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Difference sets and doubly transitive actions on Hadamard matrices. (English) Zbl 1242.05038
Summary: Non-affine groups acting doubly transitively on a Hadamard matrix have been classified by Ito. Implicit in this work is a list of Hadamard matrices with non-affine doubly transitive automorphism group. We give this list explicitly, in the process settling an old research problem of Ito and Leon.
We then use our classification to show that the only cocyclic Hadamard matrices developed from a difference set with non-affine automorphism group are those that arise from the Paley Hadamard matrices.
If $$H$$ is a cocyclic Hadamard matrix developed from a difference set then the automorphism group of $$H$$ is doubly transitive. We classify all difference sets which give rise to Hadamard matrices with non-affine doubly transitive automorphism group. A key component of this is a complete list of difference sets corresponding to the Paley Hadamard matrices.
As part of our classification we uncover a new triply infinite family of skew-Hadamard difference sets. To our knowledge, these are the first skew-Hadamard difference sets to be discovered in non-abelian $$p$$-groups with no exponent restriction.
As one more application of our main classification, we show that Hall’s sextic residue difference sets give rise to precisely one cocyclic Hadamard matrix.

##### MSC:
 05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
Magma
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##### References:
 [1] Baumert, Leonard D., Cyclic difference sets, Lecture notes in math., vol. 182, (1971), Springer-Verlag Berlin · Zbl 0218.05009 [2] Beth, Thomas; Jungnickel, Dieter; Lenz, Hanfried, Design theory, vol. I, Encyclopedia math. appl., vol. 69, (1999), Cambridge University Press Cambridge · Zbl 0945.05004 [3] Bosma, W.; Cannon, J.; Playoust, C., The magma algebra system. I. the user language, J. symbolic comput., 24, 235-265, (1997) · Zbl 0898.68039 [4] Bouyukliev, Iliya; Fack, Veerle; Winne, Joost, $$2 -(31, 15, 7)$$, $$2 -(35, 17, 8)$$ and $$2 -(36, 15, 6)$$ designs with automorphisms of odd prime order, and their related Hadamard matrices and codes, Des. codes cryptogr., 51, 2, 105-122, (2009) · Zbl 1247.05031 [5] Crnković, Dean; Rukavina, Sanja, On Hadamard $$(35, 17, 8)$$ designs and their automorphism groups, J. appl. algebra discrete struct., 1, 3, 165-180, (2003) · Zbl 1046.05010 [6] de Launey, Warwick; Flannery, Dane, Algebraic design theory, Math. surveys monogr., vol. 175, (2011), American Mathematical Society Providence, RI · Zbl 1235.05001 [7] de Launey, Warwick; Stafford, Richard M., On cocyclic weighing matrices and the regular group actions of certain Paley matrices, Coding, cryptography and computer security, Lethbridge, AB, 1998, Discrete appl. math., 102, 1-2, 63-101, (2000) · Zbl 0961.05068 [8] Ding, Cunsheng; Yuan, Jin, A family of skew Hadamard difference sets, J. combin. theory ser. A, 113, 7, 1526-1535, (2006) · Zbl 1106.05016 [9] Feng, Tao, Non-abelian skew Hadamard difference sets fixed by a prescribed automorphism, J. combin. theory ser. A, 118, 1, 27-36, (2011) · Zbl 1225.05046 [10] Hall, Marshall, Note on the Mathieu group $$M_{12}$$, Arch. math. (basel), 13, 334-340, (1962) · Zbl 0109.25704 [11] Hall, Marshall, Combinatorial theory, Wiley-intersci. ser. discrete math., (1986), John Wiley & Sons Inc., A Wiley-Interscience Publication New York · Zbl 0588.05001 [12] Hering, Christoph, Transitive linear groups and linear groups which contain irreducible subgroups of prime order, Geom. dedicata, 2, 425-460, (1974) · Zbl 0292.20045 [13] Hering, Christoph, Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II, J. algebra, 93, 1, 151-164, (1985) · Zbl 0583.20003 [14] Horadam, K.J., Hadamard matrices and their applications, (2007), Princeton University Press Princeton, NJ · Zbl 1198.15001 [15] Huppert, Bertram; Blackburn, Norman, Finite groups. III, Grundlehren math. wiss., vol. 243, (1982), Springer-Verlag Berlin [16] Ito, Noboru, Hadamard matrices with “doubly transitive” automorphism groups, Arch. math. (basel), 35, 1-2, 100-111, (1980) · Zbl 0416.05021 [17] Ito, Noboru; Kimura, Hiroshi, Studies on Hadamard matrices with “2-transitive” automorphism groups, J. math. soc. Japan, 36, 1, 63-73, (1984) · Zbl 0557.05020 [18] Ito, Noboru; Leon, Jeffrey S., An Hadamard matrix of order 36, J. combin. theory ser. A, 34, 2, 244-247, (1983) · Zbl 0518.05018 [19] Jungnickel, Dieter, Difference sets, (), 241-324 · Zbl 0768.05013 [20] Kantor, William M., Automorphism groups of Hadamard matrices, J. combin. theory, 6, 279-281, (1969) · Zbl 0206.02103 [21] Kimberley, Marion E., On collineations of Hadamard designs, J. lond. math. soc. (2), 6, 713-724, (1973) · Zbl 0264.05016 [22] Moorhouse, G. Eric, The 2-transitive complex Hadamard matrices, Preprint · Zbl 0677.51009 [23] Mordell, L.J., The Diophantine equations $$2^n = x^2 + 7$$, Ark. mat., 4, 455-460, (1962) · Zbl 0106.03602 [24] Muzychuk, Mikhail, On skew Hadamard difference sets, (2010) [25] Padraig Ó Catháin, Group Actions on Hadamard matrices, M. Litt. thesis, National University of Ireland, Galway, 2008. [26] Ó Catháin, Padraig; Röder, Marc, The cocyclic Hadamard matrices of order less than 40, Des. codes cryptogr., 58, 1, 73-88, (2011) · Zbl 1246.05033 [27] Ó Catháin, Padraig; Stafford, Richard M., On twin prime power Hadamard matrices, Cryptogr. commun., 2, 2, 261-269, (2010) · Zbl 1225.05055
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