Open problems in Costas arrays

Abstract

A collection of open problems in Costas arrays is presented, classified into several categories, along with the context in which they arise. IntroductionCostas arrays are square arrays of dots/1s and blanks/0s, such that there exists exactly one dot per row and column (that is, they are permutation arrays), and such that a) no four dots not lying on a straight line form a parallelogram, b) no four dots lying on a straight line form two equidistant pairs, and c) no three dots lying on a straight line are equidistant. They appeared for the first time in 1965 in the context of SONAR detection ([18], and later [19] as a journal publication), when J. P. Costas, disappointed by the poor performance of SONARs, used them to describe a novel frequency hopping pattern for SONAR systems with optimal auto-correlation properties (namely auto-correlation sidelobes of height at most 1). At that stage their study was entirely empirical and application-oriented. In 1984, however, after the publication of the two main construction methods for Costas arrays by S. Golomb [58], still the only ones of general applicability available today, they officially acquired their present name and they became an object of mathematical interest and study.The present author, along with his collaborators, has published numerous journal publications over the past six years on several aspects of Costas arrays, namely their theory, their properties, their enumeration, and their applications, in which many questions about them were settled. However, many old questions, along with several new ones emerging in the course of the author's research, have remained open, defying all attempts to answer them. The most important and promising amongst those are collected in this work, in the hope that it will get researchers interested in Costas arrays and willing to contribute further with their efforts.Inevitably, the selection of the material and its presentation were influenced by the author's own preferences and views. Although a conscientious effort towards impartiality was made, the broad ground rule was set that this study is intended to be centered around Costas arrays themselves, as opposed to an object or application which Costas arrays are related to. Accordingly, the selection of the various open problems presented was based on their potential advancement of our knowledge of Costas arrays themselves, be it in the direction of their theory or their applications, rather than on the advancement of some research area (e.g. SONAR/RADAR systems or cryptography) through the introduction (or further application) of Costas arrays in it.An attempt has been made to group the problems presented into families, representing thematic units. Once more, it should be stressed that this grouping is, to some extent, subjective, as it will be seen that some problems would naturally fit in several groups: in such cases, the choice was based on the problem's main context and orientation. This is not the first time a compilation of open problems in Costas arrays is attempte...

Parallel m-dimensional relative ant colony optimization (mDRACO) for the Costas-array problem

“…Definition 2 A Costas array (Drakakis 2011) of size m is an m×m array of 1s and 0s such that there is exactly one element with value 1 in each row and column, and there are no equal displacement vectors between distinct pairs of distinct elements with value 1. (Equivalently, no three or four elements with value 1 form a (potentially degenerate) parallelogram.…”

“…The Costas-array problem is an existence problem. Whether there exist Costas arrays of size m for all positive integers m remains an open question (Drakakis 2011). The smallest m for which it is unknown whether a Costas array of size m exists is 32 (Drakakis 2011).…”

“…Costas arrays are square arrays of 1s and 0s such that there is exactly one element with value 1 in each row and column and no three or four distinct elements with value 1 form a (potentially degenerate) parallelogram (Drakakis 2011). John P. Costas first described these arrays in 1965 as representations of sequences of SONAR pulse frequencies with optimal auto-correlation properties (Costas 1984); they have since been applied in other areas, such as communication systems and cryptography (Taylor et al 2011).…”

“…Costas arrays are also objects of theoretical interest to the research community. Golomb and Taylor (1984) describe several algebraic constructions for certain classes of Costas arrays in 1984, which remain the only known Costas-array constructions (Drakakis 2011). All Costas arrays of order m ≤ 29 are known (Drakakis et al 2011), but the existence of Costas arrays for many sizes m ≥ 30 remains an open question (Drakakis 2011).…”

Parallel m-Dimensional Relative Ant Colony Optimization (mDRACO) For the Costas-Array Problem

“…(potentially degenerate) parallelogram [10]. John P. Costas first described these arrays in 1965 as representations of sequences of SONAR pulse frequencies with optimal autocorrelation properties [6]; they have since been applied in other areas, such as communication systems and cryptography [19].…”