On the minimum distance of array codes as LDPC codes

For a prime q and an integer j≤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)≥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).