## View abstract

### 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. document.getElementById('cloakd97529580a03b92cb49560ac5180a5d3').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyd97529580a03b92cb49560ac5180a5d3 = 'r&#105;d&#97;t&#111;l&#101;' + '&#64;'; addyd97529580a03b92cb49560ac5180a5d3 = addyd97529580a03b92cb49560ac5180a5d3 + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_textd97529580a03b92cb49560ac5180a5d3 = 'r&#105;d&#97;t&#111;l&#101;' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloakd97529580a03b92cb49560ac5180a5d3').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyd97529580a03b92cb49560ac5180a5d3 + '\'>'+addy_textd97529580a03b92cb49560ac5180a5d3+'<\/a>';

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.