Quadratic recursive towers of function fields over $\mathbb{F}_2$

Authors : HENNING STICHTENOTH, SEHER TUTDERE

Pages : 665-682

View : 9 | Download : 6

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :Let $\FF=insert ignore into journalissuearticles values(F_n);_{n\geq 0}$ be a quadratic recursive tower of algebraic function fields over the finite field $\F_2$, i.e. $\FF$ is a recursive tower such that $[F_n:F_{n-1}]=2$ for all $n\geq 1$. For any integer $r\geq 1$, let $\beta_rinsert ignore into journalissuearticles values(\FF);:=\lim_{n\to \infty} B_rinsert ignore into journalissuearticles values(F_n);/ginsert ignore into journalissuearticles values(F_n);$, where $B_rinsert ignore into journalissuearticles values(F_n);$ is the number of places of degree $r$ and $ginsert ignore into journalissuearticles values(F_n);$ is the genus, respectively, of $F_n/\F_2$. In this paper we give a classification of all rational functions $finsert ignore into journalissuearticles values(X,Y);\in \F_2insert ignore into journalissuearticles values(X,Y);$ that may define a quadratic recursive tower $\FF$ over $\F_2$ with at least one positive invariant $\beta_rinsert ignore into journalissuearticles values(\FF);$. Moreover, we estimate $\beta_1insert ignore into journalissuearticles values(\FF);$ for each such tower.
Keywords : Towers of algebraic function fields, genus, number of places