Families of multi-dimensional arrays with optimal correlations between all members

Families of sequences with low off-peak autocorrelation and low cross-correlation are highly valued in spread-spectrum communication. Digital watermarking has an equal need for diverse families of orthogonal multi-dimensional (nD) arrays, where each array has optimal correlation properties. In this reported work, a 1D discrete projection method is used to construct new families of nD orthogonal arrays of size pⁿ, with p a 4k - 1 prime. Finite field algebra and Hadamard matrices are applied to analyse these arrays. The periodic autocorrelation of each array is `perfect´ (p² - 1 peak value, with -1 off-peak for p × p arrays). The cross-correlation between any pair of the p members of each 2D family has the lowest possible values, 0 or ±p. The arrays can be synthesised for arbitrarily large p and outperform Kasami sequences. The alphabet values for these optimal arrays can be roots of unity or signed integers. The aperiodic autocorrelation of the p × p arrays can attain a merit factor of above 3 at shift (p/4, p/4), consistent with Golay´s conjecture in 1D.