Coverings of curves with asymptotically many rational points

Ihara defined the quantity A(q), which is the lim sup as g approaches ∞ of the ratio N_q(g)/g, where N_q(g) is the maximum number of rational points a curve of genus g defined over a finite field F_q may have. A(q) is of great relevance for applications to algebraic-geometric codes. It is known that A(q) ≤ 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.