Designs, Codes and Cryptography, 10, 237–250 (1997) c  1997 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Unitals and Unitary Polarities in Symmetric Designs RUDOLF MATHON Department of Computer Science, University of Toronto, Toronto, Ontario, Canada M5S1A4 TRAN VAN TRUNG Institute for Experimental Mathematics, University of Essen, Ellernstrasse 29, 45326 Essen, Germany Communicated by: D. Jungnickel Received May 10, 1995; Revised January 2, 1996; Accepted January 23, 1996 Dedicated to Hanfried Lenz on the occasion of his 80 th birthday. Abstract. We extend the notion of unital as well as unitary polarity from finite projective planes to arbitrary symmetric designs. The existence of unitals in several families of symmetric designs has been proved. It is shown that if a unital in a point-hyperplane design PG d−1 (d , q ) exists, then d = 2 or 3; in particular, unitals and ovoids are equivalent in case d = 3. Moreover, unitals have been found in two designs having the same parameters as the PG 4 (5, 2), although the latter does not have a unital. It had been not known whether or not a nonclassical design exists, 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. Keywords: unital, unitary polarity, symmetric design 1. Introduction Unitals in projective planes are incidence structures obtained from the set of absolute points and non-absolute lines of unitary polarities. To be precise, let P be a projective plane of order n = s 2 . A polarity σ of P having s 3 + 1 absolute points is called a unitary polarity. In this case, each non-absolute line contains s + 1 absolute points, and each absolute line contains, of course, a unique absolute point. In other words the set of absolute points and non-absolute lines of σ forms a 2 − (s 3 + 1, s + 1, 1) design U , which is called a unital. Results about unitals and unitary polarities for projective planes can be found in [3, 6, 11]. From combinatorial point of view, a unital in a projective plane P of order n = s 2 is a set U of s 3 + 1 points having the property: Through each point p ∈ U there is exactly one line (tangent) in P meeting U in one point, namely p, and there are s 2 lines (secants) in P each having s + 1 points in common with U . (*) Therefore, without taking unitary polarities into account, a unital in a projective plane may be defined by the property (*). A unital is also defined to be a 2 − (s 3 + 1, s + 1, 1) design, with s > 1, see for instance [22]. This implies that a unital is not necessarily derived from a unitary polarity or even not embedded in a projective plane. Actually, this fact is shown in a paper of Brouwer [5]. Furthermore, Mathon [16] has constructed a first unital of block