List Decoding of Lifted Gabidulin Codes via the Pl\"ucker Embedding

TL;DR

This paper presents a novel list decoding method for lifted Gabidulin codes in the Grassmannian using the Plücker embedding, representing code subsets as linear block codes to efficiently identify close codewords.

Contribution

It introduces a new decoding approach leveraging Plücker coordinates, enabling list decoding of lifted Gabidulin codes in the Grassmannian.

Findings

Representation of lifted Gabidulin codes as linear block codes in Plücker coordinates

Decoding involves parity-check equations from block codes and ball descriptions

Method improves decoding capabilities for network coding applications

Abstract

Codes in the Grassmannian have recently found an application in random network coding. All the codewords in such codes are subspaces of $\F_q^n$ with a given dimension. In this paper, we consider the problem of list decoding of a certain family of codes in the Grassmannian, called lifted Gabidulin codes. For this purpose we use the Pl\"ucker embedding of the Grassmannian. We describe a way of representing a subset of the Pl\"ucker coordinates of lifted Gabidulin codes as linear block codes. The union of the parity-check equations of these block codes and the equations which arise from the description of a ball around a subspace in the Pl\"ucker coordinates describe the list of codewords with distance less than a given parameter from the received word.