Nonintersecting subspaces based on finite alphabets

Two subspaces of a vector space are here called "nonintersecting" if they meet only in the zero vector. Motivated by the design of noncoherent multiple-antenna communications systems, we consider the following question. How many pairwise nonintersecting M_t-dimensional subspaces of an m-dimensional vector space V over a field F can be found, if the generator matrices for the subspaces may contain only symbols from a given finite alphabet A⊆F? The most important case is when F is the field of complex numbers C; then M_t is the number of antennas. If A=F=GF(q) it is shown that the number of nonintersecting subspaces is at most (q^m-1)/(q^Mt-1), and that this bound can be attained if and only if m is divisible by M_t. Furthermore, these subspaces remain nonintersecting when "lifted" to the complex field. It follows that the finite field case is essentially completely solved. In the case when F=C only the case M_t=2 is considered. It is shown that if A is a PSK-configuration, consisting of the 2^r complex roots of unity, the number of nonintersecting planes is at least 2^r(m-2) and at most 2^r(m-1)-1 (the lower bound may in fact be the best that can be achieved).