Enumerative Coding for Line Polar Grassmannians with applications to codes

TL;DR

This paper introduces an efficient enumeration method for points in orthogonal and symplectic line Grassmannians, enabling improved encoding, decoding, and error correction for related Grassmann codes.

Contribution

It presents a novel enumeration technique for line polar Grassmannians, facilitating practical coding applications in error correction.

Findings

Efficient enumeration algorithms for orthogonal and symplectic line Grassmannians.

Enhanced encoding and decoding strategies for line polar Grassmannian codes.

Potential improvements in error correction performance.

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.