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Let F ( x ) = x 2 2 k + 2 k + 1 be a power function over the finite field GF ( 2 n ) , where k is a positive integer and n = 4 k . It is known by a recent result of Bracken and Leander that F ( x ) is a highly nonlinear differentially 4-uniform function. Building upon this work, in this paper we determine the differential spectrum of F ( x ) . [ABSTRACT FROM AUTHOR]

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. So far, most of previous results obtained were for cyclic codes with no more than three nonzeros. Recently, the authors of [37] constructed a class of cyclic codes with arbitrary number of nonzeros, and computed the weight distribution for several cases. In this paper, we determine the weight distribution for a new family of such codes. This is achieved by introducing certain new methods, such as the theory of Jacobi sums over finite fields and subtle treatment of some complicated combinatorial identities. [ABSTRACT FROM AUTHOR]

Abstract: Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in Ma et al. (2011) [10], Ding et al. (2011) [6], Wang et al. (2012) [15], Xiong (2012) [16,17]. In this paper we solve one more special case. The problem of finding the weight distribution is transformed into a problem of evaluating certain character sums over finite fields, which in turn can be solved by using the Jacobi sums directly. [Copyright &y& Elsevier]

Abstract: Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in Ma et al. (2011) , Ding et al. (2011) , Wang et al. (2011) . In this paper we provide a slightly different approach toward the general problem and use it to solve one more special case. We make extensive use of standard tools in number theory such as characters of finite fields, the Gauss sums and the Jacobi sums to transform the problem of finding the weight distribution into a problem of evaluating certain character sums over finite fields, which on the special case is related with counting the number of points on some elliptic curves over finite fields. Other cases are also possible by this method. [Copyright &y& Elsevier]

Abstract: For the finite field of q elements (q odd) and a quadratic non-residue (that is, a non-square) , we define the distance function on the upper half plane . For two sets with , and a non-trivial additive character ψ on , we give the following estimate which (assuming that ) is non-trivial if as . [Copyright &y& Elsevier]