Galois towers over non-prime finite fields
Volume 164 / 2014
Acta Arithmetica 164 (2014), 163-179
MSC: 11R32, 11R58, 11G20, 11G09.
DOI: 10.4064/aa164-2-6
Abstract
We construct Galois towers with good asymptotic properties over any non-prime finite field $\mathbb F_{\ell }$; that is, we construct sequences of function fields $\mathcal {N}=(N_1 \subset N_2 \subset \cdots )$ over $\mathbb F_{\ell }$ of increasing genus, such that all the extensions $N_i/N_1$ are Galois extensions and the number of rational places of these function fields grows linearly with the genus. The limits of the towers satisfy the same lower bounds as the best currently known lower bounds for the Ihara constant for non-prime finite fields. Towers with these properties are important for applications in various fields including coding theory and cryptography.
