# Optimal Space-Time Block Code Designs Based on Irreducible Polynomials of Degree Two

The main of this paper is to prove that in terms of normalized density, a space-time block code based on an irreducible quadratic polynomial over the Eisenstein integers is an optimal space-time block code compared with any quadratic space-time block code over the ring of integers of imaginary quadratic fields. In addition we find the optimal design of space-time block codes based on an irreducible quadratic polynomial over some rings of imaginary quadratic fields.

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## i Introduction

The use of several antennas at both the transmitter and receiver ends of a wireless channel increases the data rate. Good transmission over such channels can only be achieved through Space-Time coding. We consider the coherent case where the receiver has perfect knowledge of all the channel coefficients.

There have been several kinds of space-time block code designs, for example, orthogonal space-time code designs [1, 2, 3, 4], unitary space-time designs [5, 6, 7], lattice based diagonal space-time code designs using algebraic number theory [8, 9, 10] and lattice based space-time block code designs from algebraic number fields, that have attracted much attention [11, 12, 13, 14, 15, 16].

A lattice is a discrete finitely generated free abelian subgroup

of a real or complex finite dimensional vector space, called the ambient space. In the space-time setting, a natural ambient space is the space

of complex matrices. Due to the symmetric situation, we only consider full-rank lattices that have a basis consisting of matrices that are linearly independent over We can form a matrix having rows consisting of the real and imaginary parts of all the basis elements.

From the pairwise error probability (PEP) point of view

[17] the performance of a space-time code is dependent on two parameters: full diversity and minimum determinant (or coding gain or diversity product or determinant distance, or minimum product distance). Since we are considering lattices with full-diversity, the challenge is to maximize the second parameter.

By [18] the normalized minimum determinant of is defined as

 δ(Λ)=detmin(Λ)|det(G)|12n

where is a generator matrix of and . We can as well use the normalized density of the lattice

 ρ(Λ)=detmin(Λ)2n|det(G)|.

Since we can conclude that maximizing the normalized minimum determinant is equivalent to maximizing the normalized density of the corresponding lattice, i.e., minimizing the absolute value of the generator matrix

In [19] and [20], a space-time code design of full diversity is proposed by using cyclotomic extensions and in [21] is proposed space-time block codes designs based on quadratic extension. In this paper we present a structure of space-time block code with full diversity similar to the one defined in [21], over the ring of integers of a imaginary quadratic field for two-transmitter antennas. Here, contrary to what is presented in [21] the informations symbols are not only in or .

Our main contribution of this paper is to prove that the space-time block code design over proposed in [21] is an optimal space-time block code compared to any space-time block code over where is any imaginary quadratic field. We also find optimal space-time block codes designs over the ring of integers of with and The optimality is the sense that the normalized density (or normalized minimum determinant) is maximal when the mean transmission signal power is fixed. When we will see that the optimal code obtained has the same normalized minimum determinant of the Silver code [22].

The motivation to consider information symbols in another quadratic imaginary field instead and , which are considered in most communication problems, is that there are some recently researches that consider as base field a general ring of algebraic integer, for example [23] and [24]. Moreover, the lattice reduction has also been generalized to a ring of imaginary quadratic integers [25] and only the rings from where takes the values and can be used to define Lovász condition [26].

This paper is organized as follows. In Section II, we define real and complex lattices and briefly introduce the normalized density to composed lattices. In Section III, we propose the space-time block code designs scheme and some properties. In Section IV, we give a criterion to compare two space-time block codes and some results that will be useful in the proofs of the next section. In Section V, we dedicate to optimal space-time block codes designs over the ring of integers of imaginary quadratic fields. We present optimal codes over the ring of integers of with a positive squarefree integer. We also prove that the space-time block code over is an optimal space-time block code compared to any space-time block code over the ring of integers of any imaginary quadratic field.

## ii Real and Complex Lattices

In this section, we first define real and complex lattices, and next we describe how a complex lattice can be represented by a real lattice.

###### Definition ii.1

An -dimensional real lattice is a subset in

 Λn(M) = {(x1,⋯,xn)t=M(z1,⋯,zn)t∣ zk∈Z for 1≤k≤n},

where stands for the transpose and is an real matrix of full rank and called the generator matrix of the real lattice and .

