Trellis-Based Check Node Processing for Low-Complexity Nonbinary LP Decoding

Linear Programming (LP) decoding is emerging as an attractive alternative to decode Low-Density Parity-Check (LDPC) codes. However, the earliest LP decoders proposed for binary and nonbinary LDPC codes are not suitable for use at moderate and large code lengths. To overcome this problem, Vontobel et al. developed an iterative Low-Complexity LP (LCLP) decoding algorithm for binary LDPC codes. The variable and check node calculations of binary LCLP decoding algorithm are related to those of binary Belief Propagation (BP). The present authors generalized this work to derive an iterative LCLP decoding algorithm for nonbinary linear codes. Contrary to binary LCLP, the variable and check node calculations of this algorithm are in general different from that of nonbinary BP. The overall complexity of nonbinary LCLP decoding is linear in block length; however the complexity of its check node calculations is exponential in the check node degree. In this paper, we propose a modified BCJR algorithm for efficient check node processing in the nonbinary LCLP decoding algorithm. The proposed algorithm has complexity linear in the check node degree. We also introduce an alternative state metric to improve the run time of the proposed algorithm. Simulation results are presented for $(504, 252)$ and $(1008, 504)$ nonbinary LDPC codes over $\mathbb{Z}_4$.