Low-density parity-check (LDPC) codes are among the most prominent error-correction schemes in today’s data-driven world. 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 MC2) method to perform this FL optimization, addressing the removal of short cycles. While we focus on partitioning and lifting of SC codes, our MC2approach can effectively work for other procedures and/or other code designs. 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 MC2method theoretically as we prove the invariance of the Markov chain where each state represents a possible partitioning or lifting arrangement, i.e., sample, that has a specific probability. Via our simulations, we then fit the distribution of the number of cycles resulting from a given arrangement on a Gaussian distribution. By analyzing the mean, 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 MC2approach to reach a local minimum as well as the size of the Markov chain recurrent class through approximating the probability of getting arbitrarily close to the local minimum. Our approach is compatible with code design techniques based on gradient-descent. Experimental results show that our MC2method generates SC codes with remarkably fewer short cycles and substantial gains in error/erasure-rate performance compared with the current state-of-the-art. Moreover, to reach the same number of cycles, our MC2method requires orders of magnitude less overall time compared with the available literature methods.

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.