Neighbour-transitive codes and partial spreads in generalised quadrangles

A code C in a generalised quadrangle \({\mathcal {Q}}\) is defined to be a subset of the vertex set of the point-line incidence graph \({\Gamma }\) of \({\mathcal {Q}}\). The minimum distance \(\delta \) of C is the smallest distance between a pair of distinct elements of C. The graph metric gives rise to the distance partition \(\{C,C_1,\ldots ,C_\rho \}\), where \(\rho \) is the maximum distance between any vertex of \({\Gamma }\) and its nearest element of C. Since the diameter of \({\Gamma }\) is 4, both \(\rho \) and \(\delta \) are at most 4. If \(\delta =4\) then C is a partial ovoid or partial spread of \({\mathcal {Q}}\), and if, additionally, \(\rho =2\) then C is an ovoid or a spread. A code C in \({\mathcal {Q}}\) is neighbour-transitive if its automorphism group acts transitively on each of the sets C and \(C_1\). Our main results (i) classify all neighbour-transitive codes admitting an insoluble group of automorphisms in thick classical generalised quadrangles that correspond to ovoids or spreads, and (ii) give two infinite families and six sporadic examples of neighbour-transitive codes with minimum distance \(\delta =4\) in the classical generalised quadrangle \({\mathsf {W}}_3(q)\) that are not ovoids or spreads.