Fast enumeration for Grassmannian space

Abstract

The Grassmanian Gq(n; k) is the set of all k-dimensional subspaces of vector space Fnq. The coding of elements of Grassmanian was considered in many papers [1]-[7], and has the application in network coding [8]-[19]. The enumerative coding of the elements of Grassmanian Gq(n; k) is association every element of the Grassmanian with its number, i. e. the number from [0;...; |Gq(n; k)| - 1]. The algorithm of enumerative coding of the elements of the Grassmanian, which has complexity O(nk(n - k) log n log log n) is presented in the paper [20]. We present the advanced algorithm of the enumerative coding of the elements of the Grassmanian, which has the complexity that does not exceed O(n2log2nloglog n). The advanced algorithm is based on the method of fast enumeration of combinatorial objects from the paper of B. Ryabko [21].