Families of unitary matrices achieving full diversity

This paper presents an algebraic construction of families of unitary matrices that achieve full diversity. They are obtained as subsets of cyclic division algebras.Comment: To appear in the proceedings of the 2005 IEEE International Symposium on Information Theory, Adelaide, Australia, September 4-9, 200

Algebraic Cayley Differential Space–Time Codes

Cayley space-time codes have been proposed as a solution for coding over noncoherent differential multiple-input multiple-output (MIMO) channels. Based on the Cayley transform that maps the space of Hermitian matrices to the manifold of unitary matrices, Cayley codes are particularly suitable for high data rate, since they have an easy encoding and can be decoded using a sphere-decoder algorithm. However, at high rate, the problem of evaluating if a Cayley code is fully diverse may become intractable, and previous work has focused instead on maximizing a mutual information criterion. The drawback of this approach is that it requires heavy optimization which depends on the number of antennas and rate. In this work, we study Cayley codes in the context of division algebras, an algebraic tool that allows to get fully diverse codes. We present an algebraic construction of fully diverse Cayley codes, and show that this approach naturally yields, without further optimization, codes that perform similarly or closely to previous unitary differential codes, including previous Cayley codes, and codes built from Lie groups

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