Fast Decoding of Multipoint Codes from Algebraic Curves

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 F-q(N) over a finite field F-q, so that those algebraic geometry codes can be decoded efficiently if the dimension N of the affine space, including the defining curve is small.