Multidimensional Costas arrays and periodic properties

Abstract

Costas arrays were first introduced for SONAR detection applications, but later became an interesting object of mathematical research. Several generalizations of Costas arrays to multiple dimensions have been proposed. In this thesis, we lay the ground for the study of multidimensional Costas arrays by proposing concepts and showing results that extend into the higher-dimensional realm most of what is known about the periodicity of two-dimensional Costas arrays. Among the most important results is, for large classes of arrays, the non-existence of multidimensional Costas arrays that preserve the Costas property in every commensurate window when extended periodically in all directions. We also extend to higher-dimensions the Golomb-Moreno Conjecture that asserts all circular Costas maps are those from the Welch construction, which had been proved in the two-dimensional case. We prove a weaker version of this conjecture in the higher-dimensional context.