The Minimal Product Parity Check Matrix and Its Application

In decoding a linear block code C by iterative decoding algorithms, these algorithms are essentially applied on a graphical representation of C such as Tanner graph and factor graph. Due to the impact of the structure of these graphs on the performance of code, we consider minimal parity check matrices(matrices with minimum number of nonzero entries) of product codes. An algorithm constructing such matrices is given. It turns out that under the given construction method, product coding technique is indeed a powerful method to construct irregular LDPC codes. Given two codes A and B, the construction method produces a Tanner graph with girth g := min {8, ga, gb} for the product code A ⊗ B, where ga and gb are the girth of Tanner graphs representing A and B, respectively. Simulation results confirm the positive practical impact of the minimality of the parity check matrices.