Towers of Global Function Fields with Asymptotically Many Rational Places and an Improvement on the Gilbert ‐ Varshamov Bound
Harald Niederreiter, Chaoping Xing · Mathematische Nachrichten · 1998
Abstract We construct infinite class field towers of global function fields with asymptotically many rational places. In this way, we improve on asymptotic bounds of Serre, Perret, Schoof, and Xing. The results can be interpreted equivalently as asymptotic bounds on the number of rational points of smooth algebraic curves over finite fields. As an application, we show an improvement on the Gilbert‐Varshamov bound for linear codes over finite fields of a sufficiently large composite nonsquare order.