Matrices with bounded correlation

This paper describes matrices constructed from cyclic shifts of a prototype column by using a shift sequence. Various shift sequences and column sequences are analysed. In particular, a large family of p×p matrices can be constructed by polynomial shift sequences. An upper bound on the off-peak autocorrelation of matrices occurs because it is not possible for more than d-1 columns to match, where d is the degree of the polynomial. The corresponding bound for cross-correlation is d. A method of generating all distinct matrices (enumeration algorithm) is presented. An example shows that it is possible for shift sequences to have high complexity, whilst preserving the ideal autocorrelation of the matrix. The matrices have applications in the watermarking of digital images, sonar arrays, frequency hopping patterns and time hopping sequences.