Two-dimensional patterns with optimal auto- and cross-correlation functions

We consider a set of sonar arrays of the same size with an additional constraint that the non-periodic two-dimensional cross-correlation value for any two distinct arrays is limited to either 1 or O. These find applications in various multiuser communication systems such as multiuser radar and sonar systems and/or fiber-optic CDMA networks. We define an optimal pair of sonar arrays to be a pair of sonar arrays having an ideal cross-correlation function and having the minimum possible number of columns for a fixed number of rows. We first prove that an optimal pair having a rows can have at most n columns