5 Perfect space--time block codes.

This chapter is devoted to the definition and construction of perfect Space-Time block codes. We will now assemble the three preceding chapters, using the algebraic techniques presented in Chapter 4, to build Space-Time block codes (STBC) satisfying the design criteria explained in Chapters 2 and 3. We illustrate the code constructions for small numbers of antennas, namely up to six antennas. These constructions have been originally presented in [21, 44].

5.1 Definition of Perfect Space Time Codes

Let us start by recalling what are the targeted code features. The two main design parameters for coherent Space Time codes are:

* Full diversity. The rank criterion tells for square STBCs that full diversity is obtained if the determinant of the difference of two distinct codewords is nonzero: d et([X.sub.i] - [X.sub.j]) [not equal to] 0, [X.sub.i] z[not equal to] [X.sub.j][member of] C.

* Minimum determinant. The coding gain is given by the minimum determinant min

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In order to further improve the performance of Space Time codes, we ask for two more properties, motivated in Chapters 2 and 3.

* Non-vanishing determinant. We say that a code has a non-vanishing determinant if, prior to SNR normalization, there is a lower bound on the minimum determinant that does not depend on the constellation size.

* Shaping. In order to optimize the energy efficiency of the codes, a shaping constraint on the signal constellation is introduced. The Q-QAM or Q-HEX to be sent are normalized according to the power at the transmitter. However, since we use linear STBCs, what is transmitted on each layer (see Definition 4.15) is a linear combination of information symbols, which may change the energy of the signal. Each layer can be written as Mv, where v is the vector containing the QAM or HEX information symbols, while M is a matrix that encodes the symbols into each layer. In order to get energy efficient codes, we ask the matrix M to be unitary. We will refer to this type of constellation shaping as cubic shaping, since a unitary matrix applied on a vector containing discrete values can be interpreted as generating points in a lattice. For example, if we use QAM symbols, we get the [Z.sup.n] (cubic) lattice.

There is another property on which we do insist though it has not been stated yet, since it follows from the shaping constraint. However, let us make it explicit here.

* Uniform average energy transmitted per antenna. The ith antenna of the system will transmit a signal [x.sub.ij] at time j. We ask that on average, the energy of each codeword entry is constant, in order to have a balanced repartition of the energy at the transmitter.

Having motivated the properties that an STBC should have to maximize its performance, we are now able to give the definition of a perfect Space-Time block code.

Definition 5.1 A square [n.sub.t] x [n.sub.t] STBC is called a perfect code if and only if:

* It is a full rate linear code using [n.sup.2.sub.t] information symbols either QAM or HEX.

* The minimum determinant of the infnite code is nonzero (so that in particular the rank criterion is satisfed).

* The energy required to send the linear combination of the information symbols on each layer is similar to the energy used for sending the symbols themselves (we do not increase the transmitted energy in encoding the information symbols).

* It induces uniform average transmitted energy per antenna in all T time slots, i.e., all the coded symbols in the code matrix have the same average energy.

In the rest of this chapter, we will give the construction of perfect codes for 2, ... , 6 antennas.

Remark 5.1 In Chapter 4, we have introduced as many algebraic techniques needed as possible, starting from no algebra background. This allows to explain almost all that is required to build the SpaceTime codes, apart the non-norm element [gamma], the element such that none of its powers are a norm, needed to construct a division cyclic algebra. The techniques involved are far beyond the scope of this tutorial. For the sake of completeness, Subsection 5.2.1 gives one example of such techniques, in the simplest case, when the algebra is of degree 2. For more general dimensions, the interested reader may refer for example to [44] if [gamma] is a root of unity, or more generally to [20, 36], and it has been shown in [21] that dividing a non-norm element by its conjugate still gives a non-norm element.

5.2 The Golden Code

The Golden code is a 2 x 2 perfect code. It has been found independently in [7, 15]. Its name, Golden code, comes from its algebraic construction [7], which involves the Golden number [1+[square root of (50)]/2

The Golden code is built using the cyclic algebra

A = (L = Q(i, [square root of(5)])/Q (i), [sigma] (i)

with [sigma]: [square root(5)] [right of] [square root(5)] We have that

[O.sub.L]={a + b [theta] | a,b [member of] Q(i)},

where [theta] = [1 + [square root(5)]/2. Before shaping, a codeword from this algebra is of the form:

[a + b[theta] c + d [theta] i(c + d[sigma]-([theta])) a + b[sigma]-(thata])] ,

with a, b, c, d [member of] Z |i|. By definition, the codebook obtained is linear and full rate (since it contains the four information symbols a, b, c, d). It is also fully diverse since i is not a norm (see next subsection for a proof), and thus, .A is a cyclic division algebra.

