Effective Application of Normalized Min-Sum Decoding for Short BCH Codes

Abstract: This paper introduces an enhanced normalized min-sum decoder designed to address the performance and complexity challenges associated with developing parallelizable decoders for short BCH codes in high-throughput applications. The decoder optimizes the standard parity-check matrix using heuristic binary summation and random cyclic row shifts, resulting in a Tanner graph with low density, controlled redundancy, and minimized length-4 cycles. The impact of row redundancy and rank deficiency in the dual code's minimum-weight codewords on decoding performance is analyzed. To improve convergence, three random automorphisms are applied simultaneously to the inputs, with the resulting messages merged at the end of each iteration. Extensive simulations demonstrate that, for BCH codes with block lengths of 63 and 127, the enhanced normalized min-sum decoder achieves a 1-2 dB performance gain and 100X faster convergence compared to existing parallel and iterative decoders. Additionally, a hybrid decoding scheme is proposed, which selectively activates order statistics decoding when the enhanced normalized min-sum decoder fails. This hybrid approach is shown to approach maximum-likelihood performance while retaining the advantages of the normalized min-sum decoder across a broad SNR range.