Low-delay, High-rate Non-square Complex Orthogonal Designs

The maximal rate of a non-square complex orthogonal design for $n$ transmit antennas is $1/2+\frac{1}{n}$ if $n$ is even and $1/2+\frac{1}{n+1}$ if $n$ is odd and the codes have been constructed for all $n$ by Liang (IEEE Trans. Inform. Theory, 2003) and Lu et al. (IEEE Trans. Inform. Theory, 2005) to achieve this rate. A lower bound on the decoding delay of maximal-rate complex orthogonal designs has been obtained by Adams et al. (IEEE Trans. Inform. Theory, 2007) and it is observed that Liang's construction achieves the bound on delay for $n$ equal to 1 and 3 modulo 4 while Lu et al.'s construction achieves the bound for $n=0,1,3$ mod 4. For $n=2$ mod 4, Adams et al. (IEEE Trans. Inform. Theory, 2010) have shown that the minimal decoding delay is twice the lower bound, in which case, both Liang's and Lu at al.'s construction achieve the minimum decoding delay. % when $n=2$ mod 4. For large value of $n$, it is observed that the rate is close to half and the decoding delay is very large. A class of rate-1/2 codes with low decoding delay for all $n$ has been constructed by Tarokh et al. (IEEE Trans. Inform. Theory, 1999). % have constructed a class of rate-1/2 codes with low decoding delay for all $n$. In this paper, another class of rate-1/2 codes is constructed for all $n$ in which case the decoding delay is half the decoding delay of the rate-1/2 codes given by Tarokh et al. This is achieved by giving first a general construction of square real orthogonal designs which includes as special cases the well-known constructions of Adams, Lax and Phillips and the construction of Geramita and Pullman, and then making use of it to obtain the desired rate-1/2 codes. For the case of 9 transmit antennas, the proposed rate-1/2 code is shown to be of minimal-delay.