On the lines of special block designs

Abstract

For every choice of integers d, i, and s with i ¦ d and d > i > s > 0 and for every prime power q, a block design $\text{B}$ with the following properties can be constructed:

• (a) $\text{B}$ admits an abelian translation group which is transitive on the points of $\text{B}$.

• (b) $\text{B}$ has the parameters v = q^d, k = qⁱ, and λ = 2.

• (c) At least one line of $\text{B}$ consists of q^s points.

• (d) The points of $\text{B}$ may be regarded as the points of the affine geometry $\text{U}$ = AG(d, q), the blocks as certain i-dimensional linear manifolds of $\text{U}$; the incidence of $\text{B}$ is induced by $\text{U}$.

If s ∤ i or if $\text{i > s >}\text{i}\text{2}$, then any design $\text{B}$ which satisfies condition (a) to (d) possesses lines (in the general sense, see e.g., Dembowski [4]) with different numbers of points.