# Space-time coding

**Inter-symbol Interference multiple access channels:** We studied the multiple-access channel where users employ time-reversed space-time block codes (TR-STBC) which is suitable for ISI channels. One of the contributions is to identify key algebraic properties of the TR-STBC codes which allows us to design and analyze multiuser detectors. Using these, we showed that a diversity order of $2M_r(\nu+1)$ is achievable at full transmission rate for each user, when we have $M_r$ receive antennas, $M_t=2$ transmit antennas per user, channel memory of $\nu$ and an optimal multiuser maximum-likelihood (ML) decoder is used. Due to the decoding complexity of the ML detector we study the algebraic structure of linear multiuser detectors which utilize the properties of the STBC. We also do this when we have finite block length constraints using properties of quaternionic block circulant matrices.

**Rate-growth of linear detectors:** In (Telatar, 1995; Foschini, 1996), it was shown that multiple transmit and receive antennas can provide linear growth in reliable transmission rates. One of the questions we asked was whether the rate advantages shown by Foschini et al can be obtained by using simpler receiver structures, such as linear receivers (akin to multiuser detection schemes). We showed that the linear growth in the rate with the number of antennas, asymptotically as the number of antennas being large, can also be achieved by such low-complexity schemes. However, the linear growth assumes that the channel gain becomes unbounded resulting in unbounded achievable rates. Consequently we examined the normalized channel (where the average gain is unity), and found that the mutual information grows linearly with signal-to-noise ratio (SNR) as the diversity elements become large.This is analogous to the infinite bandwidth result in Gaussian channels (Gallager, Information theory and reliable communications, 1968). These results demonstrate that the rate advantages of multiple antennas can be still obtained using lower complexity linear detection schemes and such schemes have a rich literature in the context of multiuser detection (Verdu, Multiuser detection, 1998).

**Non-intersecting subspaces over finite alphabet:** In the high SNR regime of non-coherent space-time codes, we communicate using subspaces in which the codewords lie. The reliability (diversity order) is determined by the maximal intersection between the subspaces of any two codewords in the codebook. Moreover, in many applications we are constrained to use a specific transmit alphabet ({\em e.g.,} QPSK). Therefore we formulated a question on what is the maximal rate we can get with these alphabet constraints when we desire to have the maximal diversity order. This translates to a mathematical question on the number of pairwise nonintersecting $M_t$-dimensional subspaces of an $m$-dimensional vector space $V$ over a field $\mathbb{F}$ if the generator matrices for the subspaces may contain only symbols from a given finite alphabet. Note that, two subspaces of a vector space are here called “nonintersecting” if they meet only in the zero vector. We have a complete answer if the alphabet is GF(q); we show that the number of nonintersecting subspaces is at most $(q^m-1)/(q^{M_t}-1)$, and that this bound can be attained if and only if $m$ is divisible by $M_t$. When we have PSK constellations, we have examined the case when we have $M_t=2$ tramsmit antennas, and have shown that 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).

**Quaternionic space-time codes:** We designed full-rate, full-diversity orthogonal space-time block codes for 4 transmit antennas based on quaternionic algebra. This code is non-linear but can be designed to have no constellation expansion for QPSK modulation. The quaternionic structure can be further used to reduce the complexity of the optimal decoder. We also used this code to construct non-coherent differential codes.