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：由于空间耦合码在渐近性能和有限长度下的优异表现，其作为低密度奇偶校验码的一类正受到越来越多的关注。多维空间耦合码通过重排操作连接多个空间耦合码副本构建而成，旨在缓解各类非均匀性问题并提升数据存储与传输系统的性能。随着多维空间耦合码设计中自由度的增加，设计过程的复杂度也随之增长，导致合理利用这些自由度变得愈发困难。本文提出了一种基于梯度下降算法的概率性框架用于多维空间耦合码的设计，通过该框架可设计更优的多维码并解决上述挑战。具体而言，我们以表征多维空间耦合码设计的概率分布矩阵元素为变量，推导了码字图表示中待优化的短环期望数量表达式。进而求解局部最优概率分布，将其作为有限长度算法优化器的初始点以生成最终的多维空间耦合码。我们提供了理论分析与算法实现，实验结果表明：相较于现有最优方案，我们设计的码字（简称为GD-MD码）具有显著更低的短环数量。此外，算法在数次迭代内即可收敛至最优解，验证了通过局部最优GD-MD分布缩小搜索空间所实现的复杂度降低效果。