Abstract
We construct several new cyclic (v; k1, k2, k3; λ) difference families, with v ≡ 3 (mod 4) a prime and λ = k1 + k2 + k3 − (3v − 1)/4. Such families can be used in conjunction with the well-known Paley-Todd difference sets to construct skew-Hadamard matrices of order 4v. Our main result is that we have constructed for the first time the examples of skew Hadamard matrices of orders 4 · 239 = 956 and 4 · 331 = 1324.
References
[1] L. D. Baumert, Cyclic difference sets. Lecture Notes in Mathematics, Vol. 182 Springer-Verlag, Berlin-New York 1971. 10.1007/BFb0061260Search in Google Scholar
[2] D. Ž. Ðokovic, Some new D-optimal designs. Australas. J Combin. 15 (1997), 221–231. Search in Google Scholar
[3] D. Ž. Ðokovic, Skew-Hadamard matrices of orders 188 and 388 exist. International Mathematical Forum, 22 (2008), 1063– 1068 Search in Google Scholar
[4] D. Ž. Ðokovic, Skew-Hadamard matrices of orders 436, 580, and 988 exist, J. Combin. Designs, 16 (2008), 493–498. 10.1002/jcd.20180Search in Google Scholar
[5] D. Ž. Ðokovic, Supplementary difference sets with symmetry for Hadamard matrices. Operators and Matrices, 3 (2009), 557–569. Search in Google Scholar
[6] D. Ž. Ðokovic, O. Golubitsky and I. S. Kotsireas, Some new orders of Hadamard and skew-Hadamard matrices. J. Combin. Designs, 22 (2014), 270–277. Search in Google Scholar
[7] D. Ž. Ðokovic and I. S. Kotsireas, New results on D-optimal matrices. J. Combin. Designs, 20 (2012), 278–289. 10.1002/jcd.21302Search in Google Scholar
[8] D. Ž. Ðokovic and I. S. Kotsireas, D-optimal matrices of orders 118, 138, 150, 154 and 174. In: C. J. Colbourn (ed.) Algebraic Design Theory and Hadamard Matrices, pp. 71–82, ADTHM, Lethbridge, Alberta, Canada, July 2014. Springer Proceedings in Mathematics & Statistics, vol. 133. Springer 2015. 10.1007/978-3-319-17729-8_6Search in Google Scholar
[9] W. Duke, Some old problems and new results about quadratic forms, Not. Amer. Math. Soc. 44 (1997), 190–196. Search in Google Scholar
[10] R. J. Fletcher, C. Koukouvinos and J. Seberry, New skew-Hadamard matrices of order 4.59 and new D-optimal designs of order 2.59, Discrete Math. 286 (2004), 251–253. 10.1016/j.disc.2004.05.009Search in Google Scholar
[11] D. Jungnickel, A. Pott, K. W. Smith, Difference sets, in Handbook of combinatorial designs. Edited by C. J. Colbourn and J. H. Dinitz. Second edition. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2007. Search in Google Scholar
[12] C. Koukouvinos, S. Stylianou, On skew-Hadamard matrices, Discrete Math. 308 (2008), 2723–2731. 10.1016/j.disc.2006.06.037Search in Google Scholar
[13] J. Seberry, M. Yamada, Hadamardmatrices, sequences, and block designs. In Contemporary design theory, 431–560,Wiley- Intersci. Ser. Discrete Math. Optim., Wiley, New York, 1992, Search in Google Scholar
[14] D. R. Stinson, Combinatorial designs. Constructions and analysis. Springer-Verlag, New York, 2004. Search in Google Scholar
[15] J. H. van Lint, R. M. Wilson, A course in combinatorics. Cambridge University Press, Cambridge, 1992. Search in Google Scholar
©2016 Dragomir Ž. Ðokovic* and Ilias S. Kotsireas
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.