List-decoding of AG codes without genus penalty

07/09/2023

∙

by   Peter Beelen, et al.

∙

0

∙

share

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 ℓ, 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 𝒪̃(sℓ^ωμ^ω-1p^e(n+g)) field operations, where n is the code length.