Tower of algebraic function fields with maximal Hasse-Witt invariant and tensor rank of multiplication in any extension of $\mathbb{F}_2$ and $\mathbb{F}_3$

TL;DR

This paper improves bounds on the tensor rank of multiplication in extensions of finite fields, specifically for fields with 2 and 3 elements, by leveraging algebraic function fields with maximal Hasse-Witt invariant.

Contribution

It introduces a novel approach using algebraic function fields with maximal Hasse-Witt invariant and field descent to improve bounds for tensor rank in extensions of _2 and _3.

Findings

Significant improvement in bounds for tensor rank in _2 and _3 extensions.

Utilization of Garcia-Stichtenoth towers with maximal Hasse-Witt invariant.

Application of field descent method to optimize bounds.

Abstract

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 $\mathbb{F}_3$.