Optimal Threshold-Based Multi-Trial Error/Erasure Decoding with the Guruswami-Sudan Algorithm

TL;DR

This paper develops optimal threshold-based multi-trial decoding strategies for Reed-Solomon codes using the Guruswami-Sudan algorithm, balancing error and erasure tradeoffs to minimize residual errors.

Contribution

It provides formulae for optimal exploitation of erasure options in decoders with arbitrary error/erasure tradeoff factors, improving decoding efficiency.

Findings

BMD decoders with more trials can outperform GS decoders with fewer trials in residual error probability.

Optimal threshold strategies significantly reduce residual errors compared to non-optimized approaches.

BMD decoders are computationally more efficient than GS decoders for similar error performance.

Abstract

Traditionally, multi-trial error/erasure decoding of Reed-Solomon (RS) codes is based on Bounded Minimum Distance (BMD) decoders with an erasure option. Such decoders have error/erasure tradeoff factor L=2, which means that an error is twice as expensive as an erasure in terms of the code's minimum distance. The Guruswami-Sudan (GS) list decoder can be considered as state of the art in algebraic decoding of RS codes. Besides an erasure option, it allows to adjust L to values in the range 1<L<=2. Based on previous work, we provide formulae which allow to optimally (in terms of residual codeword error probability) exploit the erasure option of decoders with arbitrary L, if the decoder can be used z>=1 times. We show that BMD decoders with z_BMD decoding trials can result in lower residual codeword error probability than GS decoders with z_GS trials, if z_BMD is only slightly larger than z_GS. This is of practical interest since BMD decoders generally have lower computational complexity than GS decoders.