to give a geometric description of some inversive planes, indeed, of all known inversive planes. An ovoid in PG(3,q) is a set of q2 + 1 points, no three collinear. It can be shown that every plane (which is a hyperplane since the geometric dimension is 3) of PG(3,q) meets an ovoid O in either 1 or q + 1 points. The plane sections of size q + 1 of O are the blocks of an inversive plane of order q. Any inversive plane arising this way is called egglike. All known inversive planes are egglike. An example of an ovoid is the elliptic quadric, the set of zeros of the quadratic form x1x2 + f(x3, x4), where f is an irreducible quadratic form in two variables over GF(q). [f(x,y) = x2 + xy + y2 for example]. If q is an odd power of 2, another type of ovoid is known – the Suzuki–Tits ovoid. Theorem. Let q be a positive integer, at least 2. (a) If q is odd, then any ovoid is projectively equivalent to the elliptic quadric in a projective geometry PG(3,q); so q is a prime power and there is a unique egglike inversive plane of order q. (But it is unknown if non-egglike ones exist.) (b) if q is even, then q is a power of 2 and any inversive plane of order q is egglike (but there may be some unknown ovoids). Partially balanced designs (PBIBDs) An n-class association scheme consists of a set X of size v together with a partition S of X × X into n + 1 binary relations, R0, R1, ..., Rn. A pair of elements in relation Ri are said to be ith–associates. Each element of X has ni ith associates. Furthermore: and is called the Identity relation. Defining , if R in S, then R* in S If , the number of such that and is a constant depending on i, j, k but not on the particular choice of x and y. An association scheme is commutative if for all i, j and k. Most authors assume this property. A partially balanced incomplete block design with n associate classes (PBIBD(n)) is a block design based on a v-set X with b blocks each of size k and with each element appearing in r blocks, such that there is an association scheme with n classes defined