Some Constructions and Mathematical Properties of Zero-Correlation-Zone Sonar Sequences

In this paper, we propose the zero-correlation-zone (ZCZ) of radius r on two-dimensional m×n sonar sequences and define the (m,n,r) ZCZ sonar sequences. We also define some new optimality of an (m,n,r) ZCZ sonar sequence which has the largest r for given m and n. Because of the ZCZ for perfect autocorrelation, we are able to relax the distinct difference property of the conventional sonar sequences, and hence, the autocorrelation of ZCZ sonar sequences outside ZCZ may not be upper bounded by 1. We may sometimes require such an ideal autocorrelation outside ZCZ, and we define ZCZ-DD sonar sequences, indicating that it has an additional distinct difference (DD) property. We first derive an upper bound on the ZCZ radius r in terms of m and n≥m. We next propose some constructions for (m,n,r) ZCZ sonar sequences, which leads to some very good constructive lower bound on r. Furthermore, this construction suggests that for m and r, the parameter n can be as large as possible indefinitely. We present some exhaustive search results on the existence of (m,n,r) ZCZ sonar sequences for some small values of r. For ZCZ-DD sonar sequences, we prove that some variations of Costas arrays construct some ZCZ-DD sonar sequences with ZCZ radius r=2. We also provide some exhaustive search results on the existence of (m,n,r) ZCZ-DD sonar sequences. Lots of open problems are listed at the end.