Arithmetic fuchsian groups and space time block codes

Arithmetic fuchsian groups and space time block codes^*

E.D. Carvalho^I; A.A. Andrade^II; R. Palazzo Jr.^III; J. Vieira Filho^IV

^IDepartment of Mathematics, FEIS-UNESP, Ilha Solteira, SP, Brazil. E-mail: [e-mail]

^IIDepartment of Mathematics, IBILCE-UNESP, São José do Rio Preto, SP, Brazil. E-mail: [e-mail]

^IIIDepartment of Telematics, FEEC-UNICAMP, Campinas, SP, Brazil. E-mail: [e-mail]

^IVDepartment of Electrical Engineering, FEIS-UNESP, Ilha Solteira, SP, Brazil. E-mail: [e-mail]

ABSTRACT

In the context of space time block codes (STBCs) the theory of arithmetic Fuchsian groups is presented. Additionally, in this work we present a new class of STBCs based on arithmetic Fuchsian groups. This new class of codes satisfies the property full-diversity, linear dispersion and full-rate.

Mathematical subject classification: 18B35, 94A15, 20H10.

Key words: Arithmetic Fuchsian group, division algebra, quaternion order and space time codes.

1 Introduction

Within the context of digital communication systems, multiple-input-multiple-output (MIMO) wireless links, that is, systems that using multiple antennas at the transmitter and the receiver, has emerged. It is used to combat fading from diversity technique, i.e, different replicas of the same information symbol may be transmitted over independent channels and are the available at the receiver side. Notices, in this process, the signal is lost only when all its copies are lost. From then on, both the data rate and the performance are improved by many orders of magnitude with no extra cost of spectrum. This is also the main reason that the MIMO attracts and motivated much researcher on signal processing. The key feature of a multiple-antenna system is it ability in exploit the turn multiple-path propagation, which is traditionally regarded as a disadvantage to wireless communications, into benefit to the users and result in a diversity again.

In this paper, the diversity can also be obtained at the transmitter by spacing the transmit antennas sufficiently and introducing a code (called Space Time Block Codes-STBCs) between the transmitted symbols over M transmit antennas (space) and T symbol periods (time).

Alamouti code [1] was the first practical SBTC proposed in the literature, based with provides full transmit diversity for systems which works with two antennas. The Alamouti code is given by codeword matrices, such that, thematrix operations representing multiplication in the Hamilton quaternions. It is also one of the most successful STBCs because of its simple structure and it is great performance and simple decoding. Tarokh et al. [2] proved that STBC achieves a pairwise error probability (PEP) that is inversely proportional to SNR^MN, where SNR denotes the signal-to-noise ratio, M the number oftransmit antennas and N the number of receive antennas. In this same work, Tarokh showed that the main code design criterion for the STBCs is the rank criterion, i.e, the rank of the difference of two distinct codeword matrices hasto be maximal. If this property is satisfied the STBCs is called fully diverse.

Full rate (i.e, the number of transmitted signals corresponds to the number of information symbols to be sent) and full diversity codes for the 2×2 MIMO systems, were first constructed by Damen et al. [3], using algebraic number theory. Hassibi in [4] introduced linear dispersion space time block codes (LD-STBCs), i.e, if two codeword matrices X₁,X₂ belong to the code then X₁± X₂∈ and X₁X₂∈ . The idea of LD-STBCs is to spread the information symbols over space and time. Oggier et al. [5] reformulate the rank criterion for LD-STBCs, when the codeword matrices are square, saying the STBC is fully diverse if

| det (X_i - X_j) |²≠ 0, for all X_i ≠ X_j ∈ .

By linearity, it follows that |det(X)|²≠ 0 for all nonzero codeword X ∈ .

Division algebras have been proposed [5], [6], [7], [8] as a new tool for constructing STBCs, since they are non-commutative algebras that naturally yield linear fully diverse codes. However, for determining precisely these algebras are division algebras can be a nontrivial problem. Katok [9] characterized some particular classes of 2×2 matrices space M₂() isomorphic to Hamilton quaternion (division algebra). The construction of this matrix space are based on the existence of the arithmetic Fuchsian groups, i.e, discrete subgroups of PSL(2,) obtained by some arithmetic construction in the hyperbolic plane.

From then on, we proposed one arithmetic construction of the arithmeticFuchsian Groups Γ from the self-dual tessellations {4g,4g}, with g > 2, where g denotes the genus of the compact surface, it has the hyperbolic plane as universal covering. This systematic procedure anable us to construction new class of 2×2 STBCs. Additionally, we will show this new class STBCs satisfies the properties of linear dispersion, full rate and full diversity codes. In fact, according to our best knowledge the theory of arithmetic Fuchsian groups required for giving this notion has never been considered before in this area.

