On the Geometry of Balls in the Grassmannian and List Decoding of Lifted Gabidulin Codes

TL;DR

This paper investigates the geometric structure of balls in the Grassmannian space under different metrics, and applies these insights to improve list decoding of lifted Gabidulin codes in network coding.

Contribution

It characterizes the balls in the Grassmannian with respect to subspace and injection metrics using Plücker embedding and rational parametrization, and applies these to list decoding of lifted Gabidulin codes.

Findings

Describes the structure of Grassmannian balls with respect to two metrics.

Provides algebraic representations of lifted Gabidulin codes.

Proposes a method for list decoding using algebraic equations.

Abstract

The finite Grassmannian $\mathcal{G}_{q}(k,n)$ is defined as the set of all $k$-dimensional subspaces of the ambient space $\mathbb{F}_{q}^{n}$. Subsets of the finite Grassmannian are called constant dimension codes and have recently found an application in random network coding. In this setting codewords from $\mathcal{G}_{q}(k,n)$ are sent through a network channel and, since errors may occur during transmission, the received words can possible lie in $\mathcal{G}_{q}(k',n)$, where $k'\neq k$. In this paper, we study the balls in $\mathcal{G}_{q}(k,n)$ with center that is not necessarily in $\mathcal{G}_{q}(k,n)$. We describe the balls with respect to two different metrics, namely the subspace and the injection metric. Moreover, we use two different techniques for describing these balls, one is the Pl\"ucker embedding of $\mathcal{G}_{q}(k,n)$, and the second one is a rational parametrization of the matrix representation of the codewords. With these results, we consider the problem of list decoding a certain family of constant dimension codes, called lifted Gabidulin codes. We describe a way of representing these codes by linear equations in either the matrix representation or a subset of the Pl\"ucker coordinates. The union of these equations and the equations which arise from the description of the ball of a given radius in the Grassmannian describe the list of codewords with distance less than or equal to the given radius from the received word.