A Note on Optimal Towers over Finite Fields

Abstract

Recently, under the influence of Elkies' conjecture [2], the optimal recursive towers of algebraic function fields (of one variable) over the finite fields with square cardinality are studied [3, 5, 6]. In this paper, we define the limit $$ \lim_{i \to \infty} \text{(the number of places of degree $n$ in $F_{i}/\mathbf{F}_{q}$)/(genus of $F_{i}$)} $$ of a tower $F_{0} \subseteq F_{1} \subseteq F_{2} \subseteq \cdots$ over the finite field $\F_{q}$. Using this limit, we prove that all the proper constant field extensions of all the optimal towers over the finite fields with square cardinality are not optimal, and we show a simple criterion whether a tower is optimal or not. Moreover, we give many new recursive towers of finite ramification type.