An explicit construction for large sets of infinite dimensional qq-Steiner systems

Let $V$ be a vector space over the finite field ${\mathbb F}_q$. A $q$-Steiner system, or an $S(t,k,V)_q$, is a collection ${\mathcal B}$ of $k$-dimensional subspaces of $V$ such that every $t$-dimensional subspace of $V$ is contained in a unique element of ${\mathcal B}$. A large set of $q$-Steiner systems, or an $LS(t,k,V)_q$, is a partition of the $k$-dimensional subspaces of $V$ into $S(t,k,V)_q$ systems. In the case that $V$ has infinite dimension, the existence of an $LS(t,k,V)_q$ for all finite $t,k$ with $1<t<k$ was shown by Cameron in 1995. This paper provides an explicit construction of an $LS(t,t+1,V)_q$ for all prime powers $q$, all positive integers $t$, and where $V$ has countably infinite dimension.Comment: 5 page

Elusive Codes in Hamming Graphs

We consider a code to be a subset of the vertex set of a Hamming graph. We examine elusive pairs, code-group pairs where the code is not determined by knowledge of its set of neighbours. We construct a new infinite family of elusive pairs, where the group in question acts transitively on the set of neighbours of the code. In our examples, we find that the alphabet size always divides the length of the code, and prove that there is no elusive pair for the smallest set of parameters for which this is not the case. We also pose several questions regarding elusive pairs

Using mixed dihedral groups to construct normal Cayley graphs, and a new bipartite 22-arc-transitive graph which is not a Cayley graph

A \emph{mixed dihedral group} is a group $H$ with two disjoint subgroups $X$ and $Y$, each elementary abelian of order $2^n$, such that $H$ is generated by $X\cup Y$, and $H/H'\cong X\times Y$. In this paper we give a sufficient condition such that the automorphism group of the Cayley graph \Cay(H,(X\cup Y)\setminus\{1\}) is equal to $H: A(H,X,Y)$, where $A(H,X,Y)$ is the setwise stabiliser in \Aut(H) of $X\cup Y$. We use this criterion to resolve a questions of Li, Ma and Pan from 2009, by constructing a $2$-arc transitive normal cover of order $2^{53}$ of the complete bipartite graph \K_{16,16} and prove that it is \emph{not} a Cayley graph.Comment: arXiv admin note: text overlap with arXiv:2303.00305, arXiv:2211.1680

A family of (2)-groups and an associated family of semisymmetric, locally (2)-arc-transitive graphs

A mixed dihedral group is a group (H) with two disjoint subgroups (X) and (Y), each elementary abelian of order (2^n), such that (H) is generated by (Xcup Y), and (H/H\u27cong Xtimes Y). In this paper, for each (ngeq 2), we construct a mixed dihedral (2)-group (H) of nilpotency class (3) and order (2^a) where (a=(n^3+n^2+4n)/2), and a corresponding graph (Sigma), which is the clique graph of a Cayley graph of (H). We prove that (Sigma) is semisymmetric, that is, ({mathop{rm Aut}}(Sigma)) acts transitively on the edges but intransitively on the vertices of (Sigma). These graphs are the first known semisymmetric graphs constructed from groups that are not (2)-generated (indeed (H) requires (2n) generators). Additionally, we prove that (Sigma) is locally (2)-arc-transitive, and is a normal cover of the `basic\u27 locally (2)-arc-transitive graph ({rmbf K}_{2^n,2^n}). As such, the construction of this family of graphs contributes to the investigation of normal covers of prime-power order of basic locally (2)-arc-transitive graphs – the `local\u27 analogue of a question posed by C. H. Li