Abstract
Let D be an affine difference set of order n in an abelian group G relative to a subgroup N. Set \({\tilde{H}}\) = H \ {1, ω}, where H = G/N and \({\omega=\prod_{\sigma\in H}\sigma}\) . Using D we define a two-to-one map g from \({\tilde{H}}\) to N. The map g satisfies g(σ m) = g(σ)m and g(σ) = g(σ −1) for any multiplier m of D and any element σ ∈ \({\tilde{H}}\) . As applications, we present some results which give a restriction on the possible order n and the group theoretic structure of G/N.
Similar content being viewed by others
References
Arasu K.T.: Cyclic affine planes of even order. Discrete Math. 76, 177–181 (1989)
Arasu K.T., Jungnickel D.: Affine difference sets of even order. J. Combin. Theory Ser. A 52, 188–196 (1989)
Arasu K.T., Pott A.: On quasi-regular collineation groups of projective planes. Des. Codes Cryptogr. 1, 83–92 (1991)
Arasu K.T., Pott A.: Cyclic affine planes and Paley difference sets. Discrete Math. 106(7), 19–23 (1992)
Galati J.C.: A group extensions approach to affine relative difference sets of even order. Discrete Math. 306, 42–51 (2006)
Hiramine Y.: On affine difference sets and their multipliers. Discrete Math. (accepted).
Jungnickel D.: A note on affine difference sets. Arch. Math. 47, 279–280 (1986)
Pott A.: An affine analogue of Wilbrink’s theorem. J. Combin. Theory Ser. A 55, 313–315 (1990)
Pott A.: Finite Geometry and Character. Theory Lecture Notes in Mathematics, vol. 1601. Springer-Verlag, Berlin (1995)
Pott A.: A survey on relative difference sets. In: Groups, Difference Sets, and the Monster (Columbus, OH, 1993), pp. 195–232. Ohio State Univ. Math. Res. Inst. Publ., 4, de Gruyter, Berlin (1996).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by D. Jungnickel.
Rights and permissions
About this article
Cite this article
Hiramine, Y. A two-to-one map and abelian affine difference sets. Des. Codes Cryptogr. 50, 285–290 (2009). https://doi.org/10.1007/s10623-008-9231-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-008-9231-5