Large Families of “Grey” Arrays with Perfect Auto-correlation and Optimal Cross-Correlation

Large sets of distinct 2D arrays of variable size that possess both strong auto-correlation and weak cross-correlation properties are highly valuable in many imaging and communications applications. We use the discrete Finite Radon Transform to construct p× p arrays with “perfect” correlation properties, for any prime p. Array elements are restricted to the integers { 0 , ± 1 , + 2 }. Each array exhibits perfect periodic auto-correlation, having peak correlation value p², with all off-peak values being exactly zero. Each array contains just 3 (p- 1) / 2 zero elements, the minimum number possible using this alphabet. Large families with size M= p²- 1 of such arrays can be constructed. Each of the M(M- 1) / 2 intra-family periodic cross-correlations is guaranteed to have one of the three lowest possible merit factors. These family size M can be extended to multiples of p²- 1 if we permit more than the three lowest cross-correlation levels.