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

TL;DR

This paper extends the Ding-Helleseth-Martinsens construction method to create new almost difference sets using cyclotomic classes of order 12, while also proving the non-existence for orders six and eight.

Contribution

It introduces two new classes of almost difference sets based on cyclotomic classes of order 12 and demonstrates the limitations of the Ding-Helleseth-Martinsens method for certain orders.

Findings

Constructed two classes of almost difference sets using order 12 cyclotomic classes.

Proved non-existence of Ding-Helleseth-Martinsens constructions for orders six and eight.

Extended the application of the construction method to new algebraic structures.

Abstract

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.