On the tensor rank of multiplication in any extension of $\F_2$

TL;DR

This paper establishes improved bounds on the tensor rank of multiplication in extensions of _2, leveraging advanced algebraic function field techniques to achieve the best known asymptotic bounds.

Contribution

It introduces a generalized Chudnovsky algorithm with derivative evaluations on specific places, applied to a Garcia-Stichtenoth tower over _4, to improve tensor rank bounds.

Findings

Derived new upper bounds for tensor rank in _2 extensions.

Achieved the best known asymptotic bounds for tensor rank.

Utilized algebraic function field towers and derivative evaluations effectively.

Abstract

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 $\F_{2^4}$.