Fast Decoding of AG Codes

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.