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Session S26 - Finite fields and applications

Thursday, July 15, 14:00 ~ 14:20 UTC-3

Minimal value set polynomials and towers of Garcia, Stichtenoth and Thomas

Ricardo Toledano

Universidad Nacional del Litoral, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We introduce the notions of $S_\phi$-polynomial and $S$-minimal value set polynomial where $\phi$ is a polynomial over a finite field $\mathbb{F}_q$ and $S$ is a finite subset of an algebraic closure of $\mathbb{F}_q$. We will show that the polynomials used by Garcia, Stichtenoth and Thomas in their work on good recursive tame towers are $S_\phi$-minimal value set polynomials for $\phi=x^m$, whose $S$-value sets can be explicitly computed in terms of the monomial $x^m$. As a consequence of these results we will show that the tower recursively defined by the equation $y^2=2x(x-1)$ over $\mathbb{F}_q$ with $q=3^n\geq 9$, is the only one in the whole family having limit bigger than one.

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