×

On Lander’s conjecture for difference sets whose order is a power of 2 or 3. (English) Zbl 1209.05031

Summary: Let \(p\) be a prime and let \(b\) be a positive integer. If a \((v, k, \lambda , n)\) difference set \(D\) of order \(n = p^{b }\) exists in an abelian group with cyclic Sylow \(p\)-subgroup \(S\), then \({p\in\{2,3\}}\) and \(|S| = p\). Furthermore, either \(p = 2\) and \(v \equiv \lambda \equiv 2\) (mod 4) or the parameters of \(D\) belong to one of four families explicitly determined in our main theorem.

MSC:

05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
05B25 Combinatorial aspects of finite geometries
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baumert L.D.: Cyclic difference sets. Springer Lecture Notes, vol. 182. Springer, Heidelberg (1971) · Zbl 0218.05009
[2] Beth T., Jungnickel D., Lenz H.: Design theory, 2nd edn. Cambridge University Press, Cambridge (1999) · Zbl 0945.05004
[3] Jungnickel D.: Difference sets. In: Dinitz, J.H., Stinson, D.R. (eds) Contemporary design theory: a collection of surveys, Wiley, New York (1992)
[4] Jungnickel D., Schmidt B. et al.: Difference sets: an update. In: Hirschfeld, J.W.P. (eds) Geometry, combinatorial designs and related structures. Proceeding of the first Pythagorean conference, pp. 89–112. Cambridge University Press, Cambridge (1997) · Zbl 0883.05023
[5] Lander E.S.: Symmetric designs: an algebraic approach London Mathematical Society Lecture Notes, vol. 75. Cambridge University Press, Cambridge (1983) · Zbl 0502.05010
[6] Leung K.H., Ma S.L., Schmidt B.: Nonexistence of abelian difference sets: Lander’s conjecture for prime power orders. Trans. Am. Math. Soc. 356, 4343–4358 (2004) · Zbl 1043.05025 · doi:10.1090/S0002-9947-03-03365-8
[7] Leung K.H., Schmidt B.: The field descent method. Des. Codes Cryptogr. 36, 171–188 (2005) · Zbl 1066.05037 · doi:10.1007/s10623-004-1703-7
[8] Pott A.: Finite geometry and character theory. Springer Lecture Notes, vol. 1601. Springer, Heidelberg (1995) · Zbl 0818.05001
[9] Schützenberger M.P.: A nonexistence theorem for an infinite family of symmetrical block designs. Ann. Eugen. 14, 286–287 (1949) · Zbl 0035.08801
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.