Designs and codes in affine geometry

Abstract: Classical designs and their (projective) q-analogs can both be viewed as designs in matroids, using the matroid of all subsets of a set and the matroid of linearly independent subsets of a vector space, respectively. Another natural matroid is given by the point sets in general position of an affine space, leading to the concept of an affine design. Accordingly, a t-(n, k, $\lambda$) affine design of order q is a collection B of (k-1)-dimensional spaces in the affine geometry A = AG(n-1, q) such that each (t-1)-dimensional space in A is contained in exactly $\lambda$ spaces of B. In the case $\lambda$ = 1, as usual, one also refers to an affine Steiner system S(t, k, n). In this work we examine the relationship between the affine and the projective q-analogs of designs. The existence of affine Steiner systems with various parameters is shown, including the affine q-analog S(2, 3, 7) of the Fano plane. Moreover, we consider various distances in matroids and geometries, and we discuss the application of codes in affine geometry for error-control in a random network coding scenario.