Notes on Algebraic-Geometric Codes

摘要

Ideas from algebraic geometry became useful in coding theory after Goppa's construction. He had the beautiful idea of associating to a curve X defined over F (sub g) the finite field with q elements, a code C. This code, called Algebraic-Geometric (AG) code, is constructed from two divisors D and G on X, where one of them, say D, is the sum of n distinct F (sub q)-rational points of X. It turns out that the minimum distance d of C satisfies d is greater than or equal to n-deg. (G). These notes are based on a series of lectures given in May 2003 at the Mathematical Department of KTH in Stockholm. Contents: (1) Linear codes; (2) Reed-Solomon codes; (3) Algebraic curves; (4) Algebraic-Geometric codes; (5) Bounds on linear codes; (6) One-point AG codes; and (7) MDS codes and Almost MDS codes.