An upper bound for the Garcia-Stichtenoth numbers of towers (Q819515)

Let \(F = F/\mathbb{F}_q\) be an algebraic function field in one variable over finite field \(\mathbb{F}_q\), briefly function field. Let \(g(F)\) be the genus of \(F\), and \(N(F)\) of \(\mathbb{F}_q\)-rational places. The author studies the towers \(\mathcal{F} = (F_0 , F_1, F_2, \ldots, F_m, \ldots)\) of function fields where \(F_{m+1}/F_m\) is separable extension of positive degree, and for some \(m \geq 0\), \(F_m/\mathbb{F}_q\) is non-rational and non-elliptic. \textit{A. Garcia} and \textit{H. Stichtenoth} [J. Number Theory 61, No. 2, 248--273 (1996; Zbl 0893.11047)] proved that the limit \(\lambda(\mathcal{F}) = \lambda(\mathcal{F}/\mathbb{F}_q) = \lim_{m \to \infty} N(F_m)/g(F_m)\) exists; it is called Garcia-Stichtenoth number. The author obtains the upper bound for Garcia-Stichtenoth numbers, computes this number for several towers of function fields, and examines several examples, among them the examples of optimal towers.