On full-rank parity-check matrices of product codes

Abstract

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 H₁ and H₂, are given for the product code A⊗B. It is shown that the girth of Tanner graph TG(H_i) associated with H_i, i = 1,2, is bounded below by {9_a, 9_b,8} where 9_a and 9_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.