Three-Level Cretan Matrices Constructed via Conference Matrices

Полный текст:


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.

Об авторах

N. Balonin
Saint-Petersburg State University of Aerospace Instrumentation

M. Sergeev
Saint-Petersburg State University of Aerospace Instrumentation

Ofer Hadar
Ben-Gurion University of the Negev

Jennifer Seberry
University of Wollongong

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Для цитирования: ., ., ., . . Информационно-управляющие системы. 2015;(2):4-6.

For citation: Balonin N.A., Sergeev M.B., Hadar O., Seberry J. Three-Level Cretan Matrices Constructed via Conference Matrices. Information and Control Systems. 2015;(2):4-6. (In Russ.)

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