Affine design

Let be a resolvable -design (see Tactical configuration), that is, the block set of is partitioned into parallel classes each of which in turn partitions the point set . is called affine, or affine resolvable, if there exists a constant such that any two non-parallel blocks intersect in exactly points. For proofs of the results stated below, see [a1].

The affine -designs are precisely the nets, see Net (in finite geometry), and the affine -designs coincide with the Hadamard -designs, that is, the -designs, cf. Tactical configuration. There are no non-trivial affine -designs with . Thus, the most interesting case is that of affine -designs, which are often simply called affine designs. Any affine -design satisfies the inequality , where denotes the number of blocks through a point and where denotes the number of blocks in a parallel class. Moreover, equality holds in this inequality if and only the -design is an (affine) -design. Any resolvable -design satisfies the inequality , and equality holds if and only the design is affine. In this case, all parameters of may be written in terms of the two parameters and , as follows:

and the design is denoted by .

The outstanding problem in this area is to characterize the possible pairs for which an exists. The only known pairs to date (2001) are those with and the pairs of the form for some prime power and some integer . The case corresponds to Hadamard -designs, i.e. -designs; any such design extends uniquely to a Hadamard -design, and existence — which is equivalent to that of an Hadamard matrix of order — is conjectured for all values of . The classical examples for the second case are the affine designs formed by the points and hyperplanes of the -dimensional finite affine spaces over the Galois field of order (so is a prime power here; cf. also Affine space). As to the case , a design is just an affine plane of order , see also Plane.

In general, an affine design cannot be characterized just by its parameters. For instance, the number of non-isomorphic designs with the same parameters as grows exponentially with a growth rate of at least , where . Hence, it is desirable to characterize the designs among the affine or resolvable designs. For instance, by Dembowski's theorem, a resolvable design with and in which every line (that is, the intersection of all blocks through two given points) meets every non-parallel block is isomorphic to some ; the same conclusion holds if admits an automorphism group which is transitive on ordered triples of non-collinear points. See [a1], Sec. XII.3, for proofs and further characterizations. In particular, there is a wealth of results characterizing the classical affine planes and other interesting classes of affine planes; for example, a result of Y. Hiramine [a2] states that any finite affine plane that admits a collineation group acting primitively on points is a translation plane (cf. Plane; Primitive group of permutations). Detailed studies of translation planes may be found in [a3] and [a4].

References

[a1]	T. Beth, D. Jungnickel, H. Lenz, "Design theory" , Cambridge Univ. Press (1999) (Edition: Second)
[a2]	Y. Hiramine, "Affine planes with primitive collineation groups" J. Algebra , 128 (1990) pp. 366–383
[a3]	M.J. Kallaher, "Affine planes with transitive collineation groups" , North-Holland (1981)
[a4]	H. Lüneburg, "Translation planes" , Springer (1980)