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

The algebraic rigidity of BCH codes challenges the development of parallelizable and efficient decoders for high-throughput applications. To address this, we propose a hybrid scheme combining normalized min-sum and order statistics decoding, achieving near-maximum likelihood performance for short BCH codes while retaining the benefits of the normalized min-sum decoder over a wide SNR range. First, a heuristic method constructs a parity-check matrix with low density, appropriate redundancy, and fewer length-4 cycles through binary sum and random row cyclic shifts, forming a solid foundation for decoder design. The impact of row redundancy and rank deficiency in the dual code's minimum-weight codewords on frame error rate is analyzed. In the revised normalized min-sum decoder, three types of random automorphisms enhance decoding diversity, while aggregated messages accelerate convergence. For BCH codes of length 63 and 127, the proposed approach achieves a 1-2 dB bit error rate advantage over parallelizable alternatives or requires up to two orders of magnitude fewer iterations than other iterative rivals. These results highlight its effectiveness in hybrid decoding for ultra-reliable, low-latency communications.