In this paper, the study of vague structures of Γ-semigroups is initiated by introducing the concepts of vague subsemigroup, vague bi-ideal and vague (1, 2)-ideal in a Γ-semigroup and some fundamental results are obtained. Then the concept of vague weakly completely prime ideals in Γ-semigroups has been introduced and studied directly and via operator semigroups

We give a construction of 3-class and 4-class association schemes from s-nonlinear and differentially 2s-uniform functions, and a construction of p-class association schemes from weakly regular p-ary bent functions, where p is an odd prime.

let $g$ be a finite group‎. ‎a subgroup‎ ‎$h$ of $g$ is called an $mathcal h $ -subgroup in‎ ‎$g$ if $n_g (h)cap h^gleq h$ for all $gin‎ ‎g$. a subgroup $h$ of $g$ is called a weakly‎ $mathcal h^ast $-subgroup in $g$ if there exists a‎ ‎subgroup $k$ of $g$ such that $g=hk$ and $hcap‎ ‎k$ is an $mathcal h$-subgroup in $g$. we‎ ‎investigate the structure of the finite group $g$ under the‎ ‎assump...

Exponential sums and character sums have many applications in number theory and combinatorial number theory. Many famous and interesting problems have been studied by means of exponential sums techniques: Goldbach ternary conjecture (Vinogradov), Waring Problem (Hardy, Littlewood, Vaughan, Vinogradov, Wooley), Distribution of prime numbers (Ford, Hardy, Korobov, Linnik,...) as well as many othe...

For a prime p with p ≡ 3 (mod 4) and an odd number m, the Bentness of the p-ary binomial function fa,b(x) = Tr1 (ax pm−1) + Tr1(bx pn−1 4 ) is characterized, where n = 2m, a ∈ Fpn , and b ∈ Fp2 . The necessary and sufficient conditions of fa,b(x) being Bent are established respectively by an exponential sum and two sequences related to a and b. For the special case of p = 3, we further characte...

Let p be a prime number. In this paper, we study cyclic codes over the ring Z p [u, v]/u 2 , v 2 , uv − vu. We find a unique set of generators for these codes. We also study the rank and the Hamming distance of these codes. We obtain all except one ternary optimal code of length 12 as the Gray image of the cyclic codes over the ring Z p [u, v]/u 2 , v 2 , uv − vu. We also characterize the p-ary...

It is well-known that for each prime power q and for each d ∈ 2N, there exists a circulant weighing matrix of order q d+1−1 q−1 and weight q . We extend this result to show that there exist φ(d+1) 2 inequivalent circulant weighing matrices of order q d+1−1 q−1 and weight q , where φ is the Euler totient function. Further, we obtain a bound on the magnitude of the values taken by the cross-corre...

We construct new binary and ternary self-orthogonal linear codes. In order to do this we use an equivalence between the existence of a self-orthogonal linear code over a prime eld with a prescribed minimum distance and the existence of a solution of a certain system of Diophantine linear equations. To reduce the size of the system of equations we restrict the search for solutions to solutions w...

This paper specifies some conditions as to when an integer m is locally represented by a positive definite diagonal integer-matrix ternary quadratic form Q at a prime p. We use quadratic Gauss sums and a version of Hensel’s Lemma to count how many solutions there are to the equivalence Q(~x) ≡ m (mod p) for any k ≥ 0. Given that m is coprime to the determinant of the Hessian matrix of Q, we can...