| Denote by A(q) the upper limit of the ratio of the maximum number of points of an algebraic curve (smooth, projective, absolutely irreducible) defined over the finite field Fq to the genus. Vluaduct and Drinfeld established the bound Aq≤q -1 . When q is a square, it is known that Aq=q -1 . When q is odd, the exact value of A( q) is unknown. There are general lower bounds by, among others, Niederreiter and Xing, and Serre and Temkine which complement each other. The bulk of the thesis is devoted to improving these bounds. These general bounds are also an important contribution to coding theory because they imply improvements of the Gilbert-Varshamov bound which is an asymptotic bound commonly used to measure the performance of long error-correcting codes.;In chapter 1 we give the necessary basic background and motivation together with a summary of the main results. In chapter 2 we prove a necessary and sufficient condition for tower of function fields to be asymptotically good. This improves a previous result of Garcia and Stichtenoth. In chapter 3 improvements of various lower bounds of A(q) are presented. Finally, in chapter 4 we give lower bounds for A( p) for small primes p. |
