Symmetric Hadamard matrices of order 116 and 172 exist

Olivia Di Matteo; Dragomir Ž. Ðoković; Ilias S. Kotsireas

Special Matrices (2015)

  • Volume: 3, Issue: 1, page 227-234, electronic only
  • ISSN: 2300-7451

Abstract

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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.

How to cite

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Olivia Di Matteo, Dragomir Ž. Ðoković, and Ilias S. Kotsireas. "Symmetric Hadamard matrices of order 116 and 172 exist." Special Matrices 3.1 (2015): 227-234, electronic only. <http://eudml.org/doc/275944>.

@article{OliviaDiMatteo2015,
abstract = {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.},
author = {Olivia Di Matteo, Dragomir Ž. Ðoković, Ilias S. Kotsireas},
journal = {Special Matrices},
language = {eng},
number = {1},
pages = {227-234, electronic only},
title = {Symmetric Hadamard matrices of order 116 and 172 exist},
url = {http://eudml.org/doc/275944},
volume = {3},
year = {2015},
}

TY - JOUR
AU - Olivia Di Matteo
AU - Dragomir Ž. Ðoković
AU - Ilias S. Kotsireas
TI - Symmetric Hadamard matrices of order 116 and 172 exist
JO - Special Matrices
PY - 2015
VL - 3
IS - 1
SP - 227
EP - 234, electronic only
AB - 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.
LA - eng
UR - http://eudml.org/doc/275944
ER -

References

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  2. [1] N. A. Balonin, Jennifer Seberry, Visualizing Hadamard matrices: the propus construction, preprint 15pp (submitted 6 Aug 2014). 
  3. [2] N. A. Balonin, Jennifer Seberry, A review and new symmetric conferencematrices, Informatsionno-upravliaiushchie sistemy, 2014, 8470; 4 (71), 2–7. 
  4. [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. 
  5. [4] D. Ž. Ðoković and I. S. Kotsireas, New results on D-optimal matrices. J. Combin. Designs, 20 (2012), 278–289. 
  6. [5] D. Ž. Ðoković and I. S. Kotsireas, Compression of periodic complementary sequences and applications, Des. Codes Cryptogr. 74 (2015), 365–377. 
  7. [6] Y. J. Ionin and M. S. Shrikhande, Combinatorics of Symmetric Designs. New Mathematical Monographs, 5. Cambridge University Press, Cambridge, 2006. Zbl1114.05001
  8. [7] R. Mathon, Symmetric conference matrices of order pq2 + 1. Canad. J. Math., 30 (1978), 321–331. Zbl0385.05018
  9. [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. Zbl1317.05003
  10. [9] R. J. Turyn, An infinite class of Williamson matrices, J. Combinatorial Theory Ser. A 12 (1972), 319–321. Zbl0237.05008
  11. [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. Zbl0992.94028
  12. [11] M. Xia, T. Xia, J. Seberry and J. Wu, An infinite series of Goethals–Seidel arrays, Discrete Applied Mathematics 145 (2005) , 498–504. Zbl1057.05019
  13. [12] O. Di Matteo, Parallelizing quantum circuit synthesis. MSc thesis, University of Waterloo (2015). 

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