This work is organized as follows. In Section 2, we present the concepts of Fuchsian group and quaternion order. In Section 3, we determine the Fuchsian groups Γ_4g from hyperbolic tessellation. In section 4, we shown the Fuchsian groups Γ_4g are derived from quaternion algebras. In Section 5, we present a new class of codes via arithmetic Fuchsian groups. Some conclusions are presented in Section 6.

2 Arithmetic Fuchsian Groups and Quaternion Order

Let F be a totally real number field of degree n > 2 over and _F be the ring of algebraic integers of F. Let { σ₁,..., σ_n} the n different embedding F into .

The quaternion algebra = (t,s)_F is defined as the 4-dimensional vector space over F, with a basis {1,i,j,ij}, satisfying the conditions i² = t, j² = s, ij = -ji and (ij)² = -ts, where t,s ∈ = F-{0}. The quaternion algebra = (t,s)_F can be embedded in M(2,F()), i.e, there is a linear map such that

where s = r₁r₂. There exists -isomorphism ρ_i,

with 2 < i < n, where A is non-ramified in ρ₁ and ramified in the remaining ρ_i's.

The element = x₀ - x₁i - x₂j - x₃ij ∈ is called conjugate of the element x = x₀ + x₁i + x₂j + x₃ij ∈ . The reduced trace and the reduced norm of an element x ∈ are defined as Trd(x) = x + and Nrd(x) = x, respectively. Thus the norm Nrd(x) is a quadratic form over F given by Nrd(x) = . An order in over F is a subring of containing _F, which is finitely generated as an _F-module such that F = .

We consider the upper-half plane

² = {z ∈ :Im(z) > 0} equipped with a Riemannian metric

With this metric,

² is a model of the hyperbolic plane.

Let G be the group formed of all Möubius transformations, T:→,given by T(z) = , where a, b, c, d ∈ and ad - bc = 1. To this transformation the following pair of matrices are associated

Hence,

where SL(2,) is the group of real matrices with determinant equal to 1 and I₂ denotes the 2×2 identity matrix. A Fuchsian group Γ is a discrete subgroup of PSL(2, ), that is, Γ consists of isometries that preserving orientation and acting on ² by homeomorphisms [9] and [12].

For each order in , consider ¹ as the set ¹ = {x ∈ : Nrd(x) = 1}. Note that ¹ is a multiplicative group. We observe that a Fuchsian group maybe obtained by the isomorphism ρ₁ given by the Equation (1) applied in ¹. In fact, if x ∈ ¹, then Nrd(x) = det( ρ₁(x)) = 1. From this, it follows that ρ₁(¹) is a subgroup of SL(2,). Therefore, the derived group from the quaternion algebra = (t, s)_F whose order is , denoted by Γ(,), is given by

The group Γ(,) is a Fuchsian group [10]. If Γ is a subgroup of Γ(,) with finite index, then Γ is a Fuchsian group derived from a quaternion algebra , also called Arithmetic Fuchsian Group. The Möubius transformation is given by f(z) = , whose matrix associated is given by

maps

² to the Poincaré disc ² (another Euclidean model for the hyperbolic plane). The action of PSL(2,) on ² transforms to an action of PSU(1,1) on ², since

The group PSU(1,1) consists of orientations preserving isometries T : ² → ², acting on ² by homomorphisms. The isometries T are given by T(z) = , where a, c ∈ and |a|² - |c|² = 1. For each of these transformations the following pair of matrices are associated

Theorem 2.1. [9 ] Let Γ be a Fuchsian group. Then, Γ is derived from a quaternion algebra over a totally real number field F if and only if Γ satisfies the following conditions:

(1) If F = (tr(T)), where T ∈ Γ, then F is a number field of finite degree and tr( Γ) is in _F, the ring of algebraic integers of F.

(2) If σ is an embedding of F in different from the identity, then σ(tr( Γ)) is bounded in .

