On full-rank parity-check matrices of product codes

Given full-rank parity-check matrices H-A and H-B for linear binary codes A and B, respectively, two full-rank parity-check matrices, denoted H-1 and H-2, are given for the product code A circle times B. It is shown that the girth of Tanner graph TG(H-i) associated with H-i, i = 1, 2, is bounded below by {g(a), g(b), 8} where g(a) and g(b) are the girths of TG(H-A) and TG(H-B), respectively. It turns out that the product of m >= 2 single parity-check codes is either cycle-free or has girth 8, and a necessary and sufficient condition for having the latter case is provided.