Goppa codes: Algorithmic problems

Abstract

Consider linear codes C ~ with dim=k, wgt=d, rate R=~n ' relative minimum distance 6=~ . An important relation between the parameters of n C is given by the well-kmown Gilbert-Var~amov lower bound, i.e. with ~q(X) = xlogq(X-1) xlogqX (l-x) logq(1-x) (x ~(0,I)), ~q(0) = ~q(1) 0 one has : for [n,k,d]-codes C there holds I R~ ~q(6) as n ~ (asymptotically not less). An infinite family of codes will be very good if its members actually satisfy the estimate sharpened to strictly greater. V.D. Goppa, followed meanwhile by several authors, applied a combination of coding theory and algebro-geometric concepts which supplied a conceptual proof of the existence of such families (work of Goppa; Tsfasman, Vladut, Zink; cp. also Manin, Hirschfeld). The basic idea is to use Riemann-Roch's Theorem. Namely, let XI~q be a (smooth) projective curve, genus = g . Let Pl,...,Pn be ~ -rational points, let D = Zp ; and let G be an ~ -rational diviq q sor disjoint from D with deg G = a and 2g 2 < a < n+g-1. The ~ -q