Belief Propagation Decoding for Short-Length Codes Based on Sparse Tanner Graph

In this paper, we present a novel approach to improve belief propagation (BP) decoding performance by sparsifying the Tanner graph. Our proposed method first constructs a low-weight parity-check (LWPC) matrix with increased rows based on the original parity-check matrix. Then, all 4-cycles in the LWPC matrix are eliminated by adding auxiliary variable nodes and corresponding parity-check rows, resulting in the generalized LWPC (G-LWPC) matrix. Compared to the original matrix, the sparsity is greatly improved without altering the encoding constraints. Simulation results show that BP decoding based on the G-LWPC matrix is particularly effective for short code lengths. For a (128,20) polar code with 24-bit cyclic redundancy check, our proposed G-LWPC matrix reduces the density from 17.4% to 1.3%, and with 10 iterations, BP decoding can achieve the performance of successive cancellation list decoding with a list size of 32.