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

Stéphane Ballet (1) , Julia Pieltant (2, 3)

1 I2M - Institut de Mathématiques de Marseille
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
3 GRACE - Geometry, arithmetic, algorithms, codes and encryption

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 $\mathbb{F}_q$ , with q a power of p. 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–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 $\mathbb{F}_2$ and $\mathbb{F}_3$.

Mots clés: Algebraic function field; tower of function fields; tensor rank; algorithm; finite field

Domaines: Géométrie algébrique [math.AG]; Théorie des nombres [math.NT]; Théorie de l'information et codage [math.IT]; Théorie de l'information [cs.IT]

Stéphane Ballet, Julia Pieltant. Tower of algebraic function fields with maximal Hasse-Witt invariant and tensor rank of multiplication in any extension of $\mathbb{F}_2$ and $\mathbb{F}_3$. Journal of Pure and Applied Algebra, 2018, 222 (5), pp.1069-1086. ⟨10.1016/j.jpaa.2017.06.007⟩. ⟨hal-01063511v2⟩