Sub-quadratic Decoding of One-point Hermitian Codes

TL;DR

This paper introduces the first two decoding algorithms for one-point Hermitian codes with sub-quadratic complexity, utilizing advanced computer algebra techniques for improved efficiency.

Contribution

It presents novel sub-quadratic algorithms for decoding one-point Hermitian codes, including a Guruswami-Sudan based method and a probabilistic Power decoding approach.

Findings

First sub-quadratic decoding algorithms for these codes.

Algorithms leverage state-of-the-art polynomial matrix minimisation.

Achieve similar asymptotic complexities for different decoding strategies.

Abstract

We present the first two sub-quadratic complexity decoding algorithms for one-point Hermitian codes. The first is based on a fast realisation of the Guruswami-Sudan algorithm by using state-of-the-art algorithms from computer algebra for polynomial-ring matrix minimisation. The second is a Power decoding algorithm: an extension of classical key equation decoding which gives a probabilistic decoding algorithm up to the Sudan radius. We show how the resulting key equations can be solved by the same methods from computer algebra, yielding similar asymptotic complexities.