Families of Matrices with Good Auto and Cross-Correlation

Abstract

We construct a family of \(p-1\) square \(p \times p\) matrices (\(p\) is any prime) whose periodic cross-correlation values are uniformly \(-p, 0, +p\) between all pairs of the matrices in the family. For every one of the matrices in the family, all the off-peak autocorrelation values are \(-p\) and \(0\), while the single peak value is \(p(p-1)\). For \(p = 127\) (where the values \(-p, 0, +p\) are below \(1\%\) of the size \(p^2\) of the matrices) utilization of this construction has resulted in the superimposed embedding of twelve of the matrices (as watermarks) in the standard image “Lenna” and their subsequent retrieval without recourse to the unmarked image.