An Upper Bound on the Minimum Distance of Array Low-Density Parity-Check Codes

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.