A recursive construction for the dual polar spaces DQ(2n,2)

New combinatorial constructions for the near hexagons I3I3 and DQ(6,2) in terms of ordered pairs of collinear points of the generalized quadrangle W(2)W(2) were given by Sahoo [B.K. Sahoo, New constructions of two slim dense near hexagons. Discrete Math. (in press)]. Replacing W(2)W(2) by an arbitrary dual polar space of type DQ(2n,2), n≥2n≥2, we obtain a generalization of these constructions. By using a construction alluded to in [B. De Bruyn, A new geometrical construction for the near hexagon with parameters (s,t,T2)=(2,5,{1,2})(s,t,T2)=(2,5,{1,2}), J. Geom. 78 (2003) 50–58.] we show that these generalized constructions give rise to near 2n2n-gons which are isomorphic to InIn and DQ(2n,2). In this way, we obtain a recursive construction for the dual polar spaces DQ(2n,2), n≥2n≥2, different from the one given in [B.N. Cooperstein, E.E. Shult, Combinatorial construction of some near polygons, J. Combin. Theory Ser. A 78 (1997) 120–140].