On cubic Kummer towers of Garcia, Stichtenoth and Thomas type

MARÍA CHARA; RICARDO TOLEDANO

Maringá

Congreso; Escola de Algebra; 2014

Universidade Estadual de Maringá

It is well known the importance of asymptotically good recursive towers in coding theory and some other branches of information theory. Among the class of recursive towers there is an important one, namely the class of Kummer type towers which are recursively defined by equations of the form $y^m=f(x)$ for some suitable exponent $m$ and rational function $f(x)\in K(x)$. A particular case was studied by Garcia, Stichtenoth and Thomas in \cite{GST97} where a Kummer tower over a finite field $\ff_q$ with $q\equiv 1\mod m$ is recursively defined by an equation of the form \begin{equation}\label{kummergst97} y^m=x^df(x)\,, \end{equation} where $f(x)$ is a polynomial of degree $m-d$ such that $f(0)\neq 0$, $\gcd(d,m)=1$ and its leading coefficient is an mth-power in $\ff_q$. The authors showed that they have positive splitting rate and, assuming the existence of a subset $S_0$ of $\ff_q$ with certain properties, the good asymptotic behavior of such towers can be deduced together with a concrete non trivial lower bound for their limit. Later Lenstra showed in \cite{Le02} that in the case of an equation of the form \eqref{kummergst97} over a prime field, there is not such a set $S_0$ satisfying the above conditions of Garcia, Stichtenoth and Thomas. Because of Lenstra's result it seems reasonable to expect that many Kummer towers defined by equations of the form \eqref{kummergst97} have infinite genus. However, to the best of our knowledge there are not examples of such towers in the literature. The aim of this talk is to classify those asymptotically good Kummer type towers considered by Garcia, Stichtenoth and Thomas in \cite{GST97} recursively defined by an equation of the form \begin{equation}\label{cubicgst} y^3=x^df(x)\,, \end{equation} over a finite field $\ff_q$ where $q\equiv 1\mod 3$, $d=1,2$ and $f(t)\in\ff_q[t]$ is a polynomial whose leading coefficient is a cubic power in $\ff_q$. It was shown in \cite{GST97} that in the case $d=1$ there are choices of the polynomial $f$ giving good asymptotic behavior and even optimal behavior. For instance if $f(x)=x^2+x+1$ then the equation \eqref{cubicgst} defines an optimal tower over $\ff_4$, a finite field with four elements (see \cite[Example 2.3]{GST97}). The quadratic case (i.e $m=2$ in \eqref{kummergst97}) is already included in the extensive computational search of good quadratic tame towers performed in \cite{MaWu05}.