Almost difference sets and their sequences with optimal autocorrelation

Abstract

Almost difference sets have interesting applications in cryptography and coding theory. In this paper, we give a wellrounded treatment of known families of almost difference sets, establish relations between some difference sets and some almost difference sets, and determine the numerical multiplier group of some families of almost difference sets. We also construct six new classes of almost difference sets, and four classes of binary sequences of period 0 (mod 4) with optimal autocorrelation. We have also obtained two classes of relative difference sets and four classes of divisible difference sets (DDSs). We also point out that a result due to Jungnickel can be used to construct almost difference sets and sequences of period 4 with optimal autocorrelation.

Nonlinear functions and difference sets on group actions

“…We will also investigate the existence and constructions of G-perfect nonlinear functions and G-bent functions. Several known results in [2,6,10,17] are direct consequences of our results. …”

“…Applications of Theorem 2.6 will also be discussed, and several known results in [2,6,17] are obtained as direct consequences. Throughout the paper, the following notation will be used.…”

On the radiation properties of ADS-thinned dipole arrays

“…Firstly, binary sequences derived from difference set (DSs) have been exploited for designing both linear and planar layouts [19,21]. Such an approach has been successively generalized to deal with a larger set of geometrical configurations thanks to the properties of almost difference set (ADSs) [20,22] and the existence of suitable ADS construction techniques [23,24,25] as well as of sequence repositories [26]. The main advantages coming from analytic designs are (a) the availability of a-priori PSL bounds just knowing the descriptors of the sequences at hand; (b) the numerical efficiency of the synthesis approach; (c) the regular behavior of the arising pattern sidelobes also well below those achievable with cut-andtry random placements [20,22].…”

Polarization-Agile Ads-Interleaved Planar Arrays

“…Indeed, DSs have the fundamental drawback that they exist (or can be designed) only for specific lengths and thinning factors [22][23][24]. In order to overcome this limitation while achieving satisfactory results (i.e., close to those of DSs), the utilization of sets with suboptimal (i.e., three-level) autocorrelation properties has been proposed in the field of combinatorial mathematics [25,26]. The so-called Almost Difference Sets (ADSs) [25,26] have actually been successfully applied in several array design problems comprising thinned [27][28][29][30], interleaved [31,32], and M IM O [33] layouts.…”

“…In order to overcome this limitation while achieving satisfactory results (i.e., close to those of DSs), the utilization of sets with suboptimal (i.e., three-level) autocorrelation properties has been proposed in the field of combinatorial mathematics [25,26]. The so-called Almost Difference Sets (ADSs) [25,26] have actually been successfully applied in several array design problems comprising thinned [27][28][29][30], interleaved [31,32], and M IM O [33] layouts. Hybrid ADSbased design methodologies have been proposed as well for wireless communications [34] and radio-astronomy applications [35].…”