An Artin-Schreier tower of function fields in even characteristic

M. CHARA; H. NAVARRO OYOLA; R. TOLEDANO

Montreal

Congreso; Mathematical Congress of the Americas 2017; 2017

Let $\mathbb{F}_2$ be a finite field with two elements. In 2006 Beleen, Garcia and Stichtenoth proved that any recursive tower of function fields over $\mathbb{F}_2$, defined by $g(Y ) = f(X)$ with $g(T), f(T) \in {F}_2(T)$ and $\deg f = \deg g = 2$ is given by the Artin-Schreier equation $$Y^2 + Y =\frac{1}{(1/X)^2 + (1/X) + b}+ c$$ with $b, c \in \mathbb{F}_2$. They checked that all the posible cases were already considered in previous works, except when $b = c = 1$. In fact, they left as an open problem to determine whether this tower is asymptotically good or not over $\mathbb{F}_{2^s}$, for any positive integer $s$. In this talk we will discuss the asymptotic behavior of this tower.