Tower of algebraic function fields with maximal Hasse-Witt invariant and tensor rank of multiplication in any extension of \mathbbF₂ and \mathbbF₃

2014/09/11 by Stéphane Ballet, Ballet, Stéphane, Julia Pieltant +1
Computer Science · Mathematics · #11Y16 #12E20 #14H05 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #cs.IT #math.AG #math.IT #msc:11Y16 #msc:12E20 #msc:14H05

paper · pdf · doi:10.48550/arxiv.1409.3440

arxiv created 2014/09/11 · openalex publication_date 2014/09/11 · arxiv updated 2014/09/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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 \mathbbF₂ and \mathbbF₃. 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 \mathbbF₂ and \mathbbF₃.

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