Fast Decoding of Multipoint Codes from Algebraic Curves

Abstract

Multipoint codes are a broad class of algebraic geometry codes derived from algebraic functions, which have multiple poles and/or zeros on an algebraic curve. Thus, they are more general than one-point codes, which are an important class of algebraic geometry codes in the sense that they can be decoded efficiently using the Berlekamp-Massey-Sakata algorithm. We present a fast method for decoding multipoint codes from a plane curve, particularly a Hermitian curve. Our method with some adaptation can be applied to decode multipoint codes from a general algebraic curve embedded in the N-dimensional affine space Fq <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sup> over a finite field F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> , so that those algebraic geometry codes can be decoded efficiently if the dimension N of the affine space, including the defining curve is small.