Division algebras for coherent Space-Time Coding have been introduced
by Sethuraman et al. (2003). These are non-commutative
algebras which have the nice property that one can easily associate
to an element of the algebra a matrix representation. Considering
cyclic division algebras thus naturally yields a family of invertible
matrices, or in other words, a linear code that fullfills the rank
criterion.
Introduction of Perfect Codes
We further exploit the algebraic structures of cyclic algebras to
build Space-Time Block codes that satisfy the following properties:
they have full rate, full diversity, non-vanishing constant
minimum determinant for increasing spectral efficiency, uniform average
transmitted energy per antenna and good shaping.
We call these codes perfect space-time block codes .
We give algebraic constructions of perfect STBCs for 2, 3, 4 and 6 antennas.
F. E. Oggier, G. Rekaya,
J.-C. Belfiore and E. Viterbo.
"Perfect Space-Time Block Codes",
Allerton 2004, invited paper.
and
F. E. Oggier, G. Rekaya,
J.-C. Belfiore and E. Viterbo.
"Perfect Space-Time Block Codes"[.ps],
IEEE Trans. on Information Theory, vol. 52, n.9, September
2006.
On a better understanding of Perfect Codes
Perfect codes were presented for 2, 3, 4 and 6 antennas.
We show that they actually only exist in dimensions 2, 3, 4 and 6.
G. Berhuy, F. Oggier.
"On the Existence of Perfect Space-Time Codes "
[.ps],
submitted, June 2006.
We now address the question of the optimality of the known perfect
codes. Are there other perfect codes that could perform better than
the known ones? In order to answer this question, we start by
studying the simplest case where we have only 2 antennas.
F. Oggier.
"On the Optimality of the Golden Code ",
ITW 2006, Chengdu .
Investigations for 4 antennas codes are presented in
F. Oggier, G. Berhuy.
"On Improving 4x4 Space-Time Codes ",
Asilomar conference, 2006.