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

TL;DR

This paper introduces a gradient descent-based probabilistic framework for designing multi-dimensional spatially-coupled LDPC codes, leading to codes with fewer detrimental objects and better error-rate performance.

Contribution

It proposes a novel gradient descent approach to optimize the design of high-performance MD-SC codes, addressing complexity challenges and improving code quality.

Findings

GD-MD codes have fewer detrimental objects than state-of-the-art.

GD-MD codes show significant error-rate performance improvements.

The framework effectively addresses complex cycle concatenations.

Abstract

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.