Combinatorial construction of some near polygons (Q1356022)

A near \(2n\)-gon is a linear incidence system \((P,L)\) satisfying: (i) every line contains at least two points, (ii) the point graph of \((P,L)\) is connected with diameter \(n\), (iii) for each point line pair \((p,l)\), there is a unique point \(q\) on \(l\) nearest \(p\). Note that near 4-gon is the same thing as a generalized quadrangle. The authors give a recipe which has one ingredient, a rank two incidence geometry with three points on a line and produces a geometry of the same type. The recipe is applied recursively, starting with the geometry consisting of one point and no lines, to obtain the family of dual polar spaces of type \(Sp(2n,2)\). Applying the construction to the Hamming near \(2n\)-gon on \(3^n\) points, the authors obtain some new near \(2n+2\)-gons.