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The original algorithm of D.V. Chudnovsky and G.V. Chudnovsky for the multiplication in extensions of finite fields provides a bilinear complexity which is uniformly linear with respect to the degree of the extension. Recently, Randriambololona generalized the method, allowing asymmetry in the interpolation procedure. The aim of this article is to make effective this method. We first make explicit this generalization in order to construct the underlying asymmetric algorithms. Then, we propose a generic strategy to construct these algorithms using places of higher degrees and without derivated evaluation. Finally, we provide examples of three multiplication algorithms along with their Magma implementation: in F 16 13 using only rational places, in F 4 5 using also places of degree two, and in F 2 5 using also places of degree four. [ABSTRACT FROM AUTHOR]

We obtain new uniform bounds for the symmetric tensor rank of multiplication in finite extensions of any finite field Fp or Fp2 where p denotes a prime number ≥5. In this aim, we use the symmetric Chudnovsky-type generalized algorithm applied on sufficiently dense families of modular curves defined over Fp2 attaining the Drinfeld-Vladuts bound and on the descent of these families to the definition field Fp. These families are obtained thanks to prime number density theorems of type Hoheisel, in particular a result due to Dudek (Funct Approx Commmentarii Math, 55(2):177-197, 2016). [ABSTRACT FROM AUTHOR]

We propose new results on low weight codewords of affine and projective generalized Reed-Muller (GRM) codes. In the affine case we prove that if the cardinality of the ground field is large compared to the degree of the code, the low weight codewords are products of affine functions. Then, without this assumption on the cardinality of the field, we study codewords associated to an irreducible but not absolutely irreducible polynomial, and prove that they cannot be second, third or fourth weight depending on the hypothesis. In the projective case the second distance of GRM codes is estimated, namely a lower bound and an upper bound on this weight are given. [ABSTRACT FROM AUTHOR]