One-point extensions of finite inversive planes

A one-point extension of an egglike inversive plane of order n exists if, and only if, n is 2 or 3. Hence there are no 4-(18,6,1) designs and no 4-(66,10,1) designs. An inversive plane of order n is a one-point extension of an affine plane of ordern, that is, a 3-(n + 1,n + 1,1) design. An ovoid of PG(3, q), q > 2, is a set of q2 + 1 points, no three collinear. An ovoid of PG(3, 2) is a set of 5 points, no four coplanar. Every plane of PG(3, q) meets an ovoid in either 1 or q + 1 points. The incidence structure whose points are those of an ovoid in PG(3, q), whose blocks are the plane sections of size q + 1 of the ovoid, and whose incidence is given by set membership, is an inversive plane of order q. An inversive plane of this form is called egglike. Theorem 1 [3,6.2.14] An inversive plane of even order is egglike. In consequence, its order is a power of two. A one-p oint extension of an inversive plane of order n is a 4-( n 2 + 2, n+ 2,1) design. Lemma If a 4-( n 2 + 2, n + 2,1) design exists, then n 2,3,4,8 or 13. Proof: By the standard divisibility conditions [5,p.7] for designs, n + 2 divides n(n2 + 1)(n2 + 2). Hence n + 2 divides 60. It follows that n is 2,3,4,8,10,13,18,28 or 58. By Theorem 1, n is 2,3,4,8 or 13. III