We generalize staircase codes and tiled diagonal zipper codes, preserving their key properties while allowing each coded symbol to be protected by arbitrarily many component codewords rather than only two. This generalization which we term "higher-order staircase codes" arises from the marriage of two distinct combinatorial objects: difference triangle sets and finite-geometric nets, which have typically been applied separately to code design. We demonstrate one possible realization of these codes, obtaining powerful, high-rate, low-error-floor, and low-complexity coding schemes based on simple iterative syndrome-domain decoding of coupled Hamming component codes. We anticipate that the proposed codes could improve performance--complexity--latency tradeoffs in high-throughput communications applications, most notably fiber-optic, in which classical staircase codes and zipper codes have been applied. We consider the construction of difference triangle sets having minimum scope and sum-of-lengths, which lead to memory-optimal realizations of higher-order staircase codes. These results also enable memory reductions for early families of convolutional codes constructed from difference triangle sets.

翻译：我们推广了阶梯码与平铺对角拉链码，在保持其关键特性的同时，允许每个编码符号受任意多个分量码字保护，而不仅仅是两个。这种我们称为“高阶阶梯码”的推广源于两种不同组合对象的结合：差分三角集与有限几何网，这两者在编码设计中通常被分别应用。我们展示了这些码的一种可能实现方式，基于耦合汉明分量码的简单迭代校验子域译码，获得了性能强大、高码率、低错误平层且低复杂度的编码方案。我们预期所提出的码能够改善高吞吐量通信应用（尤其是光纤通信，其中经典阶梯码与拉链码已被应用）中性能-复杂度-延迟的权衡。我们研究了具有最小作用域与长度和的差分三角集的构造，这导致了高阶阶梯码的内存最优实现。这些结果也使得基于差分三角集构建的早期卷积码家族能够实现内存的降低。