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.