Remarks on polarity designs

摘要

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 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. 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 in ∏ in 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 geomdim_qD 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 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_qD whenever D is one of the (exponentially many) "distorted" designs constructed in Jungnickel and Tonchev (Des. Codes Cryptogr. 51:131-140, ; Des. Codes Cryptogr. 55:131-140, )-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 _qD = 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_pD = 4d for the polarity design 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.