×

zbMATH — the first resource for mathematics

A group extensions approach to relative difference sets. (English) Zbl 1045.05021
Summary: A new approach to (normal) relative difference sets (RDSs) is presented and applied to give a new method for recursively constructing infinite families of semiregular RDSs. Our main result (Theorem 7.1) shows that any metabelian semiregular RDS gives rise to an infinite family of metabelian semiregular RDSs. The new method is applied to identify several new infinite families of non-abelian semiregular RDSs, and new methods for constructing generalized Hadamard matrices are given. The techniques employed are derived from the general theory of group extensions.

MSC:
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] and Groups and Representations, Springer, New York, 1995. · doi:10.1007/978-1-4612-0799-3
[2] Arasu, J Combin Theory Ser A 71 pp 316– (1995)
[3] Arasu, J Combin Theory Ser A 94 pp 118– (2001)
[4] Arasu, Discrete Math 147 pp 1– (1995)
[5] and Design Theory. Vol. 1 (2nd edn.), Cambridge University Press, Cambridge, 1999.
[6] and (Eds.), The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, 1996. · Zbl 0836.00010 · doi:10.1201/9781420049954
[7] Dembowski, Math Z 99 pp 53– (1967)
[8] Elliott, Illinois J Math 10 pp 517– (1966)
[9] Flannery, J Algebra 192 pp 749– (1997)
[10] Galati, J Combin Des 11 pp 307– (2003)
[11] A group extensions approach to relative difference sets, Ph.D. Thesis, RMIT University, Melbourne, Australia, 2003.
[12] Ganley, J Combin Theory Ser A 19 pp 134– (1975)
[13] Hiramine, Geom Dedicata 54 pp 13– (1995)
[14] Hiramine, J Algebra 142 pp 414– (1991)
[15] Horadam, J Combin Des 8 pp 330– (2000)
[16] Horadam, Discrete Appl Math 102 pp 115– (2000)
[17] Characteristic functions of relative difference sets, correlated sequences and Hadamard matrices, In: et al. (Eds.), AAECC-13, Lecture notes in computer science, Vol. 1719, Springer, Berlin, 1999, pp. 346-354. · Zbl 0956.05025
[18] Presentations of Groups, (2nd edn)., London Mathematical Society Student Texts, Vol. 15, Cambridge University Press, Cambridge, 1997.
[19] Jungnickel, Canad J Math 34 pp 257– (1982) · Zbl 0465.05011 · doi:10.4153/CJM-1982-018-x
[20] Central semiregular relative difference sets fixed by inversion, Ph.D. Thesis, RMIT University, Melbourne, Australia, 2002.
[21] Perera, Des Codes Cryptogr 15 pp 187– (1998)
[22] Finite geometry and character theory, Springer, Berlin, 1995. · Zbl 0818.05001 · doi:10.1007/BFb0094449
[23] A survey on relative difference sets?, In: Groups, difference sets, and the monster, proceedings of a special research quarter at the Ohio State University, Walter de Gruyter, Berlin, 1996.
[24] A Course in the theory of groups (2nd edn.), Springer, New York, 1996. · doi:10.1007/978-1-4419-8594-1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.