3 Fuchsian Groups from Fundamental Polygon

_4g

Let S_g be the fundamental group of a compact closed surface of genus g. The presentation of Fuchsian group is given by

with [a_i, b_i] = a_ib_ia_i^-1b_i^-1. Let us consider a regular polygon _g with 4g edges and angles between adjacent edges equal to 2 π/4g. Hence, the corresponding fundamental region of self-dual tessellations {4g, 4g} of the hyperbolic plane. Considering the Poincaré model ², and assuming that 0 ∈ ² is the barycenter of _g. Now, we determine the generators of the Fuchsian group , where edge-pairing generators of a regular polygon _g with 4g edges (fundamental region of Γ_4g) are hyperbolic transformations, T_i (whose trace tr(T_i) associated to T_i is such that tr(T_i) > 2), where g is the genus of compact surface ²/, and whose hyperbolic area is µ(²/) = 4 π(g - 1). If , where i = 1,...,g, are the hyperbolic transformations determined by matrices A_i,B_i, such that and , then the group Γ_4g generated by , where i = 1,...,g, is canonically isomorphic to S_4g [9]. We can find an explicit formula for the matrices A_i and B_i that generates the transformations and , for i = 1,...,g. Following exactly the same procedures done by Katok [9] for the case g = 2 we have the following result.

Proposition 3.1. The elements a, c of the matrix

are given by

and the remaining generator matrices are given by A_i = C⁴ⁱ A₁C^-4i and B_i = C⁴ⁱ⁺¹ A₁C⁴ⁱ⁺¹, for all i = 1,...,g, where C is the rotation matrix given by

Example 3.1. If g = 2, then the matrix A₁ associated to generator transformation ∈ Γ₈ is given by

and the other matrices A₂, B₁ and B₂ are given by conjugation.

Example 3.2. If g = 3, then the matrix A₁ associated to generator transformation ∈ Γ₁₂ is given by

where q = and the other matrices A₂, A₃, B₁, B₂ and B₃ are given by conjugation.

Now, taking the corresponding real matrices of PSL(2,) by isometries f: ²→ ² given by f(z) = , we obtain the following isomorphism

where P is the invertible matrix given by the Equation (2). Then by consequence of the Equation (4), it follows that there is an equivalence between the matrix spaces and . Thus, f() = P^-1

P is a subgroup of PSL(2,), where g = 2,3, and the generator matrices are given by P^-1A_iP = D_i and P^-1B_iP = E_i. In particular, if A₁∈ f( Γ₈) then

and if A₁∈ f( Γ₁₂) then

where p = .

Remark 3.1. If we compute all the generator matrices M = D_i or M = E_i, for i = 1,...,g, of f( Γ_4g) it is easy to check that

(1) if g = 2, then

where, a, b, c, d ∈ [] and , and

(2) if g = 3, then

where, a, b, c, d ∈ [] and .

Also, it is easy to show the product of these matrices are of the type M and belong to the group f( Γ).

4 Fuchsian Groups derived from Quaternion Algebras

In this section, we present a construction that is similar to ones given by Katok [9]. Let F = ( θ) field extension of degree 2 and σ₂ : F → be the non-identity homomorphism belong to Galois group Gal(F/) given by σ₂( θ) = - θ. Thus, ψ₂ : K → , defined by ψ₂() = i is an isomorphism, where K = F(). We consider now a quaternion algebra [ Γ] over F = ( θ) given by

Thus,

Therefore,

where

Ψ : [ Γ] → M(2,),

is an embedding given by

Consequently, ⊗ ~ , [9].

Lemma 4.1. If~and¹ = {x ∈ : Nrd(x) = 1} then Trd(¹) is bound in .

Proof. If x = x₀ + x₁i + x₂j + x₃ij ∈ ¹, where i² = j² = (ij)² = -1, and Nrd(x) = = 1, then |x₀| < 1, and hence Trd(x) = 2x₀∈ [-2,2]. Since the converse statement is obviously true it follows that Trd(x) = 2x₀∈ [-2,2].

Theorem 4.1.If g = 2, then the group f ( Γ₈) is derived from the quaternion algebra over the totally real number field().

Proof. Following the same procedures done by Katok [9] for the case g = 2 , we first show that the conditions (1) and (2) of Theorem 2.1 are satisfied by the elements of f( Γ₈). From Remark 3.1, the elements of f( Γ₈) are given by

where x₀, x₁, x₃, x₄∈ [] and tr(M) = x₀ = a₁+ a₂∈ []. In this way, it follows that (tr(f( Γ₈))) = (a₁ + a₂) = (), and tr(M) ∈ []. Since () is a totally real quadratic extension of , it follows that the condition (1) of Theorem 2.1 is satisfied. Let σ₂:() → () be the non-identity embedding defined by σ₂() = -. From Remark 3.1, it follows that the generators of Γ₈ and therefore all elements of f( Γ₈) are embedded into M₂(K), where K = ()(). Thus, σ₂ extends to an isomorphism ψ₂ : K → , where

Following exactly the same procedures done by Katok [9], the elements of f( Γ₈) are mapped into matrices in M₂() of type

M = , with a, b ∈ ψ₂(K),

where we denote this set by ⊕≈ . Now, if T ∈ f( Γ), then tr(T) = a + and by Lemma 4.1, it follows that ψ₂(a)+ ψ₂() ∈ [-2,2]. However, a + ∈ K. In this way, ψ₂(a) + ψ₂() = ψ₂(a + ) = σ₂(a + ), that is, σ₂(a + ) ∈ [-2,2]. Therefore σ₂(tr(f( Γ))) is bound in .

