Syndrome Decoding of Reed-Solomon Codes Beyond Half the Minimum Distance based on Shift-Register Synthesis

TL;DR

This paper introduces a new, simple shift-register based decoding method for low-rate Reed-Solomon codes that surpasses half the minimum distance, matching the Sudan algorithm's error correction capabilities with similar performance and complexity.

Contribution

It presents a novel decoding approach using multi-sequence shift-register synthesis that is easier to understand and implement than previous algorithms, while maintaining similar performance.

Findings

Decoding radius matches Sudan algorithm's radius

Complexity comparable to Berlekamp-Massey algorithm

Performance on QSC nearly identical to Sudan algorithm

Abstract

In this paper, a new approach for decoding low-rate Reed-Solomon codes beyond half the minimum distance is considered and analyzed. Unlike the Sudan algorithm published in 1997, this new approach is based on multi-sequence shift-register synthesis, which makes it easy to understand and simple to implement. The computational complexity of this shift-register based algorithm is of the same order as the complexity of the well-known Berlekamp-Massey algorithm. Moreover, the error correcting radius coincides with the error correcting radius of the original Sudan algorithm, and the practical decoding performance observed on a q-ary symmetric channel (QSC) is virtually identical to the decoding performance of the Sudan algorithm. Bounds for the failure and error probability as well as for the QSC decoding performance of the new algorithm are derived, and the performance is illustrated by means of examples.