New infinite families of $q$-analogs of group divisible designs with arbitrary block dimension
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913062803472384 |
|---|---|
| author | Wu, Yakun Zhou, Junling Wang, Xiaoran |
| author_facet | Wu, Yakun Zhou, Junling Wang, Xiaoran |
| contents | 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 \(Ω_k^{k-1}\), where $3\leq k\leq \min\left\lbrace m+1,l\right\rbrace $ and \(Ω_k^{k-1}\) 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 \(Ω_k^{k-1}\) 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23558 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | New infinite families of $q$-analogs of group divisible designs with arbitrary block dimension Wu, Yakun Zhou, Junling Wang, Xiaoran Combinatorics 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 \(Ω_k^{k-1}\), where $3\leq k\leq \min\left\lbrace m+1,l\right\rbrace $ and \(Ω_k^{k-1}\) 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 \(Ω_k^{k-1}\) 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. |
| title | New infinite families of $q$-analogs of group divisible designs with arbitrary block dimension |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.23558 |