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

## TL;DR

This paper introduces new mathematical properties and constructions for sonar sequences with zero-correlation zones, expanding their potential applications in signal processing.

## Contribution

The paper introduces new optimality criteria and constructions for (m,n,r) ZCZ sonar sequences, including ZCZ-DD sequences with distinct difference properties.

## Key findings

- An upper bound on the ZCZ radius r is derived in terms of m and n.
- Constructions for (m,n,r) ZCZ sonar sequences are proposed, leading to improved lower bounds on r.
- Variations of Costas arrays can construct ZCZ-DD sonar sequences with ZCZ radius r=2.

## Abstract

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.