Coverings of Curves with Asymptotically many Rational Points, Original Research Article

Ihara defined the quantity A(q), which is the lim sup as g approaches ∞ of the ratio Nq(g)/g, where Nq(g) is the maximum number of rational points a curve of genus g defined over a finite field imageq may have. A(q) is of great relevance for applications to algebraic–geometric codes. It is known that A(q)less-than-or-equals, slant√q−1 and equality holds when q is a square. By constructing class field towers with good parameters, in this paper we present improvements of lower bounds of A(q) for q an odd power of a prime.