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THREE-LEVEL CRETAN MATRICES CONSTRUCTED VIA CONFERENCE MATRICES

Purpose: Orthogonal matrices with many applications introduced by J. J. Sylvester have been become famous: 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 orthogonal 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 show that if S is a core of a symmetric conference weighing matrix, then there exists a three-level orthogonal matrix, X. We apply this result to the three-level matrices given by Paley using Legendre symbols to give a new infinite family of Cretan orthogonal matrices. An algorithmic optimization procedure is known which raises the value of the determinant. Our example is for Cretan matrices upto, say four decimal places (but could be made more). Practical relevance: The over-riding aim is to seek Cretan matrices as they have many applications in image processing (compression, masking) to statisticians undertaking medical or agricultural research, and to obtain lossless circuits for telecommunications conference networking. 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 program-complex.

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ТЕОРЕТИЧЕСКАЯ И ПРИКЛАДНАЯ МАТЕМАТИКА UDC 004.438 doi:10.15217/issn1684-8853.2015.2.4 THREE-LEVEL CRETAN MATRICES CONSTRUCTED VIA CONFERENCE MATRICES N. A. Balonina, Dr. Sc., Tech., Professor, korbendfs@mail.ru M. B. Sergeeva, Dr. Sc., Tech., Professor, mbse@mail.ru Ofer Hadarb, PhD, Professor, hadar@cse.bgu.ac.il Jennifer Seberryc, PhD, Professor of Computer Science, jennifer_seberry@uow.edu.au aSaint-Petersburg State University of Aerospace Instrumentation, 67, B. Morskaia St., 190000, Saint-Petersburg, Russian Federation bBen-Gurion University of the Negev, POB 653, 1, Ben Gurion St., Beer Sheva, 74105, Israel cCentre for Computer and Information Security Research, School of Computer Science and Software Engineering, EIS, University of Wollongong, NSW 2522, Australia Purpose: Orthogonal matrices with many applications introduced by J. J. Sylvester have been become famous: Hadamard matrices, symmetric conference matrices, and weighing matrices are the best known of these matrices with entries from the unit disk. <...> 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 show that if S is a core of a symmetric conference weighing matrix, then there exists a three-level orthogonal matrix, X. We apply this result to the three-level matrices given by Paley using Legendre symbols to give a new infinite family of Cretan orthogonal matrices. <...> An algorithmic optimization procedure is known which raises the value of the determinant. <...> Our example is for Cretan matrices upto, say four decimal places (but could be made more). <...> Practical relevance: The over-riding aim is to seek Cretan matrices as they have many applications in image processing (compression, masking) to statisticians undertaking medical or agricultural research, and to obtain lossless circuits for telecommunications conference networking. <...> Web addresses are given for other illustrations and other matrices with similar properties. <...> Keywords — Hadamard Matrices, Cretan Matrices, Conference Weighing Matrices, Orthogonal Matrices, Legendre Introduction A Cretan(n) matrix, (CM), is a orthogonal matrix of order n with moduli of entries 1, where there must be at least <...>
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