Effective bounds on class number and estimation for any step of towers of algebraic function fields over finite fields

We give new effective bounds on the class number of an algebraic function field defined over a finite field. Then we give significant examples of towers of algebraic function fields having a large class number. In particular, we estimate the genus, the number of places and the class number of function fields which are steps of towers having one or several positive Tsfasman–Vlăduţ invariants. Note that the study is not done asymptotically, but for each individual step of the towers for which we determine precise parameters.