Note on GMW Designs

Let N and n be integers such that n|N and 2 < n < N . Let q be a prime power. The difference sets introduced by Gordon, Mills and Welch in [1] produce symmetric designs with the same parameters v = (q N −1)/(q−1), k = (q N−1−1)/(q−1), λ = (q N−2−1)/(q−1) as the point-hyperplane design of PG(N − 1, q). The purpose of this note is to prove that inequivalent difference sets of this sort produce nonisomorphic designs. There are various ways to construct the GMW designs. Since we wish to study designs rather than difference sets, we will use the very nice alternative description in [4] rather than the more standard difference set point of view [1, 6]. Consider the fields V = Fq N ⊃ Fqn ⊃ Fq ; let V ◦ denote the dual of the Fqn -space V , consisting of all linear functionals f : V → Fqn . We will also view V and V ◦ as Fq -spaces, in which case we write 〈v〉 for the 1-space spanned by v ∈ V − {0} and 〈 f 〉 for the 1-space spanned by f ∈ V − {0}. Fix a ((qn − 1)/(q − 1), (qn−1− 1)/(q − 1), (qn−2− 1)/(q − 1)) difference set D in F qn/F ∗ q (hence the assumption n > 2), let D̃ denote the union in F ∗ qn of the cosets comprising D, and define the incidence structure D(N , n, D) as follows: its points are the 1-spaces 〈v〉, its blocks are the 1-spaces 〈 f 〉, and 〈v〉 and 〈 f 〉 are incident if and only if f (v) ∈ D̃∪{0}. As noted in [4], these are symmetric designs that include the “classical” ones in [1] (where D is taken to be equivalent to a difference set with corresponding symmetric design PG(n − 1, q)). Since the case in which D(N , n, D) is isomorphic to a projective space is fully handled in [1, 4], we will exclude this possibility. The statements of the following theorems deal with the fact that the same symmetric design can arise as D(N , n, D) for different values of n (which is why we have included n in the notation D(N , n, D)).