  Let X be a curve over the finite field of q elements and let N(X), g(X) be
its number of rational points and genus respectively. The Ihara constant A(q)
is defined by the limit superior of N(X)/g(X) as the genus of X goes to
infinity. In this paper, we employ a variant of Serre's class field tower
method to obtain an improvement of the best known lower bounds on A(2) and
A(3).
