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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. [ABSTRACT FROM AUTHOR]

We investigate algorithms for encoding of one-point algebraic geometry (AG) codes over certain plane curves called $C_{ab}$ curves, as well as algorithms for inverting the encoding map, which we call “unencoding”. Some $C_{ab}$ curves have many points or are even maximal, e.g. the Hermitian curve. Our encoding resp. unencoding algorithms have complexity $ \tilde { \mathcal {O}}(\text {n}^{3/2})$ resp. $ \tilde { \mathcal {O}}({\it\text { qn}})$ for AG codes over any $C_{ab}$ curve satisfying very mild assumptions, where n is the code length and q the base field size, and $ \tilde { \mathcal {O}}$ ignores constants and logarithmic factors in the estimate. For codes over curves whose evaluation points lie on a grid-like structure, for example the Hermitian curve and norm-trace curves, we show that our algorithms have quasi-linear time complexity $ \tilde { \mathcal {O}}(\text {n})$ for both operations. For infinite families of curves whose number of points is a constant factor away from the Hasse-Weil bound, our encoding and unencoding algorithms have complexities $ \tilde { \mathcal {O}}(\text {n}^{5/4})$ and $ \tilde { \mathcal {O}}(\text {n}^{3/2})$ respectively. [ABSTRACT FROM AUTHOR]

We present the first two sub-quadratic complexity decoding algorithms for one-point Hermitian codes. The first is based on a fast realization of the Guruswami–Sudan algorithm using state-of-the-art algorithms from computer algebra for polynomial-ring matrix minimization. The second is a power decoding algorithm: an extension of classical key equation decoding which gives a probabilistic decoding algorithm up to the Sudan radius. We show how the resulting key equations can be solved by the matrix minimization algorithms from computer algebra, yielding similar asymptotic complexities. [ABSTRACT FROM AUTHOR]

We derive the Wu list-decoding algorithm for generalized Reed–Solomon (GRS) codes by using Gröbner bases over modules and the Euclidean algorithm as the initial algorithm instead of the Berlekamp–Massey algorithm. We present a novel method for constructing the interpolation polynomial fast. We give a new application of the Wu list decoder by decoding irreducible binary Goppa codes up to the binary Johnson radius. Finally, we point out a connection between the governing equations of the Wu algorithm and the Guruswami–Sudan algorithm, immediately leading to equality in the decoding range and a duality in the choice of parameters needed for decoding, both in the case of GRS codes and in the case of Goppa codes. [ABSTRACT FROM PUBLISHER]