A multi-dimensional block-circulant perfect array construction

We present a N-dimensional generalization of the two-dimensional block-circulant perfect array construction by [Blake, 2013]. As in [Blake, 2013], the families of N-dimensional arrays possess pairwise good zero correlation zone (ZCZ) cross-correlation. Both constructions use a perfect autocorre-lation sequence with the array orthogonality property (AOP). This paper presents a generalization of the 2-dimensional block-circulant array construction by [Blake, 2013]. The generalized construction works in any number of dimensions, but is limited to the same size in each dimension as the original two-dimensional construction. Throughout the paper we follow the notation of [Blake, 2013]. We begin by stating the 2-dimensional construction for families of arrays with perfect autocorrelation and good cross-correlation as given in [Blake, 2013]. Construction I Let a = [a 0 , a 1 , · · · , a n−1 ] be a perfect sequence with the AOP for the divisor, d, and and c = [c(0), c(1), · · · , c(d − 1)] is a block of d perfect sequences – each of length m, where m = 0 mod d. We construct a family of arrays, S k , such that S k = [S i,j ] k = a j c(j mod d) w⌊j/d⌋+k(j mod d)+i for 0 ≤ i < n, 0 ≤ j < m, a has the AOP for the divisor d, 0 < k ≤ m, and w = m/d. This construction produces perfect arrays up to size r 2 × r 2 over r roots of unity. Each pair of distinct arrays has d 2 non-zero cross-correlation values, as d ≪ r we say the array has good ZCZ cross-correlation. The generalized N-dimensional construction is given as follows. Construction II Let a = [a 0 , a 1 , · · · , a n−1 ] be a perfect sequence with the AOP for the divisor, d, and c = [c(0), c(1), · · · , c(d − 1)] is a block of d perfect sequences – each of length m, where m = 0 mod d. We construct a family of k N-dimensional perfect arrays, S k , such that S k = S i0,i1,··· ,iN−2,j k = a j N −2 v=0 c(j mod d) w⌊j/d⌋+k(j mod d)+iv Each distinct pair of arrays from the family has d 2 non-zero cross-correlation values, regardless of the size of N. Thus, the ratio of zero to non-zero cross-correlation values …