###### Definition ii.2

An -dimensional complex lattice over a 2-dimensional real lattice is a subset of :

 Γn(G) = {(y1,⋯,yn)t=G(x1,⋯,xn)t∣ xk∈Λ2(M) for 1≤k≤n},

where stands for the transpose and is an complex matrix of full rank and called the generator matrix of the complex lattice and is a generator matrix of the lattice

Let be an complex matrix

 G=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝g11g12…g1ng21g22…g2n⋮⋮⋱⋮gn1gn2…gnn⎞⎟ ⎟ ⎟ ⎟ ⎟⎠, (1)

with and be a real matrix, which is from the real and imaginary parts of as follows:

 G=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝Re(g11)−Im(g11)…Re(g1n)−Im(g1n)Im(g11)Re(g11)…Im(g1n)Re(g1n)⋮⋮⋱⋮⋮Re(gn1)−Im(gn1)…Re(gnn)−Im(gnn)Im(gn1)Re(gn1)…Im(gnn)Re(gnn)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ (2)

where and means the real and imaginary parts of , respectively.

If over then

 (y1,⋯,yn)t=G(x1,⋯,xn)t.

Rewritten with its real part and imaginary part it follows that for Then can be rewritten as

 ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝Re(y1)Im(y1)⋮Re(yn)Im(yn)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ = G⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝Re(x1)Im(x1)⋮Re(xn)Im(xn)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ = Gdiag(M,…,M)⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝z11z12⋮zn1zn2⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,

where with

 (xk1xk2)=M(zk1zk2).

Set . We need to show that , i.e, has full rank and therefore is a real generator matrix of a -dimensional real lattice. Since is the real generator matrix of a two-dimensional real lattice We can conclude that by the following proposition.

###### Proposition 1

[19] Let be an complex matrix defined in (1) and be the real matrix defined in (2). Then .

Proposition 1 tell us that an -dimensional complex lattice over can be equivalently represented as a -dimensional real lattice Furthermore, the determinant of their generator matrices have the following relationship:

 |det(GM)| = |det(G)||det(M)|n = det(G)|2|det(M)|n.

Now we present the definition of composed complex lattice given in [20]. It is useful for -layer space-time codes.

###### Definition ii.3

An -dimensional composed complex lattice over consists of all points where each segment of length belongs to complex lattice over i.e.

 ⎛⎜⎝y(l−1)n+1⋯yln⎞⎟⎠=Gl⎛⎜⎝x(l−1)n+1⋯xln⎞⎟⎠,

Similarly to a complex lattice, an -dimensional composed complex lattice can also be represented by a -dimensional real lattice of generator matrix and the following determinant relationship holds:

 |det(GM1,⋯,Mn)|=L∏l=1|det(Gl)|2⋅|det(Ml)|n. (3)

According to the theory presented here, the normalized density can be written as follows:

 ρ(Λ)=detmin(GM1,⋯,Mn)2nL∏l=1|det(Gl)|2⋅|det(Ml)|n. (4)

In this paper we focus when and

## iii 2×2 Space-Time Block Codes

Let be a field, is an irreducible polynomial over , with , the ring of algebraic integers of . The polynomial has two roots:

Let , so and is a basis of over . Let and , be the two embeddings of to such that for any and .

###### Definition iii.1

A space-time block code based on an irreducible quadratic polynomial over , with roots defined by

 C = {X=(a+bα1c+dα1γ(c+dα2)a+bα2)∣a,b,c,d∈OF}

where the ring of integers of is a complex number chosen so that , .

This code can be defined in terms of the quaternion division algebra , with , . We refer readers to [27], [28] for a detailed exposition of the theory of simple algebras, cyclic algebras, their matrix representations and their use in space-time coding.

###### Definition iii.2

Let be an space-time block code, we define the minimum determinant of the code by

 detmin(C(F,α1,α2,γ))=minX∈C,X≠0|det(X)|.

The next lemma tells us that when we consider quadratic imaginary fields, the minimum determinant of a space-time block codes .

###### Lemma 1

If , with a positive squarefree integer then any has

###### Proof:

If then

 det(X)=(a+α1b)(a+α2b)−γ(c+α1d)(c+α2d),

and thus, taking and , we have

.

Since is not the algebraic norm of on it follows that if either or then i.e., , if Also, as , it follows that . Thus we conclude that

Now, taking such that , it follows that . Therefore, .

In order to design a space-time block code with a large normalized density and nonvanishing determinant, we consider with a positive squarefree integer.

Lattice theory has a close relation to number theory. We refer readers to [29, 30] for some concepts of lattices and rings that will be used throughout this paper.

If then with integral basis Otherwise, if then with integral basis . Then the corresponding generator matrix of 2-dimensional lattice is

 M=(100√d)orM=⎛⎝1120√d2⎞⎠,

respectively.

A two-layer space-time block code is a lattice code over and by Definition ii.3:

 (a+bα1a+bα2)=(1α11α2)(ab)and
 (c+dα1γ(c+dα2))=(1α1γγα2)(cd),

with

 G1 = (1α11α2)and G2 = (100γ)(1α11α2)=(1α1γγα2).

