Bounds on Fast Decodability of Space-Time Block Codes, Skew-Hermitian Matrices, and Azumaya Algebras
We study fast lattice decodability of house-time block codes for $n$ transmit and receive antennas, written very generally as a linear combination $add _i=1^2l s_i A_i$ , where the $s_i$ are real info symbols and also the $A_i$ are $ntimes n~ mathbb R$ -linearly freelance complicated-valued matrices. We show that the mutual orthogonality condition $A_iA_vphantom R_R_R_aj^* + A_jA_i^*=zero$ for distinct basis matrices is not solely sufficient but also necessary for fast decodability. We tend to build on this to indicate that for full-rate ( $l = n^2$ ) transmission, the decoding complexity will be no higher than $ |S|^n^2+1$ , where $ |S|$ is the dimensions of the effective real signal constellation. We have a tendency to also show that for full-rate transmission, $g$ -group decodability, as defined by Jithamithra and Rajan, is not possible for any $gge a pair of$ . We then use the idea of Azumaya algebras to derive bounds on the most range of teams into that the premise matrices can be partitioned so that the matrices in different groups are mutually orthogonal—a key measure of quick decodability. We have a tendency to show that normally, this maximum range is of the order of only the 2-adic val- e of $n$ . In the case where the matrices $A_i$ arise from a division algebra, which is most desirable for diversity, we show that the maximum number of groups is only four. Thence, the decoding complexity for this case is no better than $ |S|^lceil l/2 rceil $ for any rate $l$ .
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