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

Let F = (F-n)(n >= 0) be a quadratic recursive tower of algebraic function fields over the finite field F-2 i.e. F is a recursive tower such that [F-n : Fn-l] = 2 for all n >= 1. For any integer r >= 1, let beta(r)(F) := lim(n ->infinity)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) is an element of F-2 (X, Y) that may define a quadratic recursive tower F over F-2 with at least one positive invariant beta(r)(F). Moreover, we estimate beta(1)(F)for each such tower