On the generalization of the Costas property in higher dimensions

Drakakis, Konstantinos

doi:10.3934/amc.2010.4.1

Cited by 6 publications

(11 citation statements)

References 13 publications

(25 reference statements)

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“…)} has size 4: its elements comprise a single equivalence class of Costas arrays whose corresponding permutations are (2, 4, 5, 1, 6, 3), (3, 6, 1, 5, 4, 2), (4, 1, 6, 2, 3, 5), (5,3,2,6,1,4). We show in Section 3 that this example is a member of an infinite family of Costas cubes all of whose elements D satisfy |S(D)| = 4.…”

Section: Figure 1: Costas Cube and Its Three Projectionsmentioning

confidence: 99%

“…For example, let D = (d i,j,k ) be the 6 × 6 × 6 array given by (1,6,4), (2,4,6), (3, 1, 2), (4, 3, 1), (5,2,5), (6,5,3)} .…”

Section: Introductionmentioning

confidence: 99%

“…Represent F 3 3 as Z 3 [x]/ 1 + 2x 2 + x 3 and take φ = 2 + 2x, for which 1 − φ = 2 + x and 1 − φ −1 = x + x 2 are also primitive. From(6), the triples (i, j, k) for which…”

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confidence: 99%

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Costas Cubes

Jedwab

Yen

2018

IEEE Trans. Inform. Theory

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A Costas array is a permutation array for which the vectors joining pairs of 1s are all distinct. We propose a new three-dimensional combinatorial object related to Costas arrays: an order n Costas cube is an array (d i,j,k ) of size n × n × n over Z2 for which each of the three projections of the array onto two dimensions, namely ( i d i,j,k ) and ( j d i,j,k ) and ( k d i,j,k ), is an order n Costas array. We determine all Costas cubes of order at most 29, showing that Costas cubes exist for all these orders except 18 and 19 and that a significant proportion of the Costas arrays of certain orders occur as projections of Costas cubes. We then present constructions for four infinite families of Costas cubes. J. Jedwab is with

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Section: Figure 1: Costas Cube and Its Three Projectionsmentioning

confidence: 99%

“…For example, let D = (d i,j,k ) be the 6 × 6 × 6 array given by (1,6,4), (2,4,6), (3, 1, 2), (4, 3, 1), (5,2,5), (6,5,3)} .…”

Section: Introductionmentioning

confidence: 99%

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Costas Cubes

Jedwab

Yen

2018

IEEE Trans. Inform. Theory

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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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“…The FPGA implementation achieved speedups of up to 40× over the GPU and 4.44× over the fastest reported software implementation [9]. The first generalizations of Costas arrays were given by Drakakis in [12]. He defined various classes of multidimensional arrays by modifying the Costas property constraints and presented examples.…”

Section: Previous Workmentioning

confidence: 99%

Multidimensional Costas Arrays and Their Enumeration Using GPUs and FPGAs

Arce-Nazario

Ortiz-Ubarri

2012

International Journal of Reconfigurable Computing

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The enumeration of two-dimensional Costas arrays is a problem with factorial time complexity and has been solved for sizes up to 29 using computer clusters. Costas arrays of higher dimensionality have recently been proposed and their properties are beginning to be understood. This paper presents, to the best of our knowledge, the first proposed implementations for enumerating these multidimensional arrays in GPUs and FPGAs, as well as the first discussion of techniques to prune the search space and reduce enumeration run time. Both GPU and FPGA implementations rely on Costas array symmetries to reduce the search space and perform concurrent explorations over the remaining candidate solutions. The fine grained parallelism utilized to evaluate and progress the exploration, coupled with the additional concurrency provided by the multiple instanced cores, allowed the FPGA (XC5VLX330-2) implementation to achieve speedups of up to 30× over the GPU (GeForce GTX 580).

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On the generalization of the Costas property in higher dimensions

Cited by 6 publications

References 13 publications

Costas Cubes

Costas Cubes

Open problems in Costas arrays

Multidimensional Costas Arrays and Their Enumeration Using GPUs and FPGAs

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