Difference Sets Corresponding to a Class of Symmetric Designs

Abstract

We study difference sets with parameters (v, k, λ) = (ps(r2m - 1)/(r - 1), ps-1r2m-1, ps-2(r - 1)r2m-2), where r = (ps - 1)/(p -1) and p is a prime. Examples for such difference sets are known from a construction of McFarland which works for m = 1 and all p, s. We will prove a structural theorem on difference sets with the above parameters; it will include the result, that under the self-conjugacy assumption McFarland's construction yields all difference sets in the underlying groups. We also show that no abelian (160, 54, 18)-difference set exists. Finally, we give a new nonexistence prove of (189, 48, 12)-difference sets in ℤ3 × ℤ9 × ℤ7.