Double Periodic Arrays with Optimal Correlation for Applications in Watermarking

Abstract

Digital watermarking applications require constructions of double-periodic matrices with good correlations. More specifically we need as many matrix sequences as possible with both good auto- and cross-correlation. Furthermore it is necessary to have double-periodic sequences with as many dots as possible.

We have written this paper with the specific intention of providing a theoretical framework for constructions for digital watermarking applications.

In this paper we present a method that increases the number of sequences, and another that increases the number of ones keeping the correlation good and double-periodic. Finally we combine both methods producing families of double-periodic arrays with good correlation and many dots. The method of increasing the number of sequences is due to Moreno, Omrani and Maric. The method to increase the number of dots was started by Nguyen, Lázló and Massey, developed by Moreno, Zhang, Kumar and Zinoviev, and further developed by Tirkel and Hall. The very nice application to digital watermarking is due to Tirkel and Hall.

Finally we obtain two new constructions of Optical Orthogonal Codes: Construction A which produces codes with parameters $(n, \omega, \lambda) =(p(p-1), {{p^2 - 1} \over {2}}, [{p(p+1) \over 4}])$ and Construction B which produces families of code with parameters $(n, \omega, \lambda) = (p^2(p-1), {{p^2 - 1} \over {2}}, [{p(p+1) \over 4}])$ and family size p + 1.