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# New uniform and asymptotic upper bounds on the tensor rank of multiplication in extensions of finite fields

1 GRACE - Geometry, arithmetic, algorithms, codes and encryption
LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau], Inria Saclay - Ile de France
Abstract : We obtain new uniform upper bounds for the tensor rank of the multiplication in the extensions of the finite fields $\mathbb{F}_q$ for any prime power $q$; 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 $\mathbb{F}_q$, with an optimal ratio of $\mathbb{F}_{q^t}$-rational places to their genus, where $q^t$ is a square.
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Journal articles
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Cited literature [21 references]

https://hal.archives-ouvertes.fr/hal-00828153
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Submitted on : Friday, May 31, 2013 - 11:17:19 AM
Last modification on : Monday, January 24, 2022 - 11:43:20 AM
Long-term archiving on: : Tuesday, April 4, 2017 - 1:47:53 PM

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Julia Pieltant, Hugues Randriambololona. New uniform and asymptotic upper bounds on the tensor rank of multiplication in extensions of finite fields. Mathematics of Computation, American Mathematical Society, 2015, 84 (294), pp.2023-2045. ⟨10.1090/S0025-5718-2015-02921-4⟩. ⟨hal-00828153⟩

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