Fast Decoding of AG Codes

Abstract

We present an efficient list decoding algorithm in the style of Guruswami-Sudan for algebraic geometry codes. Our decoder can decode any such code using $\tilde{\mathcal{O}}(s{\ell }^{\omega }{\mu }^{\omega -1}(n+g))$ 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 at some chosen place; the "soft-O" notation $\tilde{\mathcal{O}}(\cdot )$ is similar to the "big-O" notation $\mathcal{O}(\cdot )$, but ignores logarithmic factors. For the interpolation step, which constitutes the computational bottleneck of our approach, we use known algorithms for univariate polynomial matrices, while the root-finding step is solved using existing algorithms for root-finding over univariate power series.