In this paper, we consider a tower of function fields [Formula: see text] over a finite field 𝔽q and a finite extension E/F0 such that the sequence [Formula: see text] is a tower over the field 𝔽q. Then we study invariants of [Formula: see text], that is, the asymptotic number of the places of degree r in [Formula: see text], for any r ≥ 1, if those of [Formula: see text] are known. We first give a method for constructing towers of function fields over any finite field 𝔽q with finitely many prescribed invariants being positive. For q a square, we prove that with the same method one can also construct towers with at least one positive invariant and certain prescribed invariants being zero. Our method is based on explicit extensions. Moreover, we show the existence of towers over a finite field 𝔽q attaining the Drinfeld–Vladut bound of order r, for any r ≥ 1 with qr a square (see [1, Problem-2]). Finally, we give some examples of non-optimal recursive towers with all but one invariants equal to zero.
On the splitting rate of rate of a tower of Artin-Schreier type
