Low-density parity-check (LDPC) codes are among the most prominent error-correction schemes. They find application to fortify various modern storage, communication, and computing systems. Protograph-based (PB) LDPC codes offer many degrees of freedom in the code design and enable fast encoding and decoding. In particular, spatially-coupled (SC) and multi-dimensional (MD) circulant-based codes are PB-LDPC codes with excellent performance. Efficient finite-length (FL) algorithms are required in order to effectively exploit the available degrees of freedom offered by SC partitioning, lifting, and MD relocations. In this paper, we propose a novel Markov chain Monte Carlo (MCMC or MC$^2$) method to perform this FL optimization, addressing the removal of short cycles. While iterating, we draw samples from a defined distribution where the probability decreases as the number of short cycles from the previous iteration increases. We analyze our MC$^2$ method theoretically as we prove the invariance of the Markov chain where each state represents a possible partitioning or lifting arrangement. Via our simulations, we then fit the distribution of the number of cycles resulting from a given arrangement on a Gaussian distribution. We derive estimates for cycle counts that are close to the actual counts. Furthermore, we derive the order of the expected number of iterations required by our approach to reach a local minimum as well as the size of the Markov chain recurrent class. Our approach is compatible with code design techniques based on gradient-descent. Numerical results show that our MC$^2$ method generates SC codes with remarkably less number of short cycles compared with the current state-of-the-art. Moreover, to reach the same number of cycles, our method requires orders of magnitude less overall time compared with the available literature methods.

翻译：低密度奇偶校验（LDPC）码是最重要的纠错方案之一，被广泛应用于增强各种现代存储、通信和计算系统。基于原模图（PB）的LDPC码在码设计中提供了许多自由度，并支持快速编码与解码。特别是，基于空间耦合（SC）和多维（MD）循环移位的码是性能优异的PB-LDPC码。为了有效利用由SC分割、提升和MD重定位所提供的自由度，需要高效的有限长度（FL）算法。本文提出了一种新颖的马尔可夫链蒙特卡洛（MCMC或MC$^2$）方法来进行这种FL优化，重点解决短环的消除问题。在迭代过程中，我们从定义的概率分布中抽取样本，该分布的概率随着前一迭代中短环数量的增加而降低。我们从理论上分析了MC$^2$方法，证明了马尔可夫链的不变性，其中每个状态代表一种可能的分割或提升配置。通过仿真，我们将给定配置产生的环数量分布拟合为高斯分布，并推导出与实际计数接近的环数量估计值。此外，我们还推导了该方法达到局部最小值所需迭代次数的期望阶数以及马尔可夫链递归类的规模。我们的方法与基于梯度下降的码设计技术兼容。数值结果表明，与当前最先进技术相比，我们的MC$^2$方法生成的SC码具有显著更少的短环数量。此外，为达到相同的环数量，我们的方法所需的总时间比现有文献方法少几个数量级。