Codes and Designs in Johnson Graphs From Symplectic Actions on Quadratic Forms

Abstract

The Johnson graph $J(v,k)$ has as vertices the $k$-subsets of $\mathcal{V}=\{1,\dots ,v\}$, and two vertices are joined by an edge if their intersection has size $k-1$. An \emph{$X$-strongly incidence-transitive code} in $J(v,k)$ is a proper vertex subset $\hat{I}“$ such that the subgroup $X$ of graph automorphisms leaving $\hat{I}“$ invariant is transitive on the set $\hat{I}“$ of `codewords', and for each codeword $\hat{I}”$, the setwise stabiliser $X_{\hat{I}}”$ is transitive on $\hat{I}”\times (\mathcal{V}\setminus \hat{I}”)$. We classify the \emph{$X$-strongly incidence-transitive codes} in $J(v,k)$ for which $X$ is the symplectic group ${\mathrm{Sp}}_{2n}(2)$ acting as a $2$-transitive permutation group of degree $2^{2n-1}\pm 2^{n-1}$, where the stabiliser $X_{\hat{I}}”$ of a codeword $\hat{I}”$ is contained in a \emph{geometric} maximal subgroup of $X$. In particular, we construct two new infinite families of strongly incidence-transitive codes associated with the reducible maximal subgroups of ${\mathrm{Sp}}_{2n}(2)$.