The Solution of the Waterloo Problem

Abstract Let D(d, q) be a classical (ν, k, λ)-Singer difference set in a cyclic group G corresponding to the complement of the point-hyperplane design of PG(d, q) (d ⩾ 1). We characterize those Singer difference sets D(d, q) which admit a “Waterloo decomposition” D = A ∪ B such that (A − B) · (A − B)(−1) = k in Z G: Theorem. D(d, q) admits a Waterloo decomposition if and only if d is even.

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