On Multiple Decoding Attempts for Reed-Solomon Codes: A Rate-Distortion Approach

TL;DR

This paper introduces a rate-distortion theory framework to analyze and optimize multiple decoding attempts of Reed-Solomon codes, improving decoding success rates and understanding performance-complexity trade-offs.

Contribution

It applies rate-distortion theory to model and analyze multiple errors-and-erasures decoding of RS codes, providing a novel analytical approach and design methodology.

Findings

Rate-distortion framework effectively models decoding success conditions.

Designed erasure patterns outperform existing algorithms in simulations.

Computed rate-distortion exponents for moderate blocklengths.

Abstract

One popular approach to soft-decision decoding of Reed-Solomon (RS) codes is based on using multiple trials of a simple RS decoding algorithm in combination with erasing or flipping a set of symbols or bits in each trial. This paper presents a framework based on rate-distortion (RD) theory to analyze these multiple-decoding algorithms. By defining an appropriate distortion measure between an error pattern and an erasure pattern, the successful decoding condition, for a single errors-and-erasures decoding trial, becomes equivalent to distortion being less than a fixed threshold. Finding the best set of erasure patterns also turns into a covering problem which can be solved asymptotically by rate-distortion theory. Thus, the proposed approach can be used to understand the asymptotic performance-versus-complexity trade-off of multiple errors-and-erasures decoding of RS codes. This initial result is also extended a few directions. The rate-distortion exponent (RDE) is computed to give more precise results for moderate blocklengths. Multiple trials of algebraic soft-decision (ASD) decoding are analyzed using this framework. Analytical and numerical computations of the RD and RDE functions are also presented. Finally, simulation results show that sets of erasure patterns designed using the proposed methods outperform other algorithms with the same number of decoding trials.