Tower of algebraic function fields with maximal Hasse-Witt invariant and tensor rank of multiplication in any extension of F2 and F3

Up until now, it was recognized that a detailed study of the p-rank in towers of function fields is relevant for their applications in coding theory and cryptography. In particular, it appears that having a large p-rank may be a barrier for a tower to lead to competitive bounds for the symmetric tensor rank of multiplication in every extension of the finite field Fq, with q a power of p. In this paper, we show that there are two exceptional cases, namely the extensions of F2 and F3. 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-Vlăduţ 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 F2 and F3.