List-decoding of AG codes without genus penalty

TL;DR

This paper introduces an improved list-decoding algorithm for algebraic geometry codes that reduces the negative impact of the genus penalty, enabling more efficient decoding without sacrificing performance.

Contribution

It presents an inseparable GS list-decoder that diminishes the genus penalty by a factor depending on the inseparability exponent, extending decoding capabilities for AG codes.

Findings

Reduces genus penalty by a factor of 1/p^e

Achieves list-decoding with complexity a0(s\u03bb^{}^{}^e(n+g))

Applicable to arbitrary AG codes

Abstract

In this paper we consider algebraic geometry (AG) codes: a class of codes constructed from algebraic codes (equivalently, using function fields) by Goppa. These codes can be list-decoded using the famous Guruswami-Sudan (GS) list-decoder, but the genus $g$ of the used function field gives rise to negative term in the decoding radius, which we call the genus penalty. In this article, we present a GS-like list-decoding algorithm for arbitrary AG codes, which we call the \emph{inseparable GS list-decoder}. Apart from the multiplicity parameter $s$ and designed list size $\ell$, common for the GS list-decoder, we introduce an inseparability exponent $e$. Choosing this exponent to be positive gives rise to a list-decoder for which the genus penalty is reduced with a factor $1/p^e$ compared to the usual GS list-decoder. Here $p$ is the characteristic. Our list-decoder can be executed in $\tilde{\mathcal{O}}(s\ell^{\omega}\mu^{\omega-1}p^e(n+g))$ field operations, where $n$ is the code length.