  Up until now, it was recognized that a large number of 2-torsion points was a
technical barrier to improve the bounds for the symmetric tensor rank of
multiplication in every extension of any finite field. In this paper, we show
that there are two exceptional cases, namely the extensions of $\mathbb{F}_2$
and $\mathbb{F}_3$. In particular, using the definition field descent on the
field with 2 or 3 elements of a Garcia-Stichtenoth tower of algebraic function
fields which is asymptotically optimal in the sense of Drinfel'd-Vladut and has
maximal Hasse-Witt invariant, we obtain a significant improvement of the
uniform bounds for the symmetric tensor rank of multiplication in any extension
of $\mathbb{F}_2$ and $\mathbb{F}_3$.
