Layered Normalized Min-Sum Decoding with Bit Flipping for FDPC Codes

Fair-density parity-check (FDPC) codes have been recently introduced demonstrating improved performance compared to low-density parity-check (LDPC) codes standardized in 5G systems particularly in high-rate regimes. In this paper, we introduce a layered normalized min-sum (LNMS) message-passing decoding algorithm for the FDPC codes. We also introduce a syndrome-guided bit flipping (SGBF) method to enhance the error-correction performance of our proposed decoder. The LNMS decoder leverages conflict graph coloring for efficient layered scheduling, enabling faster convergence by grouping non-conflicting check nodes and updating variable nodes immediately after each layer. In the event of decoding failure, the SGBF method is activated, utilizing a novel reliability metric that combines log-likelihood ratio (LLR) magnitudes and syndrome-derived error counts to identify the least reliable bits. A set of candidate sequences is then generated by performing single-bit flips at these positions, with each candidate re-decoded via LNMS. The optimal candidate is selected based on the minimum syndrome weight. Extensive simulation results demonstrate the superiority of the proposed decoder. Numerical simulations on FDPC (256, 192) code with a bit-flipping set size of T = 128 and a maximum of 5 iterations demonstrate that the proposed decoder achieves approximately a 0. 5\, dB coding gain over standalone LNMS decoding at a frame error rate (FER) of 10^-3, while providing coding gains of 0. 75-1. 5\, dB over other state-of-the-art codes including polar codes and 5G-LDPC codes at the same length and rate and also under belief propagation decoding.