Cyclic Division Algebras: A Tool
for Space-Time Coding
Foundations and Trends® in
Communications and Information Theory
Volume 4 Issue 1
DOI: 10.1561/0100000016
Cyclic Division Algebras: A Tool
for Space-Time Coding
Frédérique Oggier
Department of Electrical Engineering, California Institute of Technology, Pasadena CA-91125, frederique@systems.caltech.edu
Jean-Claude Belfiore
École Nationale Supérieure des Télécommunications, 46 rue Barrault, 75013 Paris, France, belfiore@enst.fr
Emanuele Viterbo
Dipartimento di Elettronica, Informatica e Sistemistica, Università della Calabria, Arcavacata di Rende, Italy, viterbo@deis.unical.it
SUGGESTED CITATION:
Frédérique Oggier,
Jean-Claude Belfiore and
Emanuele Viterbo (2007)
"Cyclic Division Algebras: A Tool for Space-Time Coding",
Foundations and Trends® in
Communications and Information Theory: Vol. 4: No 1, pp 1-95.
http:/dx.doi.org/10.1561/0100000016
Abstract
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.
Keywords:
| Cyclic algebras; division algebras; full diversity; golden code; non-vanishing determinant; perfect space-time codes; space-time
coding.
|