Deterministic approach for hypercomplex generalized orthogonal design (DAHOD)

Abstract

There has been much research on efficient spacetime coding during the past years. The idea of orthogonal coding has later been extended to new degrees of freedom like polarization and frequency (OFDM) as well. The majority of the research concerning code matrices still concentrates on real and complex codes. However, there has been some change in the very past where also quaternions came into play. Nevertheless, hypercomplex numbers like quaternions, octonions and others are still used rarely. One reason we saw was that there is no general idea and construction scheme for code matrices. Real-, complex- and quaternion-valued code matrices are still being constructed in the respective domain. For that reason, we took an abstract point of view. That is, we do not consider a certain type of hypercomplex numbers. Instead, we do all computation in a real matrix-representation. Moreover, we show the basic conditions that have to be fulfilled so that generalized orthogonal hypercomplex codes exist.