On a Conjecture About Linear Processing Complex Orthogonal Designs

Linear processing complex orthogonal designs (LPCODs) were introduced as a generalization of complex orthogonal designs (CODs) to increase the rate of transmission when they are used as orthogonal space-time codes. While the maximal rate of CODs for a fixed number of columns (antennas) is known, in case of LPCODs, the problem is still open and it is conjectured that the maximal rate of LPCODs is the same as the maximal rate of CODs. In this paper, we attack this conjecture by defining a new equivalence relation on LPCODs, called weak equivalence, and showing that for small number of antennas maximal rate LPCODs with minimal delays are all weakly equivalent. This supports our conjecture that all maximal rate LPCODs with minimal delays are weakly equivalent. In addition, for , we construct LPCODs with n columns which are not equivalent to any COD with respect to a previously known equivalence relation.