1 Introduction.

Algebraic coding has played an important role since the early age of coding theory. Error correcting codes for the binary symmetric channel were designed using fnite felds and codes for the additive white Gaussian channel were designed using Euclidean lattices.

The introduction of wireless communication required new coding techniques to combat the effect of fading channels. Modulation schemes based on algebraic number theory and the theory of algebraic lattices were proposed for single antenna Rayleigh fading channels thanks to their intrinsic modulation diversity.

New advances in wireless communications led to consider systems with multiple antennas at both the transmitter and receiver ends, in order to increase the data rates. The coding problem became more complex and the code design criteria for such scenarios showed that the challenge was to construct fully-diverse codes, i.e., sets of matrices such that the difference of any two distinct matrices is full rank. This required new tools, and from the algebraic side, division algebras quickly became prominent.

1.1 Division Algebra Based Codes

Division algebras are non-commutative algebras that naturally yield families of fully-diverse codes, thus enabling to design high rate, highly reliable Space Time codes, which are characterized by many optimal features, deeply relying on the algebraic structures of the underlying algebra.

The idea of using division algebras was fist introduced in [51], where so-called Brauer algebras were presented, and in [50], where it was shown that the acclaimed Alamouti code [1] can actually be built from a simple example of division algebras, namely the Hamilton quaternions. Quaternion algebras were more generally used in [6], where the notion of non-vanishing determinant was introduced.

Different code constructions appeared then in [52], based on feld extensions and cyclic algebras. In [7, 44] and then in [21], perfect codes were presented as division algebra codes which furthermore satisfy a shaping property and have a non-vanishing determinant. In [53], information lossless codes from crossed product algebras, a new family of division algebras, are presented. In [31], codes from maximal orders of division algebras are investigated. In [39] some non-cubic shaping, non-vanishing determinant codes are proposed based on cyclic division algebras.

In parallel, in [7, 15, 33, 63], the fist 2 x 2 codes achieving the diversity-multiplexing gain trade-off of Zheng and Tse [64] were found. It was furthermore shown [63] that a necessary condition to achieve the trade-off for a 2 x 2 code is actually to have a non-vanishing determinant (though not stated with this terminology). In [7], it was shown that the algebraic structure of cyclic division algebras was the key for constructing 2 x 2 non-vanishing determinant codes. In [20], it was shown more generally that division algebra codes are a class of codes that achieve the trade-off, thanks to the non-vanishing determinant. All the notions mentioned in the above short history of division algebra based codes will be explained in this work. We will focus on cyclic division algebras, a particular family of division algebras. These will be built over number felds, with base feld Q(i) or Q(j), with [i.sub.2] = -1 and [j.sub.3] = 1, which are suitable to describe QAM or HEX constellations. The notion of constellation shaping will be explained, thanks to an underlying lattice structure. We will show how this is related to the information lossless property. Furthermore, having Q(i) or Q(j) as a base feld will allow us to get the so-called non-vanishing determinant property, which will be shown to be a sufficient condition to reach the diversity-multiplexing trade-off.

1.2 Organization

This paper is organized as follows. Chapter 2 details the channel model considered. It recalls the two main code design criteria derived from the pairwise probability of error, namely: the rank criterion and the determinant criterion. It then discusses the modulations used, QAM and HEX constellations. Decoding is furthermore considered, which also enlightens the importance of the constellation shaping in the code performance.

In Chapter 3, performance of the code is considered from an information theoretic perspective. The goal is to explain the role of the diversity-multiplexing gain trade-off, as well as the information lossless property, which guarantees that a coded system will have the same capacity as an uncoded one assuming QAM input symbols.

Chapters 2 and 3 give a characterization of the properties a SpaceTime code should achieve to be efficient. Codes based on cyclic division algebras have been shown to fulfil those properties. Their construction is however involved, and it is the goal of Chapter 4 to introduce the algebra background necessary to construct those codes. No algebra background is required to read this chapter. Division algebras are introduced, as well as number fields. We also defne concepts such as algebraic norm and algebraic trace, that will be important for the code construction.

Once the algebra background is set, Chapter 5 explains the construction of the Golden code and some other Perfect Space Time block codes for small number of antennas, namely up to six.

The last chapter briefly presents future applications of those techniques, toward coding for wireless networks, and trellis/block coded modulations.