Fast Decoding of AG Codes

TL;DR

This paper introduces a fast list decoding algorithm for algebraic geometry codes, significantly improving decoding efficiency by leveraging advanced polynomial matrix and power series algorithms.

Contribution

It presents a novel, efficient list decoding method for AG codes with complexity analysis and practical algorithmic improvements.

Findings

Decodes any AG code with near-linear complexity in code length

Uses polynomial matrix algorithms for interpolation step

Employs root-finding algorithms over power series for decoding

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.