A formula for constructing curves over finite fields with many rational points (Q1284199)

In previous work [Lect. Notes Comput. Sci. 1122, 187--195 (1996; Zbl 0935.11022)], the author extended ideas of Serre and used class field theory to construct curves over finite fields with many rational points. Here, the author gives a formula for computing the order of the Galois group of an abelian extension of \(\mathbb F_q(T)\) in which all but one of the places of degree one are split. The proof involves transforming the multiplicative structure of this group into an additive one using the additive group of the ring of generalized Witt vectors. Tables are given for different values of \(q\) and \(k\) of the genus and number of rational points on the curve obtained by splitting all \(q\) rational points of the projective line over \(F_q\) different from a given one \(P\) in the ray class field of conductor \(kP\). Related results may be found in the work of \textit{C. P. Xing} and \textit{H. Niederreiter} [C. R. Acad. Sci., Paris, Sér. I, Math. 322, 651--654 (1996; Zbl 0853.11051)].