List-Decoding of AG Codes Without Genus Penalty

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 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 and $\tilde {\mathcal {O}} $ means that logarithmic factors are ignored.