No abstract

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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 E := E·F = (EF n ) n≥0 is a tower over the field F q . Then we deal with the following: What can we say about the invariants of E; i.e., the asymptotic number of the places of degree r for any r ≥ 1 in E, if those of F are known? We give a method based on explicit extensions for constructing towers of function fields over F q with finitely many prescribed invariants being positive, and towers of function fields over F q , for q a square, with at least one positive invariant and certain prescribed invariants being zero. We show the existence of recursive towers attaining the Drinfeld-Vladut bound of order r, for any r ≥ 1 with q r a square, see [1,. Moreover, we give some examples of recursive towers with all but one invariants equal to zero.

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In 2019, Kavut and Tutdere proved that the covering radii of a class of primitive binary cyclic codes with minimum distance greater than or equal to $r+2$ is $r$, where $r$ is an odd integer, under some assumptions. We here show that the covering radii $R$ of a class of primitive binary cyclic codes with minimum distance strictly greater than $\ell$ satisfy $r\leq R \leq \ell$, where $\ell,r$ are some integers, with $\ell$ being odd, depending on the given code. This new class of cyclic codes covers that of Kavut and Tutdere.