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1.
In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring \(R=\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}\), where \(v^{3}=v\), for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over \(\mathbb {F}_q\) and extend these to codes over R.  相似文献   

2.
Designs, Codes and Cryptography - Let $${\mathbb {F}}_{\!q}$$ be a finite field and $$E_b\!: y^2 = x^3 + b$$ be an ordinary (i.e., non-supersingular) elliptic curve (of j-invariant 0) such that...  相似文献   

3.
4.
We prove that every [n, k, d] q code with q ≥ 4, k ≥ 3, whose weights are congruent to 0, −1 or −2 modulo q and is extendable unless its diversity is for odd q, where .   相似文献   

5.

Catalan numbers are known to count noncrossing set partitions, while Narayana and Kreweras numbers refine this count according to the number of blocks in the set partition, and by its collection of block sizes. Motivated by reflection group generalizations of Catalan numbers and their q-analogues, this paper concerns a definition of q-Kreweras numbers for finite Weyl groups W, refining the q-Catalan numbers for W, and arising from work of the second author. We give explicit formulas in all types for the q-Kreweras numbers. In the classical types ABC, we also record formulas for the q-Narayana numbers and in the process show that the formulas depend only on the Weyl group (that is, they coincide in types B and C). In addition, we verify that in the classical types ABCD the q-Kreweras numbers obey the expected cyclic sieving phenomena when evaluated at appropriate roots of unity.

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6.
Designs, Codes and Cryptography - Applications of permutation polynomials in cryptography are closely related to their cycle structures. For example, many block ciphers use permutation polynomials...  相似文献   

7.
Let Γ6 be the elliptic curve of degree 6 in PG(5, q) arising from a non-singular cubic curve of PG(2, q) via the canonical Veronese embedding
(1) If Γ6 (equivalently ) has n GF(q)-rational points, then the associated near-MDS code has length n and dimension 6. In this paper, the case q  =  5 is investigated. For q  =  5, the maximum number of GF(q)-rational points of an elliptic curve is known to be equal to ten. We show that for an elliptic curve with ten GF(5)-rational points, the associated near-MDS code can be extended by adding two more points of PG(5, 5). In this way we obtain six non-isomorphic [12, 6]5 codes. The automorphism group of is also considered.   相似文献   

8.
In this paper we study the difference between the 2-adic valuations of the cardinalities \( \# E( \mathbb {F}_{q^k} ) \) and \( \# E( \mathbb {F}_q ) \) of an elliptic curve E over \( \mathbb {F}_q \). We also deduce information about the structure of the 2-Sylow subgroup \( E[ 2^\infty ]( \mathbb {F}_{q^k} ) \) from the exponents of \( E[ 2^\infty ]( \mathbb {F}_q ) \).  相似文献   

9.
Let be a saturated formation. We describe minimal non- -, minimal non- -, and minimal non-metabelian groups. Dedicated to L. A. Shemetkov on the occasion of his seventieth birthday.  相似文献   

10.
Let G be a finite group and a formation of finite groups. We say that a subgroup H of G is -supplemented in G if there exists a subgroup T of G such that G = TH and is contained in the -hypercenter of G/H G . In this paper, we use -supplemented subgroups to study the structure of finite groups. A series of previously known results are unified and generalized. Research of the author is supported by a NNSF grant of China (Grant #10771180).  相似文献   

11.
We construct the category of quotients of -spaces and we show that it is Abelian. This answers a question of L. Waelbroeck from 1990.  相似文献   

12.
Permutation polynomials have been an interesting subject of study for a long time and have applications in many areas of mathematics and engineering. However, only a small number of specific classes of permutation polynomials are known so far. In this paper, six classes of linearized permutation polynomials and six classes of nonlinearized permutation polynomials over are presented. These polynomials have simple shapes, and they are related to planar functions. This work was supported by Australian Research Council (Grant No. DP0558773), National Natural Science Foundation of China (Grant No. 10571180) and the Research Grants Council of the Hong Kong Special Administrative Region of China (Grant No. 612405)  相似文献   

