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 $V=\{1,\dots ,v\}$, and two vertices are joined by an edge if their intersection has size $k-1$. An X-strongly incidence-transitive code in $J(v,k)$ is a proper vertex subset Γ such that the subgroup X of graph automorphisms leaving Γ invariant is transitive on the set Γ of ‘codewords’, and for each codeword Δ, the setwise stabiliser $X_{\Delta }$ is transitive on $\Delta \times (V\setminus \Delta )$. We classify the X-strongly incidence-transitive codes in $J(v,k)$ for which X is the symplectic group ${\mathrm{Sp}}_{2n}\left(2\right)$ acting as a 2-transitive permutation group of degree $2^{2n-1}\pm 2^{n-1}$, where the stabiliser $X_{\Delta }$ of a codeword Δ is contained in a 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 ${\mathrm{Sp}}_{2n}\left(2\right)$.