Neighbour-transitive codes and partial spreads in generalised quadrangles

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