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...

We study the linear complexity of sequences over the prime field Fd introduced by Sidel’nikov. For several classes of period length we can show that these sequences have a large linear complexity. For the ternary case we present exact results on the linear complexity using well known results on cyclotomic numbers. Moreover, we prove a general lower bound on the linear complexity profile for all...

A graph G has a representation modulo r if there exists an injective map f : V (G) → {0, 1, ..., r − 1} such that vertices u and v are adjacent if and only if f(u) − f(v) is relatively prime to r. The representation number of G, rep(G), is the smallest integer such that G has a representation modulo r. In this paper we study the representation numbers of various graphs, such as the complete ter...

In this paper, for a prime power q, new cyclic difference sets with Singer parameters ððq 1=q 1Þ; ðq 1 1=q 1Þ; ðq 2 1=q 1ÞÞ are constructed by using q-ary sequences (d-homogeneous functions) of period q 1 and the generalization of GMW difference sets is proposed by combining the generation methods of d-form sequences and extended sequences. When q is a power of 3, new cyclic difference sets wit...

The main aim of this paper is to study LCD codes. Linear code with complementary dual(LCD) are those codes which have their intersection with their dual code as t0u. In this paper we will give rather alternative proof of Massey’s theorem[8], which is one of the most important characterization of LCD codes. Let LCDrn, ks3 denote the maximum of possible values of d among rn, k, ds ternary LCD cod...