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Special Matrices

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Construction of symmetric Hadamard matrices of order 4v for v = 47, 73, 113

N. A. Balonin
  • Saint-Petersburg State University of Aerospace Instrumentation, 67, B. Morskaia St., 190000, Saint-Petersburg, Russian Federation
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/ D. Ž. Ðokovic
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  • University of Waterloo, Department of Pure Mathematics and Institute for Quantum Computing, Waterloo, Ontario, N2L 3G1, Canada
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/ D. A. Karbovskiy
  • Saint-Petersburg State University of Aerospace Instrumentation, 67, B. Morskaia St., 190000, Saint-Petersburg, Russian Federation
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Published Online: 2018-01-24 | DOI: https://doi.org/10.1515/spma-2018-0002

Abstract

We continue our systematic search for symmetric Hadamard matrices based on the so called propus construction. In a previous paper this search covered the orders 4v with odd v ≤ 41. In this paper we cover the cases v = 43, 45, 47, 49, 51. The odd integers v < 120 for which no symmetric Hadamard matrices of order 4v are known are the following: 47, 59, 65, 67, 73, 81, 89, 93, 101, 103, 107, 109, 113, 119. By using the propus construction, we found several symmetric Hadamard matrices of order 4v for v = 47, 73, 113.

Keywords: Symmetric Hadamard matrices; Propus array; cyclic difference families; Diophantine equations

References

  • [1] N. A. Balonin, Y. N. Balonin, D. Ž. Ðokovic, D. A. Karbovskiy, and M. B. Sergeev, Construction of symmetric Hadamard matrices, Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2017, no. 5, pp. 2-11 (In Russian). doi: 10.15217/issn1684-8853.2017.5.2CrossrefGoogle Scholar

  • [2] R. Craigen and H. Kharaghani, Hadamard matrices and Hadamard designs, in Handbook of Combinatorial Designs, 2nd ed. C. J. Colbourn, J. H. Dinitz (eds). Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2007.Google Scholar

  • [3] O. DiMateo, D. Ž. Ðokovic, I. S. Kotsireas, Symmetric Hadamardmatrices of order 116 and 172 exist, SpecialMatrices 3 (2015), 227-234.Google Scholar

  • [4] J. Seberry and N. A. Balonin, The propus construction for symmetric Hadamard matrices, Australasian Journal of Combinatorics, 69(3) (2017), 349-357.Google Scholar

  • [5] M. Xia, T. Xia, J. Seberry and J. Wu, An infinite series of Goethals-Seidel arrays, Discrete Applied Mathematics 145 (2005), 498-504.Google Scholar

About the article

Received: 2017-10-07

Accepted: 2017-12-22

Published Online: 2018-01-24


Citation Information: Special Matrices, Volume 6, Issue 1, Pages 11–22, ISSN (Online) 2300-7451, DOI: https://doi.org/10.1515/spma-2018-0002.

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© 2018, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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