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

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