In this paper, we propose a class of codes, referred to as random staircase generator matrix codes (SGMCs), which have staircase-like generator matrices. In the infinite-length region, we prove that the random SGMC is capacity-achieving over binary-input output-symmetric (BIOS) channels. In the finite-length region, we present the representative ordered statistics decoding with local constraints (LC-ROSD) algorithm for the SGMCs. The most distinguished feature of the SGMCs with LC-ROSD is that the staircase-like matrices enable parallel implementation of the Gaussian elimination (GE), avoiding the serial GE of conventional OSD and supporting a potential low decoding latency, as implied from simulations. To analyze the performance of random SGMCs in the finite-length region, we derive the ensemble weight spectrum and invoke the conventional union bound. We also derive a partially random coding union (RCU) bound, which is tighter than the conventional one and is used as a criterion to design the SGMCs. Staircase-like generator matrices allow us to derive a series of (tighter and tighter) lower bounds based on the second-order Bonferroni inequality with the incremental number of codewords. The numerical results show that the decoding performance can match well with the proposed partially RCU bound for different code rates and different profiles. The numerical results also show that the tailored SGMCs with the LC-ROSD algorithm can approach the finite-length performance bound, outperforming the 5G low-density parity-check (LDPC) codes, 5G polar codes, and Reed-Muller (RM) codes.

翻译：本文提出一类称为随机阶梯生成矩阵码（SGMC）的编码方案，其生成矩阵呈阶梯状结构。在无限长区域，我们证明了随机SGMC在二进制输入对称输出（BIOS）信道上能够达到信道容量。在有限长区域，我们针对SGMC提出了带局部约束的代表性排序统计译码（LC-ROSD）算法。SGMC与LC-ROSD最显著的特征在于：阶梯状矩阵支持高斯消元（GE）的并行实现，避免了传统排序统计译码（OSD）的串行高斯消元，仿真结果表明这有助于实现更低的译码延迟。为分析有限长区域随机SGMC的性能，我们推导了其集合权重谱并采用传统联合界。我们还推导了部分随机编码联合（RCU）界，该界比传统联合界更紧，可作为SGMC的设计准则。基于二阶Bonferroni不等式，阶梯状生成矩阵使我们能够通过递增码字数量推导出一系列（逐渐紧化的）下界。数值结果表明，对于不同码率和不同配置文件，译码性能与所提出的部分RCU界高度吻合。这些结果同时表明：采用LC-ROSD算法的定制SGMC能够接近有限长性能界，优于5G低密度奇偶校验（LDPC）码、5G极化码及里德-穆勒（RM）码。