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Three-Level Cretan Matrices of Order 37

Purpose: This note discusses three-level orthogonal matrices which were first highlighted by J. J. Sylvester. Hadamard matrices, symmetric conference matrices, and weighing matrices are the best known of these matrices with entries from the unit disk. The goal of this note is to develop a theory of such matrices based on preliminary research results. Methods: Extreme solutions (using the determinant) have been established by minimization of the maximum of the absolute values of the elements of the matrices followed by their subsequent classification. Results: We give a three-level Cretan(37). This is new and the first time such a matrix has been found whose order is other than a 2k, k is an even integer. The methods given in this note may be used to construct many more Cretan matrices based on regular Hadamard matrices. Practical relevance: The over-riding aim is to seek Cretan(n) with absolute or relative (local) maximal determinants as they have many applications in image processing and masking. Web addresses are given for other illustrations and other matrices with similar properties. Algorithms to construct Cretan matrices have been implemented in developing software of the research programcomplex.

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теоретическая и прикладная математика UDC 004.438 doi:10.15217/issn1684-8853.2015.2.2 THREE-LEVEL CRETAN MATRICES OF ORDER 37 N. A. Balonina, Dr. Sc., Tech., Professor, korbendfs@mail.ru Jennifer Seberryb, PhD, Professor of Computer Science, jennifer_seberry@uow.edu.au M. B. Sergeeva, Dr. Sc., Tech., Professor, mbse@mail.ru aSaint-Petersburg State University of Aerospace Instrumentation, 67, B. Morskaia St., 190000, Saint-Petersburg, Russian Federation bSchool of Computer Science and Software Engineering, Faculty of Engineering and Information Science, University of Wollongong, NSW, 2522, Australia matrices, symmetric conference matrices, and weighing matrices are the best known of these matrices with entries from the unit disk. <...> The goal of this note is to develop a theory of such matrices based on preliminary research results. <...> Methods: Extreme solutions (using the determinant) have been established by minimization of the maximum of the absolute values of the elements of the matrices followed by their subsequent classification. <...> This is new and the first time such a matrix has been found whose order is other than a 2k, k is an even integer. <...> The methods given in this note may be used to construct many more Cretan matrices based on regular Hadamard matrices. <...> Practical relevance: The over-riding aim is to seek Cretan(n) with absolute or relative (local) maximal determinants as they have many applications in image processing and masking. <...> Web addresses are given for other illustrations and other matrices with similar properties. <...> Keywords — Hadamard Matrices, Regular Hadamard Matrices, Orthogonal Matrices, Cretan Matrices. <...> We note that the strict definition of an orthogonal matrix, X, of order n, is that XTX = XXT = ωIn, where In is the identity matrix of order n. <...> A Cretan(n) (CM) matrix, S, is a orthogonal matrix of order n with entries with moduli 1, where there must be at least one 1 per row and column. <...> The inner product of a row of CM(n) with itself is the weight ω. <...> The inner product of distinct rows of CM(n) is zero. <...> A µ-level Cretan(n; µ; ω) matrix, CM(n; µ; ω), has µ levels or values for its entries. <...> The over-riding aim is <...>
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