On full-rank parity-check matrices of product codes - Esmaeili, Morteza (Utilitas Mathematica, 2008)
Given full-rank parity-check matrices HA and HB 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(Hi) associated with Hi, i = 1,2, is bounded below by {9a, 9b,8} where 9a and 9b, are the girths of TG(HA) and TG(HB), 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.