Enumerative Coding for Grassmannian Space

TL;DR

This paper introduces efficient enumerative encoding and decoding methods for Grassmannian spaces, leveraging two different orders based on Ferrers diagrams and reduced row echelon forms, with complexity improvements when combined.

Contribution

It presents two novel enumerative coding techniques for Grassmannian spaces, optimizing encoding/decoding complexity using distinct subspace representations.

Findings

Ferrers diagram-based coding complexity: O(k^{5/2}(n-k)^{5/2})

Reduced row echelon form-based coding complexity: O(nk(n-k)log n log log n)

Combined methods reduce complexity on average by a constant factor

Abstract

The Grassmannian space $\Gr$ is the set of all $k-$dimensional subspaces of the vector space~\smash{$\F_q^n$}. Recently, codes in the Grassmannian have found an application in network coding. The main goal of this paper is to present efficient enumerative encoding and decoding techniques for the Grassmannian. These coding techniques are based on two different orders for the Grassmannian induced by different representations of $k$-dimensional subspaces of $\F_q^n$. One enumerative coding method is based on a Ferrers diagram representation and on an order for $\Gr$ based on this representation. The complexity of this enumerative coding is $O(k^{5/2} (n-k)^{5/2})$ digit operations. Another order of the Grassmannian is based on a combination of an identifying vector and a reduced row echelon form representation of subspaces. The complexity of the enumerative coding, based on this order, is $O(nk(n-k)\log n\log\log n)$ digits operations. A combination of the two methods reduces the complexity on average by a constant factor.