Affine design

Let $\mathcal{D} = ( V , \mathcal{B} )$ be a resolvable $t - ( v , k , \lambda )$-design (see Tactical configuration), that is, the block set of $\mathcal{D}$ is partitioned into parallel classes each of which in turn partitions the point set $V$. $\mathcal{D}$ is called affine, or affine resolvable, if there exists a constant $\mu$ such that any two non-parallel blocks intersect in exactly $\mu$ points. For proofs of the results stated below, see [a1].

The affine $1$-designs are precisely the nets, see Net (in finite geometry), and the affine $3$-designs coincide with the Hadamard $3$-designs, that is, the $3 - ( 4 \mu , 2 \mu , \mu - 1 )$-designs, cf. Tactical configuration. There are no non-trivial affine $t$-designs with $t \geq 4$. Thus, the most interesting case is that of affine $2$-designs, which are often simply called affine designs. Any affine $1$-design satisfies the inequality $r \leq ( s ^ { 2 } \mu - 1 ) / ( \mu - 1 )$, where $r$ denotes the number of blocks through a point and where $s$ denotes the number of blocks in a parallel class. Moreover, equality holds in this inequality if and only the $1$-design is an (affine) $2$-design. Any resolvable $2$-design satisfies the inequality $r \geq k + \lambda$, and equality holds if and only the design is affine. In this case, all parameters of $\mathcal{D}$ may be written in terms of the two parameters $s$ and $\mu$, as follows:

\begin{equation*} k = s \mu , v = s ^ { 2 } \mu , \lambda = \frac { s \mu - 1 } { \mu - 1 } , r = \frac { s ^ { 2 } \mu - 1 } { \mu - 1 }, \end{equation*}

and the design is denoted by $A _ { \mu } ( s )$.

The outstanding problem in this area is to characterize the possible pairs $( s , \mu )$ for which an $A _ { \mu } ( s )$ exists. The only known pairs to date (2001) are those with $s = 2$ and the pairs of the form $( q , q ^ { d - 2 } )$ for some prime power $q$ and some integer $d \geq 2$. The case $s = 2$ corresponds to Hadamard $2$-designs, i.e. $2 - ( 4 \mu - 1,2 \mu - 1 , \mu - 1 )$-designs; any such design extends uniquely to a Hadamard $3$-design, and existence — which is equivalent to that of an Hadamard matrix of order $4 \mu$ — is conjectured for all values of $\mu$. The classical examples for the second case are the affine designs $A G _ { d -1} ( d , q )$ formed by the points and hyperplanes of the $d$-dimensional finite affine spaces $A G ( d , q )$ over the Galois field $\operatorname {GF} ( q )$ of order $q$ (so $q$ is a prime power here; cf. also Affine space). As to the case $d = 2$, a design $A _ { 1 } ( s )$ is just an affine plane of order $s$, 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 $A G _ { d -1} ( d , q )$ grows exponentially with a growth rate of at least $e ^ { k .\operatorname { ln } k }$, where $k = q ^ { d - 1 }$. Hence, it is desirable to characterize the designs $A G _ { d -1} ( d , q )$ among the affine or resolvable designs. For instance, by Dembowski's theorem, a resolvable design $\mathcal{D}$ with $\lambda > 1$ and $s > 2$ in which every line (that is, the intersection of all blocks through two given points) meets every non-parallel block is isomorphic to some $A G _ { d -1} ( d , q )$; the same conclusion holds if $\mathcal{D}$ 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 $A G ( 2 , q )$ 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)