Neighbour-Transitive Codes and Partial Spreads in Generalised Quadrangles

Abstract

A code C in a generalised quadrangle Q is defined to be a subset of the vertex set of the point-line incidence graph Γ of Q. The minimum distance δ 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 , . . . , C ρ }, where ρ is the maximum distance between any vertex of Γ and its nearest element of C. Since the diameter of Γ is 4, both ρ and δ are at most 4. If δ = 4 then C is a partial ovoid or partial spread of Q, and if, additionally, ρ = 2 then C is an ovoid or a spread. A code C in 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 neighbourtransitive codes with minimum distance δ = 4 in the classical generalised quadrangle W 3 (q) that are not ovoids or spreads. * This work has been supported by the Croatian Science Foundation under the projects 6732 and 5713.1. C is equivalent to a regular spread if and only if C has covering radius 2.2. If |C| = q 2 then C has covering radius ρ = 4 and is equivalent to a spread minus a line.3. If |C| = q 2 − 1 then q = 2, 3, 5, 7 or 11 and C is equivalent to one of the codes in Example 5.4, each of which has covering radius ρ = 3.4. If |C| = q + 1 and C has covering radius ρ = 3 then C is equivalent to the set of points on a hyperbolic line.5. If q = 3 and |C| = 5 then C has covering radius ρ = 3 and is equivalent to the code given in Example 5.3.Note that an example of a code as in part 2 of Theorem 1.2 is given, for each q, by a regular spread of W 3 (q) with one line removed; see Lemma 4.3. We do not prove here that these are all such examples. However, since the automorphism group of such a code must have order divisible by q 2 (q + 1), [32, Table 1] suggests that perhaps this is the case. Hence, we pose the following question.