Cyclic Algebras for Noncoherent Differential Space–Time Coding

We investigate cyclic algebras for coding over the differential noncoherent channel. Cyclic algebras are an algebraic object that became popular for coherent space–time coding, since it naturally yields linear families of matrices with full diversity. Coding for the differential noncoherent channel has a similar flavor in the sense that it asks for matrices that achieve full diversity, except that these matrices furthermore have to be unitary. In this work, we give a systematic way to find infinitely many unitary matrices inside cyclic algebras, which holds for all dimensions. We show how cyclic algebras generalize previous families of unitary matrices obtained using the representation of fixed-point-free groups. As an application of our technique, we present families of codes for three and four antennas that achieve high coding gain.