Enumerative Encoding in the Grassmannian Space

TL;DR

This paper explores various enumerative encoding methods for subspaces in the Grassmannian space, including Ferrers diagrams, identifying vectors, and a combined approach, to improve coding efficiency in network coding applications.

Contribution

It introduces and compares three enumerative encoding techniques for Grassmannian subspaces, including a novel combined method that outperforms individual approaches.

Findings

The combined encoding method is more efficient than individual methods.

Ferrers diagram and identifying vector representations are effective for encoding.

The approaches facilitate better network coding in Grassmannian spaces.

Abstract

Codes in the Grassmannian space have found recently application in network coding. Representation of $k$-dimensional subspaces of $\F_q^n$ has generally an essential role in solving coding problems in the Grassmannian, and in particular in encoding subspaces of the Grassmannian. Different representations of subspaces in the Grassmannian are presented. We use two of these representations for enumerative encoding of the Grassmannian. One enumerative encoding is based on Ferrers diagrams representation of subspaces; and another is based on identifying vector and reduced row echelon form representation of subspaces. A third method which combine the previous two is more efficient than the other two enumerative encodings.