Abstract

We investigate the generalization of the Costas property in threeor more dimensions, and we seek an appropriate definition; the two main complicationsare a) that the number of ''dots'' this multidimensional structureshould have is not obvious, and b) that the notion of the multidimensionalpermutation needs some clarification. After proposing various alternatives forthe generalization of the definition of the Costas property, based on the definitionsof the Costas property in one or two dimensions, we also offer someconstruction methods, the main one of which is based on the idea of reshapingCostas arrays into higher-dimensional entities.

Highlights

  • Costas arrays originated in engineering as a frequency hopping pattern that optimizes the performance of radars and sonars [3, 4]; being a singular combinatorial object, they have been lately the focus of intensive study by mathematicians as well, and have started leading a second independent “life” in the mathematical literature [7, 8, 9, 10, 14]

  • We increase the level of mathematical abstraction by investigating analogs of Costas arrays in 3 or more dimensions: several challenges lie ahead, as we first need to give a satisfactory definition of the Costas property in higher dimensions, and provide some algorithms to construct such Costas cubes or “hypercubes” in general

  • We will begin by stating the definition of Costas arrays [5] in such a way as to exhibit their direct relation to Golomb rulers [16], which will be construed as the 1-dimensional analog of Costas arrays

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Summary

Introduction

Costas arrays originated in engineering as a frequency hopping pattern that optimizes the performance of radars and sonars [3, 4]; being a singular combinatorial object, they have been lately the focus of intensive study by mathematicians as well, and have started leading a second independent “life” in the mathematical literature [7, 8, 9, 10, 14]. We increase the level of mathematical abstraction by investigating analogs of Costas arrays in 3 or more dimensions: several challenges lie ahead, as we first need to give a satisfactory definition of the Costas property in higher dimensions, and provide some algorithms to construct such Costas cubes or “hypercubes” in general. We will discuss several slightly different candidate definitions in higher dimensions, and we will attempt to generalize the generation methods available to us. In particular, that it is possible to generate sparse Costas hypercubes systematically, using various construction methods, but if we want to know what the densest Costas hypercube can be, things become less clear

The definition of Costas property in 1 and 2 dimensions
The Costas property in higher dimensions
Vector permutations
Strict Costas hypercubes
The main construction method
Reshaping
Putting things together:
Construction examples
Applicability of reshaping
Older generalization attempts
The original method and its non-extendability for Costas arrays
Construction of hyper-rectangles
A further generalization
Normal bases of fields and “rotational” constructions
Examples
Density of Costas hypercubes
Monte Carlo construction of Costas hypercubes
Findings
Connection with 2 dimensions

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