Cyclic Division Algebras: A Tool for Space-Time Coding

Multiple antennas at both the transmitter and receiver ends of a wireless digital transmission channel may increase both data rate and reliability. Reliable high rate transmission over such channels can only be achieved through Space-Time coding. Rank and determinant code design criteria have been proposed to enhance diversity and coding gain. The special case of full-diversity criterion requires that the difference of any two distinct codewords has full rank. Extensive work has been done on Space–Time coding, aiming at finding fully diverse codes with high rate. Division algebras have been proposed as a new tool for constructing Space–Time codes, since they are non-commutative algebras that naturally yield linear fully diverse codes. Their algebraic properties can thus be further exploited to improve the design of good codes. The aim of this work is to provide a tutorial introduction to the algebraic tools involved in the design of codes based on cyclic division algebras. The different design criteria involved will be illustrated, including the constellation shaping, the information lossless property, the non-vanishing determinant property, and the diversity multiplexing trade-off. The final target is to give the complete mathematical background underlying the construction of the Golden code and the other Perfect Space–Time block codes.

[1]  P. Samuel.  Théorie algébrique des nombres , 1971 .

[2]  S. Lang.  Algebraic Number Theory , 1971 .

[3]  M. Best.  On the existence of perfect codes , 1978 .

[4]  G. David Forney,et al.  Efficient Modulation for Band-Limited Channels , 1984, IEEE J. Sel. Areas Commun..

[5]  G. David Forney,et al.  Multidimensional constellations. I. Introduction, figures of merit, and generalized cross constellations , 1989, IEEE J. Sel. Areas Commun..

[6]  M. Carral,et al.  Quadratic and λ-hermitian forms , 1989 .

[7]  W. Fischer,et al.  Sphere Packings, Lattices and Groups , 1990 .

[8]  Gerard J. Foschini,et al.  Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas , 1996, Bell Labs Technical Journal.

[9]  Thomas C. Hales.  Sphere packings, I , 1997, Discret. Comput. Geom..

[10]  A. Robert Calderbank,et al.  Space-Time Codes for High Data Rate Wireless Communications : Performance criterion and Code Construction , 1998, IEEE Trans. Inf. Theory.

[11]  M. J. Gans,et al.  On Limits of Wireless Communications in a Fading Environment when Using Multiple Antennas , 1998, Wirel. Pers. Commun..

[12]  Siavash M. Alamouti,et al.  A simple transmit diversity technique for wireless communications , 1998, IEEE J. Sel. Areas Commun..

[13]  Reinaldo A. Valenzuela,et al.  V-BLAST: an architecture for realizing very high data rates over the rich-scattering wireless channel , 1998, 1998 URSI International Symposium on Signals, Systems, and Electronics. Conference Proceedings (Cat. No.98EX167).

[14]  Shlomo Shamai,et al.  Fading Channels: Information-Theoretic and Communication Aspects , 1998, IEEE Trans. Inf. Theory.

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

[16]  A. Robert Calderbank,et al.  Space-Time block codes from orthogonal designs , 1999, IEEE Trans. Inf. Theory.

[17]  Emre Telatar,et al.  Capacity of Multi-antenna Gaussian Channels , 1999, Eur. Trans. Telecommun..

[18]  E. Bayer-Fluckiger.  Lattices and number Fields , 1999 .

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

[20]  Thomas L. Marzetta,et al.  Unitary space-time modulation for multiple-antenna communications in Rayleigh flat fading , 2000, IEEE Trans. Inf. Theory.

[21]  Hesham El Gamal,et al.  On the theory of space-time codes for PSK modulation , 2000, IEEE Trans. Inf. Theory.

[22]  Mohamed Oussama Damen,et al.  Lattice code decoder for space-time codes , 2000, IEEE Communications Letters.

[23]  A.J. Paulraj,et al.  Space-frequency codes for broadband fading channels , 2001, Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No.01CH37252).

[24]  Mohamed Oussama Damen,et al.  A construction of a space-time code based on number theory , 2002, IEEE Trans. Inf. Theory.

[25]  B. Sundar Rajan,et al.  An algebraic description of orthogonal designs and the uniqueness of the Alamouti code , 2002, Global Telecommunications Conference, 2002. GLOBECOM '02. IEEE.

[26]  Karim Abed-Meraim,et al.  Diagonal algebraic space-time block codes , 2002, IEEE Trans. Inf. Theory.

[27]  Babak Hassibi,et al.  High-rate codes that are linear in space and time , 2002, IEEE Trans. Inf. Theory.

[28]  B. Sundar Rajan,et al.  Full-rank, full-rate STBCs from division algebras , 2002, Proceedings of the IEEE Information Theory Workshop.

[29]  Jean-Claude Belfiore,et al.  Quaternionic lattices for space-time coding , 2003, Proceedings 2003 IEEE Information Theory Workshop (Cat. No.03EX674).

[30]  Lizhong Zheng,et al.  Diversity and multiplexing: a fundamental tradeoff in multiple-antenna channels , 2003, IEEE Trans. Inf. Theory.

[31]  Mohamed Oussama Damen,et al.  Universal space-time coding , 2003, IEEE Trans. Inf. Theory.

[32]  Xue-Bin Liang,et al.  Orthogonal designs with maximal rates , 2003, IEEE Trans. Inf. Theory.

