Enumerative Coding for Line Polar Grassmannians with applications to codes

A $k$-polar Grassmannian is the 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 $\mu$ defined on $V\mathrm{.}$ 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 $\mu$ 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 both types.