On a Class of Almost Difference Sets Constructed by Using the Ding-Helleseth-Martinsens Constructions

Pseudorandom binary sequences with optimal balance and autocorrelation have many applications in stream cipher, communication, coding theory, etc. It is known that binary sequences with three-level autocorrelation should have an almost difference set as their characteristic sets. How to construct new families of almost difference set is an important research topic in such fields as communication, coding theory and cryptography. In a work of Ding, Helleseth, and Martinsen in 2001, the authors developed a new method, known as the Ding-Helleseth-Martinsens Constructions in literature, of constructing an almost difference set from product sets of GF(2) and the union of two cyclotomic classes of order four. In the present paper, we have constructed two classes of almost difference set with product sets between GF(2) and union sets of the cyclotomic classes of order 12 using that method. In addition, we could find there do not exist the Ding-Helleseth-Martinsens Constructions for the cyclotomic classes of order six and eight.