On the Existence of Aperiodic Complementary Hexagonal Lattice Arrays

Binary (periodic) aperiodic complementary sequences have been studied extensively due to their wide range of applications in engineering, for example in optics, radar and communications. They are also linked to topics in coding theory, combinatorics and Boolean functions. Complementary sequences have been generalized either by being defined over larger alphabets or by being defined from one dimension to multi-dimensions. Recently, Ding and Tarokh introduced and constructed the aperiodic complementary twodimensional arrays over the alphabet Ap = {ζ p : 0 ≤ i ≤ p − 1, p is a prime number}, whose support set is a subset of the hexagonal lattice (mostly is a set of `-layer consecutive hexagons). They demonstrated that such complementary hexagonal lattice arrays can be used on coded aperture imagining with ideal efficiency. In this paper, we study the conditions for which such complementary hexagonal lattice arrays exist. We first show that aperiodic complementary hexagonal lattice arrays over the alphabet Ap leads to aperiodic (hence periodic) complementary sequences over the alphabet Ap = A ∗ p∪{0}. Then we make use of group ring equations to characterize periodic complementary sequences over alphabet Ap. As of independent interest, we show that, if the alphabet of the periodic complementary sequences is Ap, the notion of periodic complementary sequences is equivalent to the notion of certain relative difference family. The conditions for the existence of periodic complementary sequences over alphabet Ap are derived from this characterization. As a result, we determine the existence of a pair (triple) aperiodic complementary binary hexagonal lattice arrays whose support set is an `-layer consecutive hexagons. A table listing the existence status for a pair of complementary hexagonal lattice arrays with 1 ≤ ` ≤ 20 as well as some open problems are presented.