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August  2016, 10(3): 547-554. doi: 10.3934/amc.2016025

On the existence of Hadamard difference sets in groups of order 400

Joško Mandić 1, and  Tanja Vučičić 1,

1. 

Faculty of Science, University of Split, Rudjera Boškovića 33, Split, 21000, Croatia, Croatia

Received  May 2015 Revised  June 2016 Published  August 2016

This paper concerns the problem of the existence of Hadamard difference sets in nonabelian groups of order 400. By introducing a new construction method, we construct new difference sets in 20 groups. We survey non-existence results, verifying non-existence in 45 groups.
Keywords: difference set., Transitive incidence structure, block design.
Mathematics Subject Classification: Primary: 05B05, 05B1.
Citation: Joško Mandić, Tanja Vučičić. On the existence of Hadamard difference sets in groups of order 400. Advances in Mathematics of Communications, 2016, 10 (3) : 547-554. doi: 10.3934/amc.2016025
References:
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T. Vučičić, New symmetric designs and nonabelian difference sets with parameters $(100, 45, 20)$, J. Combin. Des., 8 (2000), 291-299. doi: 10.1002/1520-6610(2000)8:4<291::AID-JCD6>3.0.CO;2-L.  Google Scholar

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show all references

References:
[1]

J. Alexander, R. Balasubramanian, J. Martin, K. Monahan, H. Pollatsek and A. Sen, Ruling out $(160,54,18)$ difference sets in some nonabelian groups, J. Combin. Des., 8 (2000), 221-231. doi: 10.1002/1520-6610(2000)8:4<221::AID-JCD1>3.3.CO;2-Y.  Google Scholar

[2]

T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge Univ. Press, 1999.  Google Scholar

[3]

W. Bosma, J. J. Cannon, C. Fieker and A. Steel, Handbook of Magma Functions, edition 2.16, 2010. Google Scholar

[4]

P. J. Cameron and C. E. Praeger, Block-transitive t-designs I: point-imprimitive designs, Discrete Math., 118 (1993), 33-43. doi: 10.1016/0012-365X(93)90051-T.  Google Scholar

[5]

J. A. Davis and J. Jedwab, A survey of Hadamard difference sets, in Groups, Difference Sets and the Monster (eds. K.T. Arasu et al.), de Gruyter, Berlin-New York, 1996, 145-156.  Google Scholar

[6]

J. F. Dillon, Variations on a scheme of McFarland for noncyclic difference sets, J. Combin. Theory Ser. A, 40 (1985), 9-21. doi: 10.1016/0097-3165(85)90043-3.  Google Scholar

[7]

J. F. Dillon, Some REALLY beautiful Hadamard matrices, Crypt. Commun., {2} (2010), 271-292. doi: 10.1007/s12095-010-0031-1.  Google Scholar

[8]

The GAP Group, GAP - Groups, Algorithms, and Programming, version 4.4,, available online at , ().   Google Scholar

[9]

A. Golemac and T. Vučičić, New difference sets in nonabelian groups of order $100$, J. Combin. Des., 9 (2001), 424-434. doi: 10.1002/jcd.1021.  Google Scholar

[10]

E. H. Moore and H. S. Pollatsek, Difference Sets: Connecting Algebra, Combinatorics and Geometry, AMS, Providence, 2013. doi: 10.1090/stml/067.  Google Scholar

[11]

K. W. Smith, Nonabelian Hadamard difference sets, J. Combin. Theory Ser. A, 70 (1995), 144-156. doi: 10.1016/0097-3165(95)90084-5.  Google Scholar

[12]

T. Vučičić, New symmetric designs and nonabelian difference sets with parameters $(100, 45, 20)$, J. Combin. Des., 8 (2000), 291-299. doi: 10.1002/1520-6610(2000)8:4<291::AID-JCD6>3.0.CO;2-L.  Google Scholar

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=, ().   Google Scholar

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