Remarks on polarity designs

Jungnickel and Tonchev (Des. Codes Cryptogr. 51:131–140) used polarities of PG(2d − 1, q) to construct non-classical designs with a hyperplane and the same parameters and same intersection numbers as the classical designs PGd (2d, q), for every prime power q and every integer d ≥ 2. Our main result shows that these properties already characterize their polarity designs. Recently, Jungnickel and Tonchev introduced new invariants for simple incidence structures D, which admit both a coding theoretic and a geometric description. Geometrically, one considers embeddings of D into projective geometries = PG(n, q), where an embedding means identifying the points of D with a point set V inin such a way that every block of D is induced as the intersection of V with a suitable subspace of . Then the new invariant—which we shall call the geometric dimension geomdimq D of D—is the smallest value of n for which D may be embedded into the n-dimensional projective geometry PG(n, q). The classical designs PGd (n, q) always have the smallest possible geometric dimension among all designs with the same parameters, namely n, and are actually characterized by this property. We give general bounds for geomdimq D whenever D is one of the (exponentially many) “distorted” designs constructed in Jungnickel and Tonchev (Des. Codes Cryptogr. 51:131–140];