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

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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

Abstract
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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:

[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.
[2]	T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge Univ. Press, 1999.
[3]	W. Bosma, J. J. Cannon, C. Fieker and A. Steel, Handbook of Magma Functions, edition 2.16, 2010.
[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.
[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.
[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.
[7]	J. F. Dillon, Some REALLY beautiful Hadamard matrices, Crypt. Commun., {2} (2010), 271-292. doi: 10.1007/s12095-010-0031-1.
[8]	The GAP Group, GAP - Groups, Algorithms, and Programming, version 4.4, available online at http://www.gap-system.org
[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.
[10]	E. H. Moore and H. S. Pollatsek, Difference Sets: Connecting Algebra, Combinatorics and Geometry, AMS, Providence, 2013. doi: 10.1090/stml/067.
[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.
[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.
[13]	http://www.pmfst.unist.hr/~vucicic/

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.
[2]	T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge Univ. Press, 1999.
[3]	W. Bosma, J. J. Cannon, C. Fieker and A. Steel, Handbook of Magma Functions, edition 2.16, 2010.
[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.
[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.
[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.
[7]	J. F. Dillon, Some REALLY beautiful Hadamard matrices, Crypt. Commun., {2} (2010), 271-292. doi: 10.1007/s12095-010-0031-1.
[8]	The GAP Group, GAP - Groups, Algorithms, and Programming, version 4.4, available online at http://www.gap-system.org
[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.
[10]	E. H. Moore and H. S. Pollatsek, Difference Sets: Connecting Algebra, Combinatorics and Geometry, AMS, Providence, 2013. doi: 10.1090/stml/067.
[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.
[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.
[13]	http://www.pmfst.unist.hr/~vucicic/

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