The Constructions Of The Almost Difference Sets

In digital communications, including spread spectrum signals, synchronous code division multiple access (CDMA) systems for mobile communications, and encryption of data, the almost balanced periodic binary sequences with optimal two or three level autocorrelation are required. They are closely related to difference sets and almost difference sets.A k-subset D of a group (G, +) of order v is called a (v, k, λ)-differenee set if every nonzero element of G appears as a difference between a pair of elements in D exactly A times.Well, A k-subset D of a group (G, +) of order v is called a (v, k,λ, t)-almost difference set (ADS) if there are t nonzero elements of G can be represented A times as a difference between a pair of elements in D, while the other v — 1 — t nonzero elements can be represented λ+ 1 times as a difference between a pair of elements in D.Almost difference sets are closely related to cryptography, coding theory and sequences. They can be used to construct cryptographic functions with optimal nonlinearity, sequences with optimal autocorrelation, and good constant-weight codes.In this paper, we study the constructions of the almost difference sets and some necessary conditions for the existence of them in three parts. Firstly, we state the background, the basic definitions and some properties of the almost difference sets; Then we construct some families of almost difference sets by means of group, cyclotomic class and direct product; Finally, we discuss some necessary conditions for the existence of (v, k, λ, t)-almost difference sets on Zv, particularly, nonexistence results for some almost differencesets are established when t = 1 and t = v — 2; In the appendix, we list the nonexist casesv for v ≤ 50 and k≤v/2on Zv which are obtained by computer searching.