Enumerative Coding for Grassmannian Space

The Grassmannian space G_q(n, k) is the set of all k-dimensional subspaces of the vector space F_qⁿ. 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_qⁿ. One enumerative coding method is based on a Ferrers diagram representation and on an order for G_q(n, k) 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) digit operations. A combination of the two methods reduces the complexity on average by a constant factor.