A095029 An example of a (v,k,lambda)=(21,5,1) cyclic difference set.

3, 6, 7, 12, 14

Offset: 1

Author

Hugo Pfoertner, May 27 2004

Keywords

• fini
• full
• nonn

Comments

A (v,k,lambda) cyclic difference set is a subset D={d_1,d_2,...,d_k} of the integers modulo v such that {1,2,...,v-1} can each be represented as a difference (d_i-d_j) modulo v in exactly lambda different ways. Difference sets with lambda=1 (planar difference sets) have order n=k-1. The Prime Power Conjecture states that all Abelian planar difference sets have order n a prime power. It is known that no cyclic planar difference sets of nonprime power order n exist with n < 2*10^9 (see Baumert, Gordon link)

Examples

			Representation of {1,...,20}: 1=7-6, 2=14-12, 3=6-3, 4=7-3, 5=12-7, 6=12-6, 7=14-7, 8=14-6, 9=12-3, 10=21+3-14, 11=14-3, 12=21+3-12, 13=21+6-14, 14=21+7-14, 15=21+6-12, 16=21+7-12, 17=21+3-7, 18=21+3-6, 19=21+12-14, 20=21+6-7. - _Hugo Pfoertner_, Aug 13 2011
		

Links

• Leonard D. Baumert and Daniel M. Gordon, On the existence of cyclic difference sets with small parameters, arXiv:math/0304502 [math.CO], 30 Apr 2003.
• Dan Gordon, List of Cyclic Difference Sets, (2003).
• Dan Gordon, Difference Sets, searchable database.

Crossrefs

Cf. A095025 (number of cyclic difference sets with n elements), A095029-A095047 (more examples of cyclic difference set with k=5..20), A000961 (prime powers).