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 C F? The most important case is when F is the field of complex numbers 0; 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 Mt. 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 Mt = 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).
nonintersecting subspaces multiple-antenna communications noncoherent systems space-time codes