Advances in Mathematics of Communications doi:10.3934/amc.2013.7.231 Volume 7, No. 3, 2013, 231–242 ARRAYS OVER ROOTS OF UNITY WITH PERFECT AUTOCORRELATION AND GOOD ZCZ CROSS-CORRELATION Samuel T. Blake, Thomas E. Hall and Andrew Z. Tirkel School of Mathematical Sciences Building 20, Clayton Campus, Monash University Victoria, 3800, Australia (Communicated by Andrew Klapper) Abstract. We present a new construction for two-dimensional, perfect au- tocorrelation arrays over roots of unity. These perfect arrays are constructed from a block of perfect column sequences. Other blocks are constructed from the first block, to generate a block-circulant structure. The columns are then multiplied by a perfect sequence over roots of unity, which, when folded in- to an array commensurate with our block width has the array orthogonality property. The size of the arrays is commensurate with the length of the un- derlying perfect sequences. For a given size we can construct an exponential number of inequivalent perfect arrays. For each perfect array we construct a family of arrays whose pairwise cross-correlation values are almost all zero (large zero correlation zones (ZCZ)). We present experimental evidence that this construction for perfect arrays can be generalized to higher dimensions. 1. Introduction Finite sequences and arrays over complex numbers are perfect, if their periodic autocorrelation for all non-trivial cyclic shifts is zero. Sequences and arrays over roots of unity with perfect periodic autocorrelation have been known for more than 40 years [3, 19]. Their impulse-like autocorrelation has resulted in applications in communications, signal processing, and as aperture functions for electromagnetic and acoustic imaging. In this paper, we introduce and analyze a new construction of perfect two dimensional arrays. Section 2 contains the necessary definitions. Section 3 presents a brief survey of known perfect sequence and perfect array constructions, with emphasis on perfect sequences used in our construction. Section 4 describes our construction, which is based on a block of perfect column sequences replicated to form a block-circulant array, and a perfect multiplying sequence. Our new construction produces many perfect arrays of the same size. In Section 5 a subset of these is generated with the remarkable property that most of their pairwise cross-correlation values are also zero. Section 6 discusses the array sizes and bounds on the number of inequivalent arrays. Some of the arrays in our construction have the unusual property that all rows, columns and proper diagonals are perfect sequences. Our arrays can also be constructed in three and more dimensions. The paper concludes with Section 7, on potential applications of the new arrays. 2010 Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35. Key words and phrases: Perfect arrays, autocorrelation, cross-correlation, array orthogonality property, block circulant, zero correlation zone. 231 c 2013 AIMS