Representation of Subspaces and Enumerative Encoding of the Grassmannian Space

Codes in the Grassmannian space have found recently application in network coding. Representation of kdimensional subspaces of Fq 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 a Ferrers diagram representation of subspaces; and another is based on an identifying vector and a reduced row echelon form representation of subspaces. A third method which combines the previous two is more efficient than the other two enumerative encodings. Each enumerative encoding is induced by some ordering of the Grassmannian. These orderings also induce lexicographic codes in the Grassmannian. Some of these codes suggest a new method to generate error-correcting codes in the Grassmannian with larger size than the current known codes.