A Multi-Dimensional Block-Circulant Perfect Array Construction

We present a $N$-dimensional generalization of the two-dimensional block-circulant perfect array construction by \cite{Blake2013}. As in \cite{Blake2013}, the families of $N$-dimensional arrays possess pairwise \textit{good} zero correlation zone (ZCZ) cross-correlation. Both constructions use a perfect autocorrelation sequence with the array orthogonality property (AOP).

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 ] 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 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 is larger for higher dimensional arrays.
We show that S k from Construction II has perfect autocorrelation (k 1 = k 2 ) and S k1 , S k2 has good cross-correlation (k 1 = k 2 ).We now compute the cross-correlation of S k1 with S k2 , (0 Consider the case when s N −1 = 0 mod d.Let s N −1 = l d and perform the change of coordinates j = qd + r, (r < d).Then the inner summation in (7) becomes which is independent of q.Thus, (7) becomes As a has the AOP for the divisor d, the summation

Now consider the case k
As a has the AOP for the divisor d, the left double summation is zero.Thus the autocorrelation is zero for s N −1 = 0 mod d.

Now, consider the case when s
and perform the change of coordinates j = qd + r, (r < d).Then (7) becomes As the sequence a has the AOP, the summation for all r and s N −1 such that s N −1 = 0 mod d.Thus the cross-correlation of S k1 with S k2 is zero for h = 0 mod d and all v.So Construction II produces a family of m perfect k-dimensional arrays with pairwise good cross-correlation.
It has been shown that perfect N -dimensional arrays can be constructed up to size r 2 × r 2 × • • • over r roots of unity.An exponential number of such arrays exist for each size.Furthermore, families of arrays possess good cross-correlation properties.Importantly, despite the size of N , the number of non-zero crosscorrelation values is always d 2 , so the ratio of the number of non-zero cross-correlation values to the number of entries in the arrays can be made as small as desired.
The following generalization of indexND allows for computations over the quaternions: Table [aref[aref[c, Mod[j, d] The following is a perfect quaternion sequence which posesses the array orthogonality property for the divisor 4: We use this sequence to construct a perfect 16 × 16 quaternion array.
there are d solutions for s v , one for each r = 0, 1, • • • , d − 1.Thus, there are d 2 non-zero cross-correlation values for s N −1 = 0 mod d.