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Intersections of shifts of multiplicative subgroups
Mathematical notes. 2016. Vol. 100. No. 1. P. 189–198.
Using Stepanov’s method, we obtain an upper bound for the cardinality of the intersection of additive shifts of several multiplicative subgroups of a finite field. The resulting inequality is applied to a question dealing with the additive decomposability of subgroups.
Zykin A. I., Ballet S., Designs, Codes and Cryptography 2019 Vol. 87 P. 517–525
We obtain new uniform bounds for the symmetric tensor rank of multiplication in finite extensions of any finite field F_p or F_{p^2} 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 F_{p_2} attaining the Drinfeld–Vladuts bound and on the descent of these families to ...
Added: May 12, 2020
Galkin S., Rybakov S., Mathematical notes 2019 Vol. 106 No. 6 P. 1014–1018
For a family of K3 surfaces we implement a variation of a general construction of towers of algebraic curves over finite fields given in a previous paper. As a result we get a good tower over k=F_{p^2}, that is optimal if p=3. ...
Added: January 29, 2020
Galkin S., Rybakov S., / Series math "arxiv.org". 2019. No. 1910.14379.
For a family of K3 surfaces we implement a variation of a general construction of towers of algebraic curves over finite fields given in a previous paper. As a result we get a good tower over k=𝔽_{p^2}, that is optimal if p=3. ...
Added: November 6, 2019
Fedorenko Sergei Valentinovich, IEEE Signal Processing Letters 2019 Vol. 26 No. 9 P. 1320–1324
An effective calculation of the Reed-Solomon code syndrome is proposed. The method is based on the use of the partial normalized cyclic convolutions in the partial inverse cyclotomic discrete Fourier transform. The method is the best of the known algorithms, in terms of multiplicative complexity. ...
Added: September 4, 2019
Trepalin A., / Series arXiv "math". 2017.
Let X be a minimal del Pezzo surface of degree 2 over a finite field 𝔽_q. The image Γ of the Galois group Gal(\bar{𝔽}_q/𝔽_q) in the group Aut(Pic(\bar{X})) is a cyclic subgroup of the Weyl group W(E_7). There are 60 conjugacy classes of cyclic subgroups in W(E_7) and 18 of them correspond to minimal del Pezzo surfaces. In this paper we study which possibilities of these subgroups for minimal del Pezzo ...
Added: December 2, 2018
Trepalin A., / Series arXiv "math". 2018.
Let X be a del Pezzo surface of degree 2 or greater over a finite field 𝔽_q. The image Γ of the Galois group Gal(\bar{𝔽}_q / 𝔽_q) in the group Aut(Pic(\bar{X})) is a cyclic subgroup preserving the anticanonical class and the intersection form. The conjugacy class of Γ in the subgroup of Aut(Pic(\bar{X})) preserving the anticanonical class and the intersection form is a natural invariant of X. We say that the ...
Added: December 2, 2018
Trepalin A., Loughran D., / Series arXiv "math". 2019.
We completely solve the inverse Galois problem for del Pezzo surfaces of degree 2 and 3 over all finite fields. ...
Added: December 2, 2018
Sergei Valentinovich Fedorenko, IEEE Transactions on Signal Processing 2015 Vol. 63 No. 20 P. 5307–5317
A normalized cyclic convolution is a cyclic convolution when one of its factors is a fixed polynomial. Herein, a novel method for constructing a normalized cyclic convolution over a finite field is introduced. This novel method is the first constructive and best known method for even lengths. This method can be applied for computing discrete ...
Added: February 3, 2018
Sergei Valentinovich Fedorenko, IEEE Signal Processing Letters 2016 Vol. 23 No. 6 P. 824–827
A novel method for computing the discrete Fourier transform (DFT) over a finite field based on the Goertzel-Blahut algorithm is described. The novel method is currently the best one for computing the DFT over even extensions of the characteristic two finite field, in terms of multiplicative complexity. ...
Added: January 26, 2018
Vyugin I. V., Solodkova E. V., Shkredov I. D., Mathematical notes 2017 Vol. 101 No. 1 P. 58–70
A new upper bound for the additive energy of the Heilbronn subgroup is found. Several applications to the distribution of Fermat quotients are obtained. ...
Added: May 22, 2017
Avdoshin S. M., Набебин А. А., М.: ДМК Пресс, 2017.
The textbook contains necessary information about universal and classical algebras, systems of axioms for the basic algebraic structures (groupoid, monoid, semi-groups, groups, partial orders, rings, fields). The basic cryptographic algorithms are described. Error-correcting codes - linear, cyclic, BCH are considered. Algorithms for designing of such codes are given. Many examples are shown. It is put ...
Added: August 19, 2016