Union bound on error probability of linear space-time block codes

The design of practical coding techniques for the multiple antenna wireless channel is a challenging problem. A number of interesting solutions have been proposed ranging from block codes to trellis codes for the MIMO (multiple input, multiple output) channel. We consider linear block codes for the quasi-static, flat-fading, coherent MIMO channel. A linear code refers to an encoder that is linear with respect to scalar input symbols. We assume maximum likelihood decoding at the receiver. We provide a cohesive framework for analysis of linear codes in terms of a union bound on the conditional probability of symbol error. The error bound is a function of the instantaneous channel realization and does not make any assumptions on channel statistics. We show that the orthogonal block codes proposed by Tarokh, Jafarkhani and Calderbank, (see IEEE Trans. Information Theory, vol.45, no.5, p.1456-67, 1999) achieve the lowest error bound among all unitary codes and are in fact optimal