.7282582

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An array low-density parity-check (LDPC) code is a quasi-cyclic LDPC code specified by two integers q and m, where q is an odd prime and m ≤ q. The exact minimum distance, for small q and m, has been calculated, and tight upper bounds on it for m ≤ 7 have been derived. In this work, we study the minimum distance of the spatially-coupled version of these codes. In particular, several tight upper bounds on the optimal minimum distance for coupling length at least two and m = 3; 4; 5, that are independent of q and that are valid for all values of q ≥ q0where q0depends on m, are presented. Furthermore, we show by exhaustive search that by carefully selecting the edge spreading or unwrapping procedure, the minimum distance (when q is not very large) can be significantly increased, especially for m = 5.
In this work, we present an upper bound on the minimum 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 distance of these codes (denoted by d(q,m)) 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 by using the concept of a template support matrix of a codeword. The bound is 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 distance results for m ≤ 6 and low-to-moderate values of q ≤ 79.
In this paper, 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, is d(q, 6) 32. In this paper, 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.
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.