Notes on Goppa Codes

Ideas from algebraic geometry became useful in coding theory after Goppa’s construction [7]. He had the beautiful idea of associating to a (projective, geometrically irreducible, non-singular, algebraic) curve X defined over Fq, the finite field with q elements, a code C. This code is constructed from two divisors D and G on X , where one of them, say D, is the sum of n distinct Fq-rational points of X . It turns out that the minimum distance d of C satisfies d ≥ n− deg(G) .