ON THE EXISTENCE OF NON-ABELIAN (210, 77, 28), (336, 135, 54) AND (496, 55, 6) DIFFERENCE SETS

Lander [Symmetric Design: An Algebraic Approach, London Mathematical Society Lecture Note Series, Vol. 74 (Cambridge University Press, Cambridge, 1983)] showed that some abelian (v, k, λ) difference sets do not exist in some groups of order v and listed some parameters of difference sets that were open for small values of k (k ≤ 50). Various authors have since studied the open cases and have either constructed the difference sets when they exist or proved their non-existence. Using restrictions imposed by the underlying normal subgroups of groups, algebraic number theory, group representations and rational idempotents of the group ring, we rule out the existence of (496, 55, 6) difference sets and show that non-abelian (210, 77, 28) and (336, 135, 54) difference sets do not exist in most groups of 210 and 336, respectively.