High-rate Information-Lossless Linear Dispersion STBCs from Group Algebra Kiran. T ECE Department Indian Institute of Science Bangalore-560012, INDIA Email: kirant@protocol.ece.iisc.ernet.in B. Sundar Rajan ECE Department Indian Institute of Science Bangalore-560012, INDIA Email: bsrajan@ece.iisc.ernet.in Abstract— For multiple-input multiple-output (MIMO) chan- nels, at high spectral efficiencies, space-time block codes (STBCs) must be designed to maximize the mutual information between the transmit and receive signals. In an uncoded scheme (spatial multiplexing or V-BLAST), it is well-known that the Gaussian input distribution maximizes the mutual information. Hassibi and Hochwald introduced a linear dispersion (LD) framework for designing space-time codes, wherein, any transmit codeword is a linear combination of a fixed set of matrices called the weight matrices. A LD space-time block code is said to be information-lossless if it does not disturb the maximum mutual information between the transmit and receive signals. In other words, a MIMO scheme using information-lossless LD space- time block codes has the same capacity as the uncoded scheme. Through computer search, information-lossless LD codes with better diversity compared to uncoded system were found by Hassibi et al., and also by Heath et al. In this paper, we give a general algebraic construction of high-rate information- lossless STBCs, both square and rectangular, by restricting the weight matrices to those which form a finite group under matrix multiplication. I. I NTRODUCTION A Linear Dispersion (LD) Space-Time Block Code (STBC) C [1] over a signal set S , is a finite set of n × l matrices (n ≤ l), where any codeword matrix belonging to the code C is of the form S(x 1 ,x 2 ,...,x Q )= Q  i=1 (x i A i + x ∗ i B i ) , where A i and B i are fixed n × l complex matrices defining the LD code, x 1 ,x 2 ...,x Q are complex scalars taking values from the signal set S and x ∗ i denotes the complex conjugate of x i . A subclass of LD codes is the purely complex LD- STBCs, where any matrix S(x 1 ,x 2 ,...,x Q ) is a complex linear combination of {A 1 , A 2 ,..., A Q } only, i.e., the ma- trices B i are all chosen to be zero matrix. We call the set {A 1 , A 2 ,..., A Q } as dispersion matrices or weight matrices. Such complex LD-STBCs have been considered in [2] also. The rate of this code is Q/l complex symbols per channel use (Q/l-scu). From the pair-wise error probability point of view, it is well known that the rank criteria and the determinant criteria are This work was partly funded by the IISc-DRDO program on Advanced Research in Mathematical Engineering through a grant to B.S. Rajan. guidelines for designing good STBCs. In [1], Hassibi et al. show that at high spectral efficiencies, while the rank criteria plays a secondary role, it is the mutual information between the transmit and receive signals that is primary with regard to both capacity as well as performance. Hence, a necessary condition for achieving capacity of a multiple-input multiple- output (MIMO) channel is to design STBCs that maximize the mutual information. A LD-STBC is said to be information- lossless if it does not disturb the maximum mutual information between the transmit and receive signals. In other words, a MIMO scheme using information-lossless LD-STBC has the same capacity as the uncoded scheme. To make this precise, we introduce the MIMO channel model. Let n t and n r be the number of transmit and receive antennas respectively. For an uncoded system, let x ∈ C nt×1 denote the vector of transmitted signals by all n t transmit antennas in one channel use, and let y ∈ C nr ×1 denote the corresponding received vector. They are related by y =  ρ n t Hx + w (1) where ρ is the signal to noise ratio at each receive antenna, H ∈ C nr ×nt is the channel matrix and w ∈ C nr ×nt is the additive noise. Both H and w have entries that are i.i.d, complex-Gaussian with zero mean and unit variance. It is assumed that H is perfectly known at the receiver, in which case the channel capacity is given by [1], [3] C(ρ, n t ,n r )= E H log 2  det  I nr + ρ n t HH H  . The above expression is the capacity for the uncoded MIMO system, when an independently chosen vector x is transmitted for every channel use. For quasi-static fading scenario, the channel matrix will not change for successive l channel uses, in which case sufficient redundancy can be introduced in the successive transmit vectors to improve the performance (in terms of probability of error) of the code. For such a case, we can write the counter part of (1) as Y =  ρ n t HS + W (2) where S is a n t × l space-time codeword, W is the n r × l complex-Gaussian additive noise matrix and Y ∈ C nr ×l is 0-7803-8794-5/04/$20.00 (C) 2004 IEEE