Similarly to the previous case, we have the next theorems.

Theorem 4.2. If g = 3, then the group f( Γ₁₂) is derived from quaternion algebra over the totally real number field ().

Theorem 4.3. If f( Γ) is a Fuchsian group whose generators are matrices in PSL(2,) of the type

where a, b, c, d ∈ _F, with ∉ , then the matrices belong to f( Γ) are identified by the elements of the quaternion order ~ of the quaternion algebra ~ (t,-1)_F.

The product of two matrices in Theorem 4.3 assumes the same form M. Furthermore, all the elements of f( Γ) may be obtained directly as the product of the generator matrices and this fact guarantee that all the elements of f( Γ) assume the same form M.

Example 4.1. Applying Theorem 4.3 and Remark 3.1 to the matrices belonging to f( Γ₈), it follows that these matrices are identified by the elements of quaternion order ₈ = .

Example 4.2. Applying Theorem 4.3 and Remark 3.1 to the matrices belonging to f( Γ₁₂), it follows that these matrices are identified by the elements of quaternion order ₁₂ = .

We will denoted by

₈(2) and ₁₂(2) the matrix spaces associated toquaternion order ₈ and ₁₂, respectively. As consequence of Equation (4)and by Theorems 4.1 and 4.2, it follows that the matrix spaces ₈(2) and ₁₂(2) are division algebras. Similarly, by Equation (4), it follows that (2) ~ P^-1

_4g(2) for g = 2,3 and P is the invertible matrix given by the Equation (2).

5 Space-Time Codes From Division Algebra

In this section, we will characterize algebraically the matrix spaces(2). First, we notice each element of matrix space _4g(2) can written as

with x, y, z and w ∈ [ θ], where θ = if M ∈ ₈(2) and θ = 3 + 2 if M ∈ ₁₂(2).

Proposition 5.1. If M ∈ _4g(2) is given by

with x = a₁ + a₂θ, y = b₁ + b₂θ, w = c₁ + c₂θ, z = d₁ + d₂θ ∈ [ θ], where θ = , θ' = if M ∈ ₈(2) and θ = , θ' = 3 + 2if M ∈ ₁₂(2), then

where a₁ + ic₁, a₂ + ic₂, d₁ + ib₁, d₂ + ib₂∈ [i].

Proof. If H ∈ _4g(2), where H = then

which concludes the proof.

Example 5.1. If M ∈ ₈(2), where

for a = a₁ + a₂, b = b₁ + b₂, c = c₁ + c₂ and d = d₁ + d₂∈ [], then

with m₁ = (a₁ + ic₁) + (a₂ + ic₂), m₄ = , m₂ = (d₁ + ib₁) + (d₂ + ib₂)), m₃ = , where denotes the complex conjugation of the element m, and a₁ + ic₁, a₂ + ic₂, d₁ + ib₁, d₂ + ib₂∈[i].

Example 5.2. If M ∈ ₁₂(2), where

for a = a₁ + a₂, b = b₁ + b₂, c = c₁ + c₂ and d = d₁ + d₂∈ [], then

where m₁ = (a₁+ ic₁) + (a₂ + ic₂), m₄ = , m₂ = (d₁ + ib₁) + (d₂ + ib₂)), m₃ = , and a₁ + ic₁, a₂ + ic₂, d₁ + ib₁, d₂ + ib₂∈[i].

5.1 Construction of Space Time Codes

In order to construction a space time code, we need a complex alphabet which can be for example belong to ring of algebraic integers [i] (QAM symbols). In the next theorem we given a construction of a space time codes ⊆ (2).

Theorem 5.1. Let F = (i) and K = F( θ'), where θ' = or . We consider the set ⊆ _4g(2). Then the set is a space time code, that satisfies the following properties:

(1)

is linear dispersion space time code,

(2)

is full rate, and

(3)

is full diversity.

Proof.

(1) If X, Y ∈ , it is easy to see X ± Y, XY ∈ . Then is a linear dispersion space time code.

(2) Notice, for each codematrix N ∈ given by the matrix of the Equation (11), it follows that 4 information symbols belongs to the [i] given by a₁+ ic₁, a₂ + ic₂, d₁ + ib₁, d₂