On invariants of towers of function fields over finite fields

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Official URL: http://dx.doi.org/10.1142/S0219498812501903

In this paper, we consider a tower of function fields F = (F-n)(n >= 0) over a finite field F-q and a finite extension E/F-0 such that the sequence epsilon := E . F = (EFn)(n >= 0) is a tower over the field F-q. Then we study invariants of epsilon, that is, the asymptotic number of the places of degree r in epsilon, for any r >= 1, if those of F are known. We first give a method for constructing towers of function fields over any finite field F-q 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 F-q attaining the Drinfeld-Vladut bound of order r, for any r >= 1 with q(r) a square (see [1, Problem-2]). Finally, we give some examples of non-optimal recursive towers with all but one invariants equal to zero.