On a problem of Garcia, Stichtenoth, and Thomas.
A tower \(\cdots\rightarrow C_n\rightarrow C_{n-1} \rightarrow \cdots\rightarrow C_1\rightarrow C_0\) of covers of curves \(C_n/\mathbb F_q\) is asymptotically good if \(\lim \#C_n(\mathbb F_q)/g(C_n)>0\), where \(\#C_n(\mathbb F_q)\) is the number of \(\mathbb F_q\)-rational points of \(C_n\) and \(g(C_n)\) is the genus of \(C_n\). Asymptotically good towers are important in coding theory, but then the curves have to be in explicit form. In this short paper the author answers a question arising from the work of \textit{A. Garcia}, \textit{H. Stichtenoth}, and \textit{M. Thomas} [Finite Fields Appl. 3, 257--274 (1997; Zbl 0946.11029)] on the construction of explicit asymptotically good towers. Garcia et al. obtained a simple explicit example over any non-prime field by applying their general result (Theorem 2.2) which asserts that an asymptotically good tower can be constructed via a polynomial \(f(t)\) with certain properties; they show that such \(f(t)\) exists over any non-prime field, but they cannot find such \(f(t)\) over a prime field. The author shows (Theorem 2) that over a prime field there does not exist any polynomial \(f(t)\) satisfying the conditions of Theorem 2.2 of Garcia et al., so their construction cannot be used to obtain asymptotically good towers over prime fields.
scientific article; zbMATH DE number 1781824
scientific article