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


Abstract: Let $\FF=(F_n)_{n\geq 0}$ 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 $\beta_r(\FF):=\lim_{n\to \infty} B_r(F_n)/g(F_n)$, where $B_r(F_n)$ is the number of places of degree $r$ and $g(F_n)$ is the genus, respectively, of $F_n/\F_2$. In this paper we give a classification of all rational functions $f(X,Y)\in \F_2(X,Y)$ that may define a quadratic recursive tower $\FF$ over $\F_2$ with at least one positive invariant $\beta_r(\FF)$. Moreover, we estimate $\beta_1(\FF)$ for each such tower.

Keywords: Towers of algebraic function fields, genus, number of places

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