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

[1]  Frédérique E. Oggier,et al.  Families of unitary matrices achieving full diversity , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[2]  Brian L. Hughes Differential Space-Time modulation , 2000, IEEE Trans. Inf. Theory.

[3]  Yindi Jing,et al.  Design of fully diverse multiple-antenna codes based on Sp(2) , 2004, IEEE Transactions on Information Theory.

[4]  Emanuele Viterbo,et al.  A universal lattice code decoder for fading channels , 1999, IEEE Trans. Inf. Theory.

[5]  Christopher Holden,et al.  Perfect Space-Time Block Codes , 2004 .

[6]  B. Sundar Rajan,et al.  Full-diversity, high-rate space-time block codes from division algebras , 2003, IEEE Trans. Inf. Theory.

[7]  J. Neukirch Algebraic Number Theory , 1999 .

[8]  Frédérique E. Oggier,et al.  Perfect Space–Time Block Codes , 2006, IEEE Transactions on Information Theory.

[9]  Babak Hassibi,et al.  On the sphere-decoding algorithm I. Expected complexity , 2005, IEEE Transactions on Signal Processing.

[10]  Bertrand M. Hochwald,et al.  Differential unitary space-time modulation , 2000, IEEE Trans. Commun..

[11]  Xiang-Gen Xia,et al.  Unitary signal constellations for differential space--Time modulation with two transmit antennas: Parametric codes, optimal designs, and bounds , 2002, IEEE Trans. Inf. Theory.

[12]  Babak Hassibi,et al.  Representation theory for high-rate multiple-antenna code design , 2001, IEEE Trans. Inf. Theory.

[13]  Frédérique E. Oggier,et al.  Cyclic Division Algebras: A Tool for Space-Time Coding , 2007, Found. Trends Commun. Inf. Theory.

[14]  B. Hassibi,et al.  Three-transmit-antenna space-time codes based on SU(3) , 2005, IEEE Transactions on Signal Processing.

[15]  Frédérique E. Oggier,et al.  Cyclic Algebras for Noncoherent Differential Space–Time Coding , 2007, IEEE Transactions on Information Theory.

[16]  I. Stewart,et al.  Algebraic Number Theory , 1992, All the Math You Missed.

[17]  Babak Hassibi,et al.  Cayley differential unitary space - Time codes , 2002, IEEE Trans. Inf. Theory.

[18]  Frédérique E. Oggier,et al.  Algebraic Number Theory and Code Design for Rayleigh Fading Channels , 2004, Found. Trends Commun. Inf. Theory.