New infinite families of $q$-analogs of group divisible designs with arbitrary block dimension

TL;DR

This paper constructs new infinite families of q-analogs of group divisible designs with arbitrary block sizes, using group actions and recursive methods, expanding the known classes of such combinatorial designs.

Contribution

It introduces a complete description of group actions on subspace sets and develops recursive constructions for new q-analog designs, including non-simple variants.

Findings

Derived numerous new infinite families of q-analogs of group divisible designs.

Established recursive methods for constructing q-analogs of pairwise balanced designs.

Produced infinite families of non-simple subspace 2-designs.

Abstract

This paper is mainly devoted to constructions of \(q\)-analogs of group divisible designs and their applications. We give a complete description of the action of \(G=\GL(m,q^l)\) on \(\Omega_k^{k-1}\), where $3\leq k\leq \min\left\lbrace m+1,l\right\rbrace $ and \(\Omega_k^{k-1}\) is the set of \(k\)-subspaces of $\GF(q)^{ml}$ whose \(\GF(q^l)\)-span has dimension \(k-1\). We do this by relating the \(G\)-orbits on \(\Omega_k^{k-1}\) to the corresponding Singer cycle orbits on subspaces of $\GF(q)^l$. From the properties of the $G$-incidence matrix between $2$-subspaces and $k$-subspaces, we obtain plenty of new infinite families of simple \(q\)-analogs of group divisible designs with arbitrary block dimension. We further establish a recursive construction for simple \(q\)-analogs of pairwise balanced designs and then produce new infinite families of such designs. We also obtain plenty of infinite families of non-simple subspace \(2\)-designs through the above two types of designs.