Improved Power Decoding of Interleaved One-Point Hermitian Codes

TL;DR

This paper introduces a new partial decoding algorithm for interleaved one-point Hermitian codes that achieves near-maximal decoding radius with high probability, improving upon previous methods and avoiding complex root-finding steps.

Contribution

The paper presents a novel decoding algorithm for interleaved Hermitian codes that extends improved power decoding techniques without requiring root-finding, surpassing previous decoding radii.

Findings

Achieves decoding error correction up to 1-( (k+g)/n )^{h/(h+1)} with high probability.

Outperforms previous best decoding radius by Kampf at all code rates.

For h=1, matches Guruswami-Sudan decoder's failure probability above its guaranteed radius.

Abstract

We propose a new partial decoding algorithm for $h$-interleaved one-point Hermitian codes that can decode-under certain assumptions-an error of relative weight up to $1-(\tfrac{k+g}{n})^{\frac{h}{h+1}}$, where $k$ is the dimension, $n$ the length, and $g$ the genus of the code. Simulation results for various parameters indicate that the new decoder achieves this maximal decoding radius with high probability. The algorithm is based on a recent generalization of Rosenkilde's improved power decoder to interleaved Reed-Solomon codes, does not require an expensive root-finding step, and improves upon the previous best decoding radius by Kampf at all rates. In the special case $h=1$, we obtain an adaption of the improved power decoding algorithm to one-point Hermitian codes, which for all simulated parameters achieves a similar observed failure probability as the Guruswami-Sudan decoder above the latter's guaranteed decoding radius.