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，而非仅两个。这一推广——我们称之为“高阶阶梯码”——源于两种不同组合对象的结合：差三角集与有限几何网，这两者通常被分别应用于码设计。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（最显著的是光纤通信，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。我们考虑了具有最小作用域与长度和的差三角集的构造，这引向了高阶阶梯码的内存最优实现。这些结果也使得基于差三角集构建的早期卷积码族能够实现内存的降低。