Estimating Weight Enumerators of Reed-Muller Codes via Sampling

Core Concepts

The author develops a sampling-based algorithm to estimate weight enumerators of Reed-Muller codes, providing close approximations to true rates.

Abstract

This paper introduces a novel algorithmic approach for estimating weight enumerators of Reed-Muller (RM) codes using sampling techniques. The method is based on a statistical physics approach and Monte-Carlo Markov Chain (MCMC) sampling. By applying this technique, the authors were able to derive approximate values for the weight enumerators of RM codes, particularly focusing on moderate-blocklength RM codes. The study highlights the importance of these estimates in understanding the weight distribution properties of RM codes and provides theoretical guarantees for the accuracy and efficiency of the proposed algorithm. The research delves into the background and definition of binary Reed-Muller (RM) codes, emphasizing their significance in various applications such as deep-space and cellular communications. It discusses the challenges in understanding the weight enumerators of RM codes and reviews previous works that have contributed to this area. The paper presents a detailed explanation of the proposed algorithm for estimating weight enumerators through sampling techniques, specifically focusing on obtaining estimates for moderate-blocklength RM codes. It describes how a statistical physics approach is utilized to estimate partition functions and generate codewords according to a Gibbs distribution. Furthermore, the study includes numerical examples showcasing comparisons between estimated rates of weight enumerators and true rates for specific RM codes like RM(9, 4) and RM(11, 5). The results demonstrate that the estimates are close to ground truth values, validating the effectiveness of the proposed algorithm. Overall, this research contributes valuable insights into efficiently estimating weight enumerators of Reed-Muller codes using innovative sampling techniques based on statistical physics principles.

Stats

For selected weights with positive weight enumerators: 𝜔 = 512, Rate Estimate: 0.2967884396 For selected weights with positive weight enumerators: 𝜔 = 516, Rate Estimate: 0.3044142654 For selected weights with positive weight enumerators: 𝜔 = 520, Rate Estimate: 0.3098708781 For selected weights with positive weight enumerators: 𝜔 = 524, Rate Estimate: 0.3117907964 For selected weights with positive weight enumerators: 𝜔 = 528, Rate Estimate: 0.3159142454

Quotes

"The crux of this approach is employing a Monte-Carlo Markov Chain (MCMC) sampler that draws codewords according to a suitably biased Gibbs distribution." "Our technique makes use of a simple statistical physics approach for estimating partition functions." "We provide theoretical guarantees on sample complexity for estimating weight enumerator algorithms."

Key Insights Distilled From

Estimating the Weight Enumerators of Reed-Muller Codes via Sampling

by Shreyas Jain... at arxiv.org 03-12-2024

https://arxiv.org/pdf/2403.05893.pdf

Deeper Inquiries

How can these sampling techniques be applied to other coding theory problems

Sampling techniques like the one described in the context can be applied to various other coding theory problems, especially those involving combinatorial counting and estimation. For instance, these techniques can be used to estimate weight distributions of other families of error-correcting codes beyond Reed-Muller codes. They can also be utilized for estimating the sizes of constrained subcodes or finding specific types of codewords within a given code structure. Additionally, sampling-based approaches could help in analyzing the performance and properties of different coding schemes under varying conditions or parameters.

What are potential limitations or drawbacks of using sampling-based approaches in coding theory

While sampling-based approaches offer a promising way to estimate weight enumerators and spectra efficiently, there are some potential limitations and drawbacks associated with their use in coding theory. One limitation is that the accuracy of estimates heavily relies on the choice of parameters such as inverse temperature 𝛽★ and sample size 𝑡. Improper selection may lead to biased estimates or inaccurate results. Another drawback is that sampling methods might struggle with high-dimensional spaces or complex code structures where generating representative samples becomes challenging. Moreover, there could be computational constraints when dealing with very large blocklengths or dimensions due to increased time complexity.

How might advancements in computational power impact the efficiency and scalability of these estimation algorithms

Advancements in computational power have a significant impact on the efficiency and scalability of estimation algorithms based on sampling techniques in coding theory. With more powerful hardware capabilities, it becomes feasible to handle larger datasets, higher-dimensional spaces, and more complex code structures effectively. This leads to faster computation times for generating samples, running simulations, and processing data sets required for estimating weight enumerators accurately. Improved computational power also enables researchers to explore a wider range of parameter values quickly without compromising precision or sacrificing accuracy in their estimations.