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 $0.5\,\mathrm{dB}$ coding gain over standalone LNMS decoding at a frame error rate (FER) of $10^{-3}$, while providing coding gains of $0.75-1.5\,\mathrm{dB}$ 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.

翻译：公平密度奇偶校验（FDPC）码是近期提出的一种编码方案，与5G系统中标准化的低密度奇偶校验（LDPC）码相比，特别是在高码率场景下，其性能表现更优。本文针对FDPC码提出了一种分层归一化最小和（LNMS）消息传递解码算法。我们还引入了一种伴随式引导比特翻转（SGBF）方法，以增强所提出解码器的纠错性能。LNMS解码器利用冲突图着色实现高效的分层调度，通过将无冲突的校验节点分组并在每层更新后立即更新变量节点，从而实现更快的收敛。当解码失败时，SGBF方法被激活，该方法采用一种新颖的可靠性度量，该度量结合了对数似然比（LLR）幅度和伴随式导出的错误计数，以识别最不可靠的比特位。随后，通过在这些位置执行单比特翻转生成一组候选序列，每个候选序列通过LNMS重新解码。基于最小伴随式权重选择最优候选。大量仿真结果证明了所提出解码器的优越性。在FDPC$(256,192)$码上进行的数值仿真，设定比特翻转集合大小为$T = 128$，最大迭代次数为$5$次，结果表明，在帧错误率（FER）为$10^{-3}$时，所提出的解码器相比独立的LNMS解码实现了约$0.5\,\mathrm{dB}$的编码增益，同时在与相同长度和码率、同样采用置信传播解码的极化码和5G-LDPC码等其他先进编码方案相比，提供了$0.75-1.5\,\mathrm{dB}$的编码增益。