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：多维空间耦合低密度奇偶校验码因其优异的渐近性能与有限长性能，正受到日益广泛的关注。这类码通过重新排列空间耦合码的副本进行连接，以缓解多种非均匀性来源，从而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自由度的增加，设计过程的复杂度增长使其难以充分开发利用这些自由度。本文提出一种基于梯度下降算法的概率性多维空间耦合码设计框架，旨在设计更优的多维码并应对上述挑战。具体而言，我们将待最小化的短环期望数量表示为码图结构中表征设计方法的概率分布矩阵元素函数，进而求得局部最优概率分布作为有限长算法优化器的初始点，最终生成多维空间耦合码。我们提供了理论分析与算法实现，实验结果表明，与现有最优方法相比，本文所提的梯度下降多维码（GD-MD码）具有显著降低的短环数量。此外，算法可在数次迭代内收敛至解，证实通过局部最优梯度下降多维分布限制搜索空间有效降低了复杂度。