Symmetric Hadamard matrices of order 116 and 172 exist

Olivia Di Matteo 1 , Dragomir Ž. Ðoković 2 ,  and Ilias S. Kotsireas 3
  • 1 University of Waterloo, Department of Physics and Astronomy, Institute for Quantum Computing
  • 2 University of Waterloo, Department of Pure Mathematics, Institute for Quantum Computing, Waterloo, Ontario, N2L 3G1, Canada
  • 3 Wilfrid Laurier University, Department of Physics & Computer Science, Waterloo, Ontario, N2L 3C5, Canada


We construct new symmetric Hadamard matrices of orders 92, 116, and 172. While the existence of those of order 92 was known since 1978, the orders 116 and 172 are new. Our construction is based on a recent new combinatorial array (GP array) discovered by N. A. Balonin and J. Seberry. For order 116 we used an adaptation of an algorithm for parallel collision search. The adaptation pertains to the modification of some aspects of the algorithm to make it suitable to solve a 3-way matching problem. We also point out that a new infinite series of symmetric Hadamard matrices arises by plugging into the GP array the matrices constructed by Xia, Xia, Seberry, and Wu in 2005.

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  • [1] N. A. Balonin, Jennifer Seberry, Visualizing Hadamard matrices: the propus construction, preprint 15pp (submitted 6 Aug 2014).

  • [2] N. A. Balonin, Jennifer Seberry, A review and new symmetric conferencematrices, Informatsionno-upravliaiushchie sistemy, 2014, 8470; 4 (71), 2–7.

  • [3] R. Craigen and H. Kharaghani, HadamardMatrices and Hadamard Designs. In Handbook of Combinatorial Designs. Edited by Charles J. Colbourn and Jeffrey H. Dinitz. Second edition. DiscreteMathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2007.

  • [4] D. Ž. Ðoković and I. S. Kotsireas, New results on D-optimal matrices. J. Combin. Designs, 20 (2012), 278–289.

  • [5] D. Ž. Ðoković and I. S. Kotsireas, Compression of periodic complementary sequences and applications, Des. Codes Cryptogr. 74 (2015), 365–377.

  • [6] Y. J. Ionin and M. S. Shrikhande, Combinatorics of Symmetric Designs. New Mathematical Monographs, 5. Cambridge University Press, Cambridge, 2006.

  • [7] R. Mathon, Symmetric conference matrices of order pq2 + 1. Canad. J. Math., 30 (1978), 321–331.

  • [8] J. Seberry Wallis, Hadamard Matrices, in W. D. Wallis, A. Penfold Street, Jennifer Seberry Wallis, Combinatorics: Room squares, sum-free sets, Hadamard matrices. Lecture Notes in Mathematics, Vol. 292. Springer-Verlag, Berlin-New York, 1972.

  • [9] R. J. Turyn, An infinite class of Williamson matrices, J. Combinatorial Theory Ser. A 12 (1972), 319–321.

  • [10] Paul C. van Oorschot and Michael J. Wiener, Parallel collision search with cryptanalytic applications, Journal of Cryptology, January 1999, Volume 12, Issue 1, 1–28.

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

  • [12] O. Di Matteo, Parallelizing quantum circuit synthesis. MSc thesis, University of Waterloo (2015).


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Special Matrices is a peer-reviewed, open access electronic journal that publishes original articles of wide significance and originality in all areas of research involving structured matrices present in various branches of pure and applied mathematics and their noteworthy applications in physics, engineering, and other sciences.