Complex Orthogonal Designs

For p×n complex orthogonal designs in k variables, where p is the number of channels uses and n is the number of transmit antennas, the maximal rate k p of the design is asymptotically half as n increases. But, for such maximal rate codes, the decoding delay p increases exponentially. To control the delay, if we put the restriction that p = n, i.e., consider only the square designs, then, the rate decreases exponentially as n increases. This necessitates the study of the maximal rate of the designs with restrictions of the form p = n+1, p = n+2, p = n+ 3 etc. In this paper, we study the maximal rate of complex orthogonal designs with the restrictions p = n+1 and p = n+2. We derive upper and lower bounds for the maximal rate for p = n + 1 and p = n + 2. Also for the case of p = n + 1, we show that if the orthogonal design admit only the variables, their negatives and multiples of these by √ −1 and zeros as the entries of the matrix (other complex linear combinations are not allowed), then the maximal rate always equals the lower bound.