On the Nonexistence of the Ding-Helleseth-Martinsens Constructions of Almost Difference Set for Cyclotomic Classes of Order 6

TL;DR

This paper proves that the Ding-Helleseth-Martinsens construction method cannot be applied to cyclotomic classes of order 6, impacting the development of sequences with optimal autocorrelation for communication systems.

Contribution

It demonstrates the nonexistence of a previously proposed construction method for almost difference sets of cyclotomic classes of order 6.

Findings

No such constructions exist for order 6 classes

Impacts the design of pseudorandom sequences with optimal autocorrelation

Clarifies limitations of the Ding-Helleseth-Martinsens method

Abstract

Pseudorandom sequences with optimal three-level autocorrelation have important applications in CDMA communication systems. Constructing the sequences with three-level autocorrelation is equivalent to finding cyclic almost difference sets as their supports. In a paper of Ding, Helleseth, and Martinsen, the authors developed a new method known as the Ding-Helleseth-Martinsens Constructions in literature to construct the almost difference set using product set between GF(2) and union sets of cyclotomic classes of order 4. In this correspondence, we show that there do not exist such constructions for cyclotomic classes of order 6.