Three-Level Cretan Matrices of Order 37

Полный текст:


Аннотация

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 program- complex.

Об авторах

N. Balonin
Saint-Petersburg State University of Aerospace Instrumentation
Россия


Jennifer Seberry
University of Wollongong
Россия


M. Sergeev
Saint-Petersburg State University of Aerospace Instrumentation
Россия


Список литературы

1. Balonin N. A., Mironovsky L. A. Hadamard Matrices of Odd Order. Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2006, no. 3, pp. 46-50 (In Russian).

2. Balonin N. A., Sergeev M. B. Local Maximum Determinant Matrices. Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2014, no. 1(68), pp. 2-15 (In Russian).

3. Balonin N. A., Seberry Jennifer. Remarks on Extremal and Maximum Determinant Matrices with Moduli of Real Entries - 1. Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2014, no. 5(72), pp. 2-4.

4. Hadamard J. Resolution d’une Question Relative aux De terminants. Bulletin des Sciences Mathematiques, 1893, vol. 17, pp. 240-246 (In French).

5. Seberry Jennifer. Regular Hadamard Matrices of Order 36. Available at: http://www.uow.edu.au/jennie/ matrices/H36/36R.html (accessed 5 February 2015).

6. Balonin N. A., Seberry Jennifer. Two Level Cretan Matrices Constructed via Singer Difference Sets. Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2014, no. 6(73), pp. 2-5.


Дополнительные файлы

Для цитирования: ., ., . . Информационно-управляющие системы. 2015;(2):2-3.

For citation: Balonin N.A., Seberry J., Sergeev M.B. Three-Level Cretan Matrices of Order 37. Information and Control Systems. 2015;(2):2-3. (In Russ.)

Просмотров: 29


Creative Commons License
Контент доступен под лицензией Creative Commons Attribution 4.0 License.


ISSN 1684-8853 (Print)
ISSN 2541-8610 (Online)