By (3), if is a generator matrix of the 8-dimensional real lattice that we denote by then

 |det(GM,M)| = 2∏l=1|det(Gl)|2⋅|det(M)|2 = |γ|2|α1−α2|4|det(M)|4,

and by (4)

 ρ(Λ−d)=1|γ|2|α1−α2|4|det(M)|4. (5)
###### Remark 1

We denote by Note that the smaller is, the better normalized density of is.

## iv Lattice Design Criterion

With the argument of Section III we present a criterion [20] to compare two space-time block codes.

###### Definition iv.1

(Determinant Criteria) Let and be two space-time block codes with

 detmin(C(F1,α1,α2,γ1) = detmin(C(F2,β1,β2,γ2)).

We say that is better than if

 cdet(C(F1,α1,α2,γ1))
 |γ1||α1−α2|2|det(M1)|2<|γ2||β1−β2|2|det(M2)|2,

where a generator matrix of and a generator matrix of .

Note that codes from same algebras may have different as we can see in the Example 1.

###### Example 1

Let and be two space-time block codes and a generator matrix of . Note that and come from the algebra and

 cdet(C1) = |γ||√−3−(−√−3)|2|det(M)|2 = |γ||2√−3|2|det(M)2|=12|γ||det(M)|2

while

 cdet(C2) = |γ|∣∣∣1+√−32−1−√−32∣∣∣2|det(M)|2 = |γ||√−3|2|det(M)|2=3|γ||det(M)|2.

Soon, .

###### Definition iv.2

Let be a set of space-time block codes, where for all . We say that is an optimal space-time block codes in , if

,

for all .

Now we stablish a strategy to minimize

In [21] it was shown that if then

 cdet(C(F,α1+p0,α2+p0,γ))=cdet(C(F,α1,α2,γ)),

i.e.,

 |det(M)|2|α1−α2|2|γ| = |det(M)|2|(α1+p0)−(α2+p0)|2|γ|. (6)

Note that the irreducible polynomial of over is , where and the irreducible polynomial of over is

 (x−(α1+p0))(x−(α2+p0)) = x2+(p−2p0)x+(q−p0p+p20) = x2+p′x+q′,

where and

We can see by (6) that minimize is the same that minimize

One strategy to minimize (triangular inequality) is first minimize

###### Proposition 2

Let , with a positive squarefree integer and its ring of integer. If then there exists such that

 |p−2p0|≤{|1+√−d2|ifd≡1(mod4)|1+√−d|ifd≡2,3(mod4).
###### Proof:

If , then with . Taking we need to check the following cases:

• if are even, then (take and ).

• if is even and

is odd, then

(take and ).

• if is odd and is even, then (take and ).

• if are odd, then (take and ).

So, given , there exists such that .

Otherwise, if , then , with . Analogous, taking and checking the same cases as above we conclude that there exists such that .

## v Optimal 2×2 Space-Time Block Codes designs

The Golden Code is a perfect space-time block code for two-transmitter antennas with minimum determinant 1 [31]. We can to verify by (5) that

In [21], the authors proved that the space-time block code

In addition, they also have proved that is an optimal space-time block code over with

As we can see, in terms of normalized density analysis the space-time block code is better than

According to these considerations, there is a question unanswered. Is an optimal space-time block code when compared with any space-time block code , with a positive squarefree integer? In order to give an answer to this question, we will see that is enough to find optimal space-time block codes over with

Suppose that there exists a space-time block code better than , i.e., there exists a space-time block code , with a positive squarefree integer, such that in other words, by Definition (iv.1)

,

where is a generator matrix of the lattice obtained via

Let’s to analyse when .

In the Section III we have seen that,

• if , then . Soon are the only satisfying .

• if then . Soon, are the only satisfying .

So, we only need to consider .

Therefore, from now on we will find the optimal space-time block codes over with in order to compare with . To do it we need to find convenient and , where is not algebraic norm of over . The corollaries bellow will be useful in this search.

###### Corollary 1

Let . If , then is not algebraic norm of over .

###### Proof:

Our goal is to show that the equation has no solution in the field of 3-adic numbers , and thus, no solution for all

Let be the valuation ring of , where denotes the 3-adic valuation of . Since , it follows that , and then . Taking obviously . Note that and By Hensel’s Lemma [32], it follows that such that , i.e., . It means that . Thus we can view the field as a subfield of . Analogously, the field can be viewed as a subfield of .

Furthermore, the norm map is then a restriction of the norm maps , where .

Thus, in order to prove our claim, it is enough to show that is not in the image of the map

Set and . Suppose on the contrary, that there are -adic numbers and such that . We first show that and are in i.e., and . Assume that at least one of them has a negative exponent 3-adic valuation, i.e.,

• if , then and . Since the non-archimedean property (see [32]) implies that .

• if , then . Since and , it follows that

Thus . Again, the non-archimedean property implies that .

In both cases,