Faster List Decoding of AG Codes

Abstract

In this article, we present a fast algorithm performing an instance of the Guruswami-Sudan list decoder for algebraic geometry codes. We show that any such code can be decoded in $\tilde {\mathcal {O}} (s^{2}\ell ^{\omega -1}\mu ^{\omega -1}(n+g) + \ell ^{\omega } \mu ^{\omega })$ operations in the underlying finite field, where n is the code length, g is the genus of the function field used to construct the code, s is the multiplicity parameter, $\ell $ is the designed list size and $\mu $ is the smallest positive element in the Weierstrass semigroup of some chosen place.