Combinatorial designs with Costas arrays properties

Etzion, Tuvi

doi:10.1016/0012-365x(91)90250-6

Cited by 11 publications

(8 citation statements)

References 8 publications

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“…Examples include Sidon sets (also known as Golomb rulers) [30,82], RADAR/SONAR sequences (also known as Golomb rectangles) [14,56,65,73,74,82,83,99], Golomb rectangles [83], honeycomb arrays [8,10,64], as well as arrays of dots where "half" of the Costas property hold, namely where all linear segments connecting pairs of dots either have distinct lengths or distinct slopes (but not necessarily both) [51,68,75,98]. Further examples include generalizations of the Costas property in higher dimensions [27,53] and in the continuum [26,29,47]. Some Costas arrays were also found found to yield Almost Perfect Nonlinear (APN) permutations [36,45], which find applications in cryptography.…”

Section: Problems Related To Variants and Generalizations Of Costas A...mentioning

confidence: 99%

Open problems in Costas arrays

Drakakis

2011

Preprint

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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...

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Section: Problems Related To Variants and Generalizations Of Costas A...mentioning

confidence: 99%

Open problems in Costas arrays

Drakakis

2011

Preprint

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“…Extensions of the Golomb and the Welch constructions to 3 dimensions were investigated in the past [6], but the objective then was slightly different: the cubes were so constructed that all the 2-dimensional "slices" along some of their directions be Costas arrays, or at least almost Costas arrays in a certain sense. In other words, although the construction was 3-dimensional, the Costas property was still investigated in 2 dimensions.…”

Section: Older Generalization Attemptsmentioning

confidence: 99%

On the generalization of the Costas property in higher dimensions

Drakakis¹

2010

Advances in Mathematics of Communications

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We investigate the generalization of the Costas property in 3 or more dimensions, and we seek an appropriate definition; the 2 main complications are a) that the number of "dots" this multidimensional structure should have is not obvious, and b) that the notion of the multidimensional permutation needs some clarification. After proposing various alternatives for the generalization of the definition of the Costas property, based on the definitions of the Costas property in 1 or 2 dimensions, we also offer some construction methods, the main one of which is based on the idea of reshaping Costas arrays into higher-dimensional entities.

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“…We call this the permutation representation of a Costas array. Using this notation, the Costas arrays in the first row of Example 1.1 have permutation representations (1,2,4,3), (4, 3, 1, 2), (2,1,3,4) and (3,4,2,1), respectively.…”

Section: Introductionmentioning

confidence: 99%

“…A Costas latin square of order n, denoted CLS(n), is a latin square of order n such that for each symbol i, 1 ≤ i ≤ n, a Costas array results if a dot is placed in the cells containing symbol i. Clearly a CLS(n) is equivalent to n disjoint Costas arrays of order n. Costas latin squares were first defined and studied by Etzion [3]. Etzion [3] defines a near Costas array of order n ≥ 2 to be an n × n array of dots and empty cells such that 1. there are n − 1 dots and n 2 − n + 1 empty cells, with at most one dot in each row and column, and 2. all the segments between pairs of dots differ in length or in slope.…”

Section: Introductionmentioning

confidence: 99%

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Packing Costas Arrays

Dinitz,

Ostergard,

Stinson

2011

Preprint

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A Costas latin square of order n is a set of n disjoint Costas arrays of the same order. Costas latin squares are studied here from a construction as well as a classification point of view. A complete classification is carried out up to order 27. In this range, we verify the conjecture that there is no Costas latin square for any odd order n ≥ 3. Various other related combinatorial structures are also considered, including near Costas latin squares (which are certain packings of near Costas arrays) and Vatican Costas squares.

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Combinatorial designs with Costas arrays properties

Cited by 11 publications

References 8 publications

Open problems in Costas arrays

Open problems in Costas arrays

On the generalization of the Costas property in higher dimensions

Packing Costas Arrays

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