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

2025-04-22

AI Summary

Designing finite-length protograph-based LDPC (PB-LDPC) codes faces two key challenges: the persistent presence of short cycles and the lack of effective methods for jointly optimizing spatial coupling (SC), lifting, and multidimensional edge reordering. Method: This paper introduces Markov Chain Monte Carlo (MCMC) into this joint optimization framework—proposing the MC² method—which employs probabilistic sampling to guide the search toward low-short-cycle configurations. We theoretically prove chain stationarity, establish a Gaussian statistical model for short-cycle counts, derive an upper bound on iterative decoding complexity, and construct a hybrid optimization framework compatible with gradient descent. Results: Experimental results show that the generated SC-LDPC codes achieve significantly fewer short cycles than state-of-the-art designs; at equivalent bit-error-rate performance, runtime is reduced by two to three orders of magnitude—effectively balancing structural degrees of freedom and practical deployment efficiency.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Security and Privacy: Large-scale security measurements

Abstract

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.

Problem

• Optimizing finite-length LDPC code design efficiently
• Removing short cycles in SC and MD LDPC codes
• Reducing computation time for cycle removal in codes

Innovation

• MCMC method for FL LDPC code design
• Removes short cycles via probability sampling
• Compatible with gradient-descent techniques