5.2.1 The Element [gamma] = i is Not a Norm in (Q(i [square root of (5)])

0We show here that the cyclic algebra .A defining the Golden code is a division algebra [7]. The proof is given for sake of completeness, but uses some tools that are beyond the scope of this work.

Proposition 5.1 Let K = (Q(i,,[square root of (5)]), then the element [gamma] = i is not a relative norm of any [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Proof. Let [Q.sub.5] denote the feld of 5-adic numbers, and[Z.sub.5] = {x [member of] [Q.sub.5]| [v.sub.5] (x) [greater than] 0} its valuation ring. The complex rationals Q(i) can be embedded in [Q.sub.5] by

i[right arrow] 2+5[Z.sub.5].

Let x = a + [square root of (5)] [member of] K with a, b [member of] Q(i), then we must show that

[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII.]

has no solution for a, b [member of] Q(i). We can lift this equation in the 5-adic feld [Q.sub.5]

[a.sup.2] - 5[b.sup.2] = 2 + 5x a, b [member of] (Q(i), X [member of] [Z.sub.5] (5.1) and show that it has no solution there. We take the valuations of both sides of (5.1)

[v.sub.5] ([a.sup.2] - 5[b.sup.2]) = [v.sub.5](2 + 5x)

to show that a and b must be in [Z.sub.5]. In fact, since x [member of] [Z.sub.5], [v.sub.5](2 + 5x) [greater than or equal to] min{[v.sub.5](2),[v.sub.5](x) + 1} = 0, and we have equality as both valuations are distinct. Now, [v.sub.5]([a.sup.2] - 5[b.sub.2]) = min{[2v.sub.5](a), [2v.sub.5] (b) + I} must be 0, hence [v.sub.5] (a) = 0 which implies a [member of] [Z.sub.5] and thus b [member of] [Z.sub.5].

We conclude by showing that

[a.sup.2] - 5[b.sup.2] = 2 + [x a, b, x [member of] [Z.sub.5]

has no solution. Reducing modulo 5[Z.sub.5] we End that 2 should be a square in the finite feld GF(5), which is a contradiction. o

5.2.2 The Lattice Z[[i].sup.2]

Let us see now how to add the shaping property on the codebook built on ,A. Equation (4.5) tells us that

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We thus look for an element a such that [|[N.sub.L/K] ([alpha])|.sup.2] = 5. In order to End such an element, we look at the factorization of 5 in [??]L:

5 = [(1 +i - i[theta]).sup.2][(1- i +i[theta]).sup.2]

We thus choose a = I + i - i9. Let us now check that we indeed get the right lattice. Its generator matrix is given by

M = ([alpha] [alpha][theta] [sigma]([alpha]) [sigma]([alpha][theta])).

A direct computation shows that MM[dagger] = 5[I.sub.2]. Thus 1/[square root of (5)] M is a unitary matrix, yielding the shaping property.

A codeword X belonging to the Golden code has thus, adding the shaping property, the form:

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where a, b, c, d are QAM symbols.

Recall that when a, b, c, d can take any value in Z[i], we say that we have an infinite code [C.sub.[infinity]] This terminology recalls the case where finite signal constellations are carved from infinite lattices.

5.2.3 The Minimum Determinant

Let us now compute the minimum determinant of the infinite code. Since [alpha][sigma]([alpha]) = 2 + i, we have

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By definition of a, b, c, d, we have that the non-trivial minimum of

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Thus the minimum determinant of the infinite code is bounded away from zero, as required.

Since an explicit computation of the determinant will not be possible in higher dimension, we show now another way of computing the minimum determinant. Since

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we have that

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Using Theorem 4.7, the determinant on the right-hand side is lower bounded by 1. Thus

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since [alpha] has been chosen such that [|[N.sub.L/K]([alpha])|.sup.2] = 5.

Note in the second row of the codeword X the factor i, which guarantees uniform average transmitted energy since [|i|.sup.2] = 1.