Families of codes exceeding the Varshamov-Gilbert bound

Abstract

The number A(q) is the superior limit of the maximum number of points of an algebraic curve defined over the finite field with q elements, divided by the genus. It has been shown by J.-P. Serre that A(q)≥c logq, where c is a positive constant. His method, based on the existence of infinite towers of Hilbert-class fields, can give better results ; we give here some new lower bounds for A(q) for certain values of q, and we deduce from these some new values of q for which there exists families of codes defined over F_q, exceeding the Varshamov-Gilbert bound.