Designs and codes in affine geometry

TL;DR

This paper explores affine designs in affine geometry, their relationship with projective q-analogs, and demonstrates the existence of affine Steiner systems, including an affine q-analog of the Fano plane, with applications in network coding.

Contribution

It introduces affine designs in matroids, establishes their connection with projective designs, and proves the existence of affine Steiner systems like the affine q-analog of the Fano plane.

Findings

Existence of affine Steiner systems, including S(2, 3, 7)

Relationship between affine and projective q-analogs of designs

Application of affine geometry codes in network error correction

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.