Arch. Math., Vol. 59, 192-196 (1992) 0003-889X/92/5902-0192 $ 2.50/0 9 1992 Birkhfiuser Verlag, Basel On the number of isomorphism classes of complete translation nets*) By DIETERJUNGNICKEL and OLIVERPFAFF 1. Introduction. Our notation will be standard (see [2] for background from Design Theory). The complete translation nets are precisely the affine 2-designs which have a transitive translation group. It was shown by Jungnickel [8] that A (d, q), the number of affine designs with the parameters of AG e_ 1 (d, q), grows exponentially with k = qe- 1. On the other hand, only the affine spaces AGa_ 1 (d, q) and 5 sporadic affine designs admit a 2-transitive automorphism group by Pfaff [13]. Furthermore the Hadamard 3-design on 12 points is the only 2-transitive affine design which is no translation net. So, what can be said about transitive affine 2-designs or more specially about complete translation nets? Let T(d, q) be the number of isomorphism classes of translation nets with the parameters of AG e_ 1 (d, q). The following theorem is fundamental: 1. i R e s u 1 t ([7]; [6]). A complete (s, #)-translation net with translation group G exists if and only if s is a prime power, ~t is a power of s and G is elementary abelian. A more elementary proof for a more general situation is contained in Jungnickel [9]. So every complete translation net has the parameters of some affine space AGe_ I (d, s). Let D be a translation net with the parameters of AGe_ ~ (d, p) for a prime p. The translation group G of D is elementary abelian by (1.1). By the well-known correspon- dence between translation nets and group constructible nets (cf. [2] p. 512) we obtain T(d, p) = 1. In the following we will apply the method of Jungnickel [8] to obtain lower bounds for T(d, pf) with f > 2 and d > 3. 2. Construction. We always assume d > 3 and q = pl with a prime p. Let S be a classical symmetric (q, qa- 2)_net, which we obtain from an affine space AG e _ t (d, q) by deleting all parallel classes which contain a hyperplane covering a fixed line L. We may assume 0 ~ L without restriction. The set of point classes of S is the factor space F:= {L+x[xeGF(q)a}. Let T be a translation net with the parameters of AGd_ 2 (d - 1, q). Without restriction we assume that GF (q)e-1 is the point set of T and *) This note will be contained in the doctoral dissertation of the second author which is written under the supervision of the first author.