Enumerative coding for line polar Grassmannians with applications to codes

Abstract

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

Line Hermitian Grassmann codes and their parameters

“…In [3], we started investigating some projective codes arising from subgeometries of the Grassmann variety associated to orthogonal and symplectic k-Grassmannians. We called such codes respectively orthogonal [3,5,6,7] and symplectic Grassman codes [4,6]. In the cases of line orthogonal and symplectic Grassmann codes, i.e.…”

“…In a forthcoming paper [8] we plan to describe and discuss algorithms for implementing encoding, decoding and error correction for line Hermitian Grassmann codes in the same spirit of [6].…”

Implementing line-Hermitian Grassmann codes

“…syndrome decoding, see [13,Chapter 1], but such an approach would be very inefficient in the case of (polar) Grassmann codes, since the parity check matrix is huge. A different, more viable, approach is what we proposed in [4] for line polar Grassmann codes of either orthogonal or symplectic type and we here extend to the Hermitian case. Suppose r x is an entry in the received vector r which we want to insure to be correct.…”

“…In the present paper we shall be concerned with point enumerators of line Hermitian Grassmannians. In this way, we continue a project started in [4] where we introduced point enumerators for line polar Grassmannians of orthogonal [7,5] and symplectic type [3].…”

Minimum distance of orthogonal line-Grassmann codes in even characteristic

“…For further details on the actual construction of orthogonal and symplectic line-Grassmann codes, we refer to [2] where some efficient algorithms for encoding, decoding and error-correction have been presented.…”