On a tower of Ihara and its limit

Towers of function fields over a fixed finite field have attracted much attention, specially for the connections with Coding Theory and Cryptography (see [TV], [NX], [Z], [GS2] and [GS3]). Ihara was the first to realize that the so-called Hasse-Weil upper bound was weak if the genus of the function field is large with respect to the cardinality of the finite field (see [Iha]). The first explicit tower (i.e., a tower having the function fields in it given by explicit polynomial equations) with an optimal asymptotic behaviour was obtained over square finite fields (see [GS]). Zink has shown the existence of towers over cubic finite fields with an exceptional asymptotic behaviour (see [Z]). The first explicit tower with this exceptional behaviour was obtained by van der Geer and van der Vlugt over the finite field with eight elements (see [GV]). Generalizations of this tower in [GV] were obtained in [BeGS] and [BaGS]. Here we study another tower F0 over cubic finite fields, also generalizing the tower in [GV]. This tower F0 was introduced by Ihara in [Ih] as a subtower of the tower in [BeGS]. A detailed exposition of F0 can be seen in [C] where it is also determined the genera of the function fields of the tower in [BaGS]. Let k be a finite field. A tower F over k is an infinite sequence of function fields Fn over k such that: