Derived and Residual Subspace Designs

TL;DR

This paper generalizes the concepts of derived and residual designs to subspace designs, proves a key theorem about their parameters, and establishes the existence of new subspace designs and limitations on certain large designs.

Contribution

It introduces a new generalization of derived and residual designs to subspace designs and proves a theorem linking parameter realizability to the existence of new designs.

Findings

Proves a $q$-analog of a theorem relating parameters of subspace designs.

Establishes the existence of several previously unknown subspace designs.

Shows that no $q$-analog of the large Witt design exists.

Abstract

A generalization of forming derived and residual designs from $t$-designs to subspace designs is proposed. A $q$-analog of a theorem by Van Trung, van Leijenhorst and Driessen is proven, stating that if for some (not necessarily realizable) parameter set the derived and residual parameter set are realizable, the same is true for the reduced parameter set. As a result, we get the existence of several previously unknown subspace designs. Some consequences are derived for the existence of large sets of subspace designs. Furthermore, it is shown that there is no $q$-analog of the large Witt design.