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

2026-03-26

AI Summary

This work proposes the first gradient descent–based probabilistic design framework, termed GD-MD, for multidimensional spatially coupled LDPC codes. Addressing the challenge of high design complexity and the difficulty in effectively suppressing harmful structures in graphical models due to excessive degrees of freedom, GD-MD optimizes a probability distribution matrix to minimize the occurrence of such detrimental structures. The framework integrates a Markov Chain Monte Carlo (MCMC) finite-length optimizer to generate high-performance codewords. Notably, it introduces gradient descent into the design of multidimensional spatially coupled codes for the first time, enabling fine-grained optimization ranging from short cycles to complex cycle-cascade configurations. Experimental results demonstrate that GD-MD significantly reduces targeted harmful structures and achieves markedly superior bit error rate performance compared to existing designs based on uniform distributions.

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.

Problem

• spatially-coupled codes
• multi-dimensional
• degrees of freedom
• detrimental objects
• LDPC codes

Innovation

• gradient descent
• multi-dimensional spatially-coupled codes
• probabilistic design framework
• detrimental object minimization
• Markov chain Monte Carlo