On primitive elements of trace equal to 1 in GF(2m)

AbstractThe object of this paper is to present a simple proof for the existence of primitive elements of trace equal 1, in GF(2m)

On the symmetry of Welch- and Golomb-constructed Costas arrays

AbstractWe prove that Welch Costas arrays are in general not symmetric and that there exist two special families of symmetric Golomb Costas arrays: one is the well-known Lempel family, while the other, although less well known, leads actually to the construction of dense Golomb rulers

On the existence of some specific elements in finite fields of characteristic 2

AbstractLet q be a power of 2, n be a positive integer, and let Fqn be the finite field with qn elements. In this paper, we consider the existence of some specific elements in Fqn. The main results obtained in this paper are listed as follows:(1)There is an element ξ in Fqn such that both ξ and ξ+ξ−1 are primitive elements of Fqn if q=2s, and n is an odd number no less than 13 and s>4.(2)For q=2s, and any odd n, there is an element ξ in Fqn such that ξ is a primitive normal element and ξ+ξ−1 is a primitive element of Fqn if either n|(q−1), and n⩾33, or n∤(q−1), and n⩾30, s⩾6

A review of Costas arrays

Costas arrays are not only useful in radar engineering, but they also present many interesting, and still open, mathematical problems. This work collects in it all important knowledge about them available today: some history of the subjects, density results, construction methods, construction algorithms with full proofs, and open questions. At the same time all the necessary mathematical background is offered in the simplest possible format and terms, so that this work can play the role of a reference for mathematicians and mathematically inclined engineers interested in the field

Primitive elements in finite fields with arbitrary trace

Arithmetic of finite fields is not only important for other branches of mathematics but also widely used in applications such as coding and cryptography. A primitive element of a finite field is of particular interest since it enables one to represent all other elements of the field. Therefore an extensive research has been done on primitive elements, especially those satisfying extra conditions. We are interested in the existence of primitive elements in extensions of finite fields with prescribed trace value. This existence problem can be settled by means of two important theories. One is character sums and the other is the theory of algebraic function fields. The aim of this thesis is to introduce some important properties of these two topics and to show how they are used in answering the existence problem mentioned above