Extended families of 2D arrays with near optimal auto and low cross-correlation

Abstract

Families of 2D arrays can be constructed where each array has perfect autocorrelation, and the cross-correlation between any pair of family members is optimally low. We exploit equivalent Hadamard matrices to construct many families of p p×p arrays, where p is any 4k-1 prime. From these families, we assemble extended families of arrays with members that exhibit perfect autocorrelation and next-to-optimally low cross-correlation. Pseudo-Hadamard matrices are used to construct extended families using p=4k+1 primes. An optimal family of 31 31×31 perfect arrays can provide copyright protection to uniquely stamp a robust, low-visibility watermark within every frame of each second of high-definition, 30fps video. The extended families permit the embedding of many more perfect watermarks that have next-to-minimal cross-correlations.