Minimum Distance Decoding of General Algebraic Geometry Codes via Lists

Algebraic geometry codes are defined by divisors D and G on a curve over a finite field F. Often, G is supported by a single F-rational point and the resulting code is called a one-point code. Recently, there has been interest in allowing the divisor G to be more general as this can result in superior codes. In particular, one may obtain a code with better parameters by allowing G to be supported by m distinct F-rational points, where m > 1. In this paper, we demonstrate that a multipoint algebraic geometry code C may be embedded in a one-point code C´. Exploiting this fact, we obtain a minimum distance decoding algorithm for the multipoint code C. This is accomplished via list decoding in the one-point code C´.