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

[34]  Gregory W. Wornell,et al.  Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks , 2003, IEEE Trans. Inf. Theory.

[35]  Frédérique E. Oggier,et al.  New algebraic constructions of rotated Z/sup n/-lattice constellations for the Rayleigh fading channel , 2004, IEEE Transactions on Information Theory.

[36]  Giuseppe Caire,et al.  Lattice coding and decoding achieve the optimal diversity-multiplexing tradeoff of MIMO channels , 2004, IEEE Transactions on Information Theory.

[37]  Emanuele Viterbo,et al.  The golden code: a 2 x 2 full-rate space-time code with non-vanishing determinants , 2004, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings..

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

[39]  J. Belfiore,et al.  The golden code: a 2×2 full-rate space-time code with nonvanishing determinants , 2004, IEEE Trans. Inf. Theory.

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

[41]  David Tse,et al.  Fundamentals of Wireless Communication , 2005 .

[42]  P. Vijay Kumar,et al.  A unified construction of space-time codes with optimal rate-diversity tradeoff , 2005, IEEE Transactions on Information Theory.

[43]  J. Belfiore,et al.  Optimal Space-Time Codes for the Amplify-and-Forward Cooperative Channel , 2005 .

[44]  B. Sundar Rajan,et al.  STBC-schemes with non-vanishing determinant for certain number of transmit antennas , 2005, ISIT.

[45]  P. Vijay Kumar,et al.  Perfect space-time codes with minimum and non-minimum delay for any number of antennas , 2005, 2005 International Conference on Wireless Networks, Communications and Mobile Computing.

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

[47]  Pranav Dayal,et al.  An optimal two transmit antenna space-time code and its stacked extensions , 2005, IEEE Transactions on Information Theory.

[48]  J. Belfiore,et al.  Partitionning the Golden Code : A framework to the design of Space-Time coded modulation , 2005 .

[49]  P. Vijay Kumar,et al.  Achieving the DMD tradeoff of the MIMO-ARQ channel , 2005, ISIT.

[50]  B. Sundar Rajan,et al.  STBC-schemes with nonvanishing determinant for certain number of transmit antennas , 2005, IEEE Transactions on Information Theory.

[51]  Philip Schniter,et al.  On the achievable diversity-multiplexing tradeoff in half-duplex cooperative channels , 2005, IEEE Transactions on Information Theory.

[52]  P. Vijay Kumar,et al.  Approximately universal optimality over several dynamic and non-dynamic cooperative diversity schemes for wireless networks , 2005, ArXiv.

[53]  P. Vijay Kumar,et al.  Explicit Space–Time Codes Achieving the Diversity–Multiplexing Gain Tradeoff , 2006, IEEE Transactions on Information Theory.

[54]  Camilla Hollanti,et al.  Optimal Matrix Lattices for MIMO Codes from Division Algebras , 2006, 2006 IEEE International Symposium on Information Theory.

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

[56]  Pramod Viswanath,et al.  Approximately universal codes over slow-fading channels , 2005, IEEE Transactions on Information Theory.

[57]  F. Oggier.  On the Optimality of the Golden Code , 2006, 2006 IEEE Information Theory Workshop - ITW '06 Chengdu.

[58]  Giuseppe Caire,et al.  Construction of Structured LaST Codes , 2006, 2006 IEEE International Symposium on Information Theory.

[59]  B. Sundar Rajan,et al.  Information-Lossless Space–Time Block Codes From Crossed-Product Algebras , 2006, IEEE Transactions on Information Theory.

[60]  Yindi Jing,et al.  Distributed Space-Time Coding in Wireless Relay Networks , 2006, IEEE Transactions on Wireless Communications.

[61]  B. Sundar Rajan,et al.  Distributed Space-Time Codes with Reduced Decoding Complexity , 2006, 2006 IEEE International Symposium on Information Theory.

[62]  Jean-Claude Belfiore,et al.  Optimal Space–Time Codes for the MIMO Amplify-and-Forward Cooperative Channel , 2005, IEEE Transactions on Information Theory.

[63]  Yi Hong,et al.  Golden Space–Time Trellis Coded Modulation , 2007, IEEE Transactions on Information Theory.

[64]  Frédérique E. Oggier,et al.  Asymptotically optimal cooperative wireless networks with reduced signaling complexity , 2007, IEEE Journal on Selected Areas in Communications.

[65]  Xiang-Gen Xia,et al.  Some Designs of Full Rate Space–Time Codes With Nonvanishing Determinant , 2007, IEEE Transactions on Information Theory.

[66]  Camilla Hollanti,et al.  Maximal Orders in the Design of Dense Space-Time Lattice Codes , 2008, IEEE Transactions on Information Theory.

[67]  Frédérique E. Oggier,et al.  An Algebraic Coding Scheme for Wireless Relay Networks With Multiple-Antenna Nodes , 2008, IEEE Transactions on Signal Processing.

[68]  Helmut Bölcskei,et al.  Space-Time Wireless Systems: From Array Processing to MIMO Communications , 2008 .

[69]  Pranav Dayal,et al.  Distributed QAM-Based Space-Time Block Codes for Efficient Cooperative Multiple-Access Communication , 2008, IEEE Transactions on Information Theory.