Exploiting the Degrees of Freedom: Multi-Dimensional Spatially-Coupled Codes Based on Gradient Descent

Spatially-coupled (SC) codes are a class of low-density parity-check (LDPC) 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 storage and 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, based on the gradient-descent (GD) algorithm, to design high performance MD codes where this challenge is addressed. In particular, we express the expected number of detrimental objects, 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 the finite-length (FL) algorithmic optimizer that produces the final MD-SC code. We adopt a recently-introduced Markov chain Monte Carlo (MCMC) FL algorithmic optimizer that is guided by the proposed GD algorithm. We apply our framework to various objects of interest. We start from simple short cycles, and then we develop the framework to address more sophisticated cycle concatenations, aiming at finer-grained optimization. We offer the theoretical analysis as well as the design algorithms. Next, we present experimental results demonstrating that our MD codes, conveniently called GD-MD codes, have notably lower numbers of targeted detrimental objects compared with the available state-of-the-art. Moreover, we show that our GD-MD codes exhibit significant improvements in error-rate performance compared with MD-SC codes obtained by a uniform distribution.

A Markov Chain Monte Carlo Method for Efficient Finite-Length LDPC Code Design

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.