Enumerative Coding for Line Polar Grassmannians with applications to codes

Abstract

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 $V.$ 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.

27 pages; revised version after review

References in corpus (5)

• Decomposable Subspaces, Linear Sections of Grassmann Varieties, and Higher Weights of Grassmann Codes
• Codes and caps from orthogonal Grassmannians
• Minimum distance of Symplectic Grassmann codes
• Minimum distance of Line Orthogonal Grassmann Codes in even characteristic
• Weight spectrum of codes associated with the Grassmannian G(3,7)

Cited by in corpus (4)

• Minimum distance of Symplectic Grassmann codes
• Minimum distance of Line Orthogonal Grassmann Codes in even characteristic
• Line Hermitian Grassmann Codes and their Parameters
• Implementing Line-Hermitian Grassmann codes