  In this paper, we obtain new bounds for the tensor rank of multiplication in
any extension of $\F_2$. In particular, it also enables us to obtain the best
known asymptotic bound. In this aim, we use the generalized algorithm of type
Chudnovsky with derivative evaluations on places of degree one, two and four
applied on the descent over $\F_2$ of a Garcia-Stichtenoth tower of algebraic
function fields defined over $\F_{2^4}$.
