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Special Matrices

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2300-7451
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Two-level Cretan matrices constructed using SBIBD

N. A. Balonin
  • Saint Petersburg State University of Aerospace Instrumentation, 67, B. Morskaia St., 190000, St. Petersburg, Russian Federation
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Jennifer Seberry
  • School of Computing and Information Technology, Faculty of Engineering and Information Sciences, University of Wollongong, NSW 2522, Australia
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2015-08-04 | DOI: https://doi.org/10.1515/spma-2015-0017

Abstract

Two-level Cretan matrices are orthogonal matrices with two elements, x and y. At least one element per row and column is 1 and the other element has modulus ≤ 1. These have been studied in the Russian literature for applications in image processing and compression. Cretan matrices have been found by both mathematical and computational methods but this paper concentrates on mathematical solutions for the first time.

We give, for the first time, families of Cretan matrices constructed using the incidence matrix of a symmetric balanced incomplete block design and Hadamard related difference sets.

Keywords: Hadamard matrices; orthogonal matrices; Cretan matrices; symmetric balanced incomplete block designs (SBIBD); difference sets

MSC: 05B20

References

  • [1] N. A. Balonin. Existence of Mersenne Matrices of 11th and 19th Orders. Informatsionno-upravliaiushchie sistemy, 2013. 2, pp. 89 – 90 (In Russian). Google Scholar

  • [2] N. A. Balonin and L. A. Mironovski. Hadamard matrices of odd order, Informatsionno-upravliaiushchie sistemy, 2006.3, pp. 46–50 (In Russian). Google Scholar

  • [3] N. A. Balonin and Jennifer Seberry. Remarks on extremal and maximum determinant matrices with real entries ≤ 1. Informatsionno-upravliaiushchie sistemy, 5, (71) (2014), p2–4. (In English). Google Scholar

  • [4] N. A. Balonin and M. B. Sergeev. On the issue of existence of Hadamard and Mersenne matrices. Informatsionnoupravliaiushchie sistemy, 2013. 5 (66), pp. 2–8 (In Russian). Google Scholar

  • [5] J. Hadamard, Résolution d’une question relative aux déterminants. Bulletin des Sciences Mathematiques. 1893. Vol. 17. pp. 240-246. Google Scholar

  • [6] La Jolla Difference Set Repository. URL www.ccrwest.org/ds.html. Viewed 2014:10:03. Google Scholar

  • [7] Jennifer Seberry and Mieko Yamada. Hadamard matrices, sequences, and block designs, Contemporary Design Theory: A Collection of Surveys, J. H. Dinitz and D. R. Stinson, eds., John Wiley and Sons, Inc., 1992. pp. 431–560. Google Scholar

About the article

Received: 2015-03-17

Accepted: 2015-07-25

Published Online: 2015-08-04


Citation Information: Special Matrices, Volume 3, Issue 1, ISSN (Online) 2300-7451, DOI: https://doi.org/10.1515/spma-2015-0017.

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©2015 N. A. Balonin and Jennifer Seberry. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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