A problem of Beelen, Garcia and Stichtenoth on an Artin-Schreier tower in characteristic two

TL;DR

This paper investigates the asymptotic properties of an Artin-Schreier tower over finite fields with characteristic two, resolving an open problem by determining conditions for its goodness or badness.

Contribution

It establishes the asymptotic behavior of the tower for even and odd s, answering an open question from 2006.

Findings

The tower is asymptotically good for even s.

The tower is asymptotically bad for odd s.

Provides new insights into Artin-Schreier towers in characteristic two.

Abstract

We study a tower of function fields of Artin-Schreier type over a finite field with $2^s$ elements. The study of the asymptotic behavior of this tower was left as an open problem by Beelen, Garc\'ia and Stichtenoth in $2006$. We prove that this tower is asymptotically good for $s$ even and asymptotically bad for $s$ odd.