Enumerative coding for line polar Grassmannians with applications to codes

A k-polar Grassmannian is a geometry having as pointset
the set of all k-dimensional subspaces of a vector space
V which are totally isotropic for a given non-degenerate
bilinear form μ defined on V . Hence it can be regarded
as a subgeometry of the ordinary k-Grassmannian. In this
paper we deal with orthogonal line Grassmannians and with
symplectic line Grassmannians, i.e. we assume k = 2 and μ
to be a non-degenerate symmetric or alternating form. We
will provide a method to efficiently enumerate the pointsets
of both orthogonal and symplectic line Grassmannians. This
has several nice applications; among them, we shall discuss an
efficient encoding/decoding/error correction strategy for line
polar Grassmann codes of either type.