Families of Multidimensional Periodic Arrays with Optimal Low Cross-Correlation

We construct new families of multidimensional periodic arrays that can be used for digital watermarking videos and images. Each member of each family is composed of columns of cyclic shifts of a Sidelnikov sequence in which the shifts are given by a multidimensional \(m_1\times m_2\times \ldots \times m_n\) “grid” that is determined by applying a logarithmic quadratic map to the elements of a direct product \(\mathbb {Z}_{m_1}\times \mathbb {Z}_{m_2}\times \ldots \times \mathbb {Z}_{m_n}\) . These arrays have peak auto-correlation on the order of \(p^{2l}\) and non-peak auto- and cross-correlation on the order of \(p^{l}\) , where \(l\ge 2\) and \(m_1m_2\ldots m_n = p^l-1,\) thereby improving the peak auto-correlation to nonpeak auto- and cross-correlations of other known constructions. Additionally, we show that the low cross-correlation values are optimal with respect to the Welch bound.