Remarks on polarity designs

Abstract

Jungnickel and Tonchev (Des. Codes Cryptogr. 51:131–140, [11]) 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 PG _d (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 (Des. Codes Cryptogr. [14] introduced new invariants for simple incidence structures ${\mathcal{D}}$, which admit both a coding theoretic and a geometric description. Geometrically, one considers embeddings of ${\mathcal{D}}$ into projective geometries Π = PG(n, q), where an embedding means identifying the points of ${\mathcal{D}}$ with a point set V in Π in such a way that every block of ${\mathcal{D}}$ is induced as the intersection of V with a suitable subspace of Π. Then the new invariant—which we shall call the geometric dimension geomdim _q ${\mathcal{D}}$ of ${\mathcal{D}}$ —is the smallest value of n for which ${\mathcal{D}}$ may be embedded into the n-dimensional projective geometry PG(n, q). The classical designs PG _d (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 geomdim _q ${ \mathcal{D}}$ whenever ${\mathcal{D}}$ is one of the (exponentially many) “distorted” designs constructed in Jungnickel and Tonchev (Des. Codes Cryptogr. 51:131–140, [11]; Des. Codes Cryptogr. 55:131–140, [12]—a class of designs with classical parameters which includes the polarity designs as a very special case. We also show that this class contains designs with the same parameters as PG _d (n, q) and geomdim _q ${\mathcal{D} = n + 1}$, for every prime power q and for all values of d and n with 2 ≤ d ≤ n−1. Regarding the polarity designs, we conjecture that their geometric dimension always satisfies our general upper bound with equality, that is, geomdim _q ${\mathcal{D} = 4d}$ for the polarity design ${\mathcal{D}}$ with the parameters of PG _d (2d, q), but we are only able to establish this result if we restrict ourselves to the special case of “natural” embeddings.