Low-Delay, High-Rate Nonsquare Complex Orthogonal Designs

The maximal rate of a nonsquare complex orthogonal design for n transmit antennas is [1/2]+[1/(n)] if n is even and [1/2]+[1/(n+1)] if n is odd and the codes have been constructed for all n by Liang (2003) and Lu (2005) to achieve this rate. A lower bound on the decoding delay of maximal-rate complex orthogonal designs has been obtained by Adams (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 (2010) have shown that the minimal decoding delay is twice the lower bound, in which case, both Liang´s and Lu et al.´s construction achieve the minimum decoding delay. 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 (1999). 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 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 nine transmit antennas, the proposed rate-[1/2] code is shown to be of minimal delay. The proposed construction results in designs with zero entries which may have high peak-to-average power ratio and it is shown that by appropriate postmultiplication, a design with no zero entry can be obtained with no change in the code parameters.