Efficient erasure decoding of Reed-Solomon codes

TL;DR

This paper introduces a practical and efficient erasure decoding algorithm for Reed-Solomon codes over binary fields, significantly improving practicality while maintaining asymptotic speed.

Contribution

The paper presents a new decoding algorithm that is both fast and practical, using Walsh transforms, unlike previous asymptotically fast but impractical methods.

Findings

Decoding time is O(q log^2 q) with small constants.

The algorithm is easily implementable.

It outperforms previous methods in practicality.

Abstract

We present a practical algorithm to decode erasures of Reed-Solomon codes over the q elements binary field in O(q \log_2^2 q) time where the constant implied by the O-notation is very small. Asymptotically fast algorithms based on fast polynomial arithmetic were already known, but even if their complexity is similar, they are mostly impractical. By comparison our algorithm uses only a few Walsh transforms and has been easily implemented.