On the minimum distance of array codes as LDPC codes

Abstract

For a prime q and an integer j less than or equal to q, the code C(q, j) is a class of low-density parity-check (LDPC) codes from array codes which has a nice algebraic structure. In this correspondence, we investigate the minimum distance d(q, j) of the code in an algebraic way. We first prove that the code is invariant under a doubly transitive group of "affine" permutations. Then, we show that d(5, 4) = 8, d(7, 4) = 8, and d(q, 4) greater than or equal to 10 for any prime q > 7. In addition, we also analyze the codewords of weight 6 in the case of j 3 and the codewords of weight 8 in C(5, 4) and C(7, 4).