Because of their excellent asymptotic and finite-length performance, spatially-coupled (SC) codes are a class of low-density parity-check codes that is gaining increasing attention. Multi-dimensional (MD) SC codes are constructed by connecting copies of an SC code via relocations in order to mitigate various sources of non-uniformity and improve performance in many data storage and data transmission systems. As the number of degrees of freedom in the MD-SC code design increases, appropriately exploiting them becomes more difficult because of the complexity growth of the design process. In this paper, we propose a probabilistic framework for the MD-SC code design, which is based on the gradient-descent (GD) algorithm, to design better MD codes and address this challenge. In particular, we express the expected number of short cycles, which we seek to minimize, in the graph representation of the code in terms of entries of a probability-distribution matrix that characterizes the MD-SC code design. We then find a locally-optimal probability distribution, which serves as the starting point of a finite-length algorithmic optimizer that produces the final MD-SC code. We offer the theoretical analysis as well as the algorithms, and we present experimental results demonstrating that our MD codes, conveniently called GD-MD codes, have notably lower short cycle numbers compared with the available state-of-the-art. Moreover, our algorithms converge on solutions in few iterations, which confirms the complexity reduction as a result of limiting the search space via the locally-optimal GD-MD distributions.

翻译：由于其优异的渐进性能和有限长度性能，空间耦合（SC）码作为一类低密度奇偶校验码正受到越来越多的关注。多维（MD）SC码通过重定位连接多个SC码副本构建，以减轻多种非均匀性来源的影响并提升数据存储与传输系统的性能。随着MD-SC码设计自由度的增加，由于设计过程的复杂度增长，合理利用这些自由度变得更加困难。本文提出一种基于梯度下降（GD）算法的MD-SC码概率性设计框架，以设计更优的多维码并应对这一挑战。具体而言，我们将码图表示中期望最小化的短环个数表示为表征MD-SC码设计的概率分布矩阵元素函数。随后我们求解局部最优概率分布，将其作为有限长度算法优化器的起点，最终生成目标MD-SC码。我们提供了理论分析与算法实现，实验结果表明：与现有最优方案相比，我们设计的MD码（简称GD-MD码）具有显著更低的短环数量。此外，我们的算法能在少量迭代中收敛至解，这证实了通过局部最优GD-MD分布约束搜索空间所带来的复杂度降低效果。