On the Sequential and Random Selection of Subspaces over a Finite Field

Let V, be an n-dimensional vector space over GF(q), and let 0 < k < n. We consider the question of generating sets of basis vectors for k-dimensional subspaces V, of V, in two ways: (a) the sequential generation of a set of basis vectors for every such subspace and (b) the random selection of such a subspace V, in such a way that all subspaces have equal a priori probabilities of being chosen. Such questions are of interest in coding theory. The answer to the first question has been known for some time and comes from the row-echelon, or " Schubert, " form of the k x IZ matrix B whose rows are the basis vectors: We fix a k-subset (a, ,..., a,) = S of {I, 2,..., n}. In the columns of B which correspond to S we enter a k x k identity matrix. In row i of B we enter 0 in all columns j > a, , i=l remaining entries of B may be assigned independently to elements of the field. Hence for the given subset S we obtain qN sets of basis vectors, each describing a different subspace V,. By varying S we obtain all of the desired V, , and of course [ 1 I: ,l= (qn _ l)(q "-l-1). .. (q-k+1-1) (qk-l)(q "-1-1) ... (q-1) = F qN'S' (1) is the total number of such Vk. Before discussing our algorithm for uniformly random selection, we mention, briefly, two others which may come immediately to mind.