Power Decoding Reed--Solomon Codes Up to the Johnson Radius

TL;DR

This paper extends Power decoding for Reed--Solomon codes to reach the Johnson radius by incorporating multiplicities, resulting in a one-pass, quasi-linear time decoding algorithm that can decode beyond the Sudan radius.

Contribution

It introduces a novel method to decode Reed--Solomon codes up to the Johnson radius using Power decoding with multiplicities, improving decoding capabilities.

Findings

Decoding up to the Johnson radius is feasible with the new method.

The algorithm is a one-pass, quasi-linear time process.

Simulation results show the decoding failure behavior.

Abstract

Power decoding, or "decoding using virtual interleaving" is a technique for decoding Reed--Solomon codes up to the Sudan radius. Since the method's inception, it has been an open question if it is possible to use this approach to decode up to the Johnson radius -- the decoding radius of the Guruswami--Sudan algorithm. In this paper we show that this can be done by incorporating a notion of multiplicities. As the original Power decoding, the proposed algorithm is a one-pass algorithm: decoding follows immediately from solving a shift-register type equation, which we show can be done in quasi-linear time. It is a "partial bounded-distance decoding algorithm" since it will fail to return a codeword for a few error patterns within its decoding radius; we investigate its failure behaviour theoretically as well as give simulation results. This is an extended version where we also show how the method can be made faster using the re-encoding technique or a syndrome formulation.