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 Mt-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 Mt is the number of antennas. If A = F = GF(q) it is shown that the number of nonintersecting subspaces is at most (qm - 1)/(qMt - 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 2r complex roots of unity, the number of nonintersecting planes is at least 2r(m-2) and at most 2r(m-1)-1 (the lower bound may in fact be the best that can be achieved. © 2005 IEEE.
Oggier, FE; Sloane, NJA; Diggavi, SN; Calderbank, AR
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