New uniform and asymptotic upper bounds on the tensor rank of multiplication in extensions of finite fields
Julia Pieltant, Hugues Randriam · Mathematics of Computation · 2015
We obtain new uniform upper bounds for the tensor rank of the multiplication in the extensions of the finite fields F q \mathbb {F}_q for any prime power q q ; moreover, these uniform bounds lead to new asymptotic bounds as well. In addition, we also give purely asymptotic bounds which are substantially better by using a family of Shimura curves defined over F q \mathbb {F}_q , with an optimal ratio of F q t \mathbb {F}_{q^t} -rational places to their genus, where q t q^t is a square.