ABSTRAC'r. Generalizing an idea of Kantor [7], Johnson and Wilke [5] introduced 'elusive' sets of functions over GF(q) to represent translation planes of order q2 that admit a coltineation group of order q2 in the linear translation complement and whose kern contains GF(q). In this paper we determine explicitly all elusive sets for q even. We obtain another translation plane of order 82 .
Recently several authors ([2]-[6]
) carried out some investigations on translation planes of order q2 that admit a collineation group of order q2 in the linear translation complement and whose kern contains GF(q). This was essentially motivated by Kantor's paper [7] on 'likeable' translation planes and also by a work of Bartolone [1] on similar questions. We restrict our attention to the case of q even. Generalizing Kantor's construction, Johnson and Wilke proved that, under some additional assumptions (see below), any such plane, not being a semifield plane, may be constructed by means of a suitable set of functions over GF(q). Sets of this kind -and their associated planes -are called 'elusive'. In this paper we determine all elusive sets. Referred to the planes, our result reads as follows: THEOREM A. Let ~z be an 'elusive' plane of even order, then ~ is either a Betten plane or a Luneburg-Tits plane or the plane rc(JV) of order 8 2 exhibited in Section 3.
In this direction Ganley already showed that every likeable plane of even order is a Betten plane. Our proof makes use of Ganley's result and of a technique, due to Menichetti [9], for the study of roots of affine polynomials over Galois fields.
The elusive plane ~(Jg') exhibited in Section 3 seems to be previously unknown.