Some Diversity Product Properties of A Family of Space-Time Trellis Codes with Asynchronous Full Diversity

In order to achieve full cooperative diversity in a relay network, most of the existing space-time coding schemes require the synchronization between terminals. A family of space-time trellis codes that achieve full cooperative diversity order without the assumption of synchronization has been recently proposed. The family is based on the stack construction by Hammons and El Gamal and its generalizations by Lu and Kumar. It has been shown that the construction of such a family is equivalent to the construction of binary matrices that have full row rank no matter how their rows are shifted. We call such matrices as shift full rank (SFR) matrices. Given such a code generated from an SFR matrix, for different delays occurring at the relays, the code corresponds to different trellis structures and thus has different diversity products in general. In this paper, we first give a pair of upper bound and lower bound of the diversity products for arbitrarily possible delays. Furthermore, for the special case of two relays and BPSK modulation, we present some detailed analyses for the relationship between the SFR generator matrices and the minimum values of all the diversity products corresponding to different delays.