13.
We calculate the Jacobi Eisenstein series of weight \(k \ge 3\) for a certain representation of the Jacobi group, and evaluate these at \(z = 0\) to give coefficient formulas for a family of modular forms \(Q_{k,m,\beta }\) of weight \(k \ge 5/2\) for the (dual) Weil representation on an even lattice. The forms we construct have rational coefficients and contain all cusp forms within their span. We explain how to compute the representation numbers in the coefficient formulas for \(Q_{k,m,\beta }\) and the Eisenstein series of Bruinier and Kuss p-adically to get an efficient algorithm. The main application is in constructing automorphic products.  相似文献   

14.
He  Shengnan  Sun  Xiaoli  Xiao  Mingqing 《Semigroup Forum》2020,101(3):680-689
Semigroup Forum - In this paper, we study the $${\mathcal {F}}$$ -transitive behaviour of the translation semigroups on complex sectors, where $${\mathcal {F}}$$ is a Furstenberg family of the...  相似文献   

15.
In this work, we completely characterize (1) permutation binomials of the form \(x^{{{2^n -1}\over {2^t-1}}+1}+ ax \in \mathbb {F}_{2^n}[x], n = 2^st, a \in \mathbb {F}_{2^{2t}}^{*}\), and (2) permutation trinomials of the form \(x^{2^s+1}+x^{2^{s-1}+1}+\alpha x \in \mathbb {F}_{2^t}[x]\), where st are positive integers. The first result, which was our primary motivation, is a consequence of the second result. The second result may be of independent interest.  相似文献   

16.
Codes over that are closed under addition, and multiplication with elements from Fq are called Fq-linear codes over . For m 1, this class of codes is a subclass of nonlinear codes. Among Fq-linear codes, we consider only cyclic codes and call them Fq-linear cyclic codes (Fq LC codes) over The class of Fq LC codes includes as special cases (i) group cyclic codes over elementary abelian groups (q=p, a prime), (ii) subspace subcodes of Reed–Solomon codes (n=qm–1) studied by Hattori, McEliece and Solomon, (iii) linear cyclic codes over Fq (m=1) and (iv) twisted BCH codes. Moreover, with respect to any particular Fq-basis of , any FqLC code over can be viewed as an m-quasi-cyclic code of length mn over Fq. In this correspondence, we obtain transform domain characterization of Fq LC codes, using Discrete Fourier Transform (DFT) over an extension field of The characterization is in terms of any decomposition of the code into certain subcodes and linearized polynomials over . We show how one can use this transform domain characterization to obtain a minimum distance bound for the corresponding quasi-cyclic code. We also prove nonexistence of self dual Fq LC codes and self dual quasi-cyclic codes of certain parameters using the transform domain characterization.AMS classification 94B05  相似文献   

17.
Journal of Fourier Analysis and Applications - In this paper we use Riesz spectral Theory and Gershgorin Theory to obtain explicit information concerning the spectrum of pseudo-differential...  相似文献   

18.
In this paper we prove that if E and F are reflexive Banach spaces and G is a closed linear subspace of the space \(\mathcal {L}_{K}(E;F)\) of all compact linear operators from E into F, then G is either reflexive or non-isomorphic to a dual space. This result generalizes (Israel J Math 21:38-49, 1975, Theorem 2) and gives the solution to a problem posed by Feder (Ill J Math 24:196-205, 1980, Problem 1). We also prove that if E and F are reflexive Banach spaces, then the space \(\mathcal {P}_{w}(^{n}E;F)\) of all n-homogeneous polynomials from E into F which are weakly continuous on bounded sets is either reflexive or non-isomorphic to a dual space.  相似文献   

19.
Periodica Mathematica Hungarica - For primes p, we investigate an $$\mathbb {F}_p$$-version of simplicial volume and compare these invariants with their siblings over other coefficient...  相似文献   

20.
For given positive integer n and ε > 0 we consider an arbitrary nonempty subset A of a field consisting of p 2 elements such that its cardinality exceeds p 2/n?ε . We study the possibility to represent an arbitrary element of the field as a sum of at most N(n, ε) elements from the nth degree of the set A. An upper estimate for the number N(n, ε) is obtained when it is possible.  相似文献   

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