A new biplane of order 9 with a small automorphism group
Journal of Combinatorial Theory Series A
Design theory
Designs from pairs of finite fields. I. A cyclic unital U(6) and other regular Steiner 2-designs
Journal of Combinatorial Theory Series A
Finite geometries
Combinatorial theory (2nd ed.)
Combinatorial theory (2nd ed.)
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We extend the notion of unital as well as unitary polarity from finite projective planes to arbitrarysymmetric designs. The existence of unitals in several families of symmetric designs has been proved. It is shownthat if a unital in a point-hyperplane design PG_{d-1}(d,q) exists, then d = 2 or 3; in particular, unitals and ovoidsare equivalent in case d = 3. Moreover, unitals have been found in two designs having the same parameters as thePG_4(5,2), although the latter does not have a unital. It had been not known whether or not a nonclassical designexists, which has a unitary polarity. Fortunately, we have discovered a unitary polarity in a symmetric 2-(45,12,3)design. To a certain extent this example seems to be exceptional for designs with these parameters.