Giuseppe Veronese and Ernst Witt—neighbours in PG(5,3)

Abstract

Let P be a point of the Veronese surface $ \nu $ in PG(5,3). Then there are four conics of $ \nu $ through P. We show that the internal points of those conics form a 12-cap which is a point model for Witt's 5-(12,6,1) design. In fact, this construction is “dual” to a similar construction that has been established in [6] recently. We give an explicit parametrization of the cap $ {\cal K} $ the domain is a dual affine plane which arises from PG(2,3) by removing one point. Thus, as a by-product, we obtain an easy approach to the extended ternary Golay code $ G_{12} $ . Finally, we discuss some other procedures which yield 12-sets of points from the Veronese surface.