Two-dimensional perfect quaternary arrays

We study two-dimensional (2-D) arrays of fourth roots of unity which have all out-of-phase periodic autocorrelations equal to zero. Generalizing the concept of a perfect binary array, we call these arrays perfect quaternary arrays. We establish connections with combinatorial design theory and exhibit large families of such arrays. Increasing the alphabet from size two to size four greatly increases the flexibility one has in choosing the dimensions for 2-D arrays with perfect periodic autocorrelation. For example, we show that the number of entries in a 2-D perfect quaternary array may be divisible by any Mersenne prime and indeed by many other primes, whereas 2-D perfect binary arrays are only known to exist with size equal to a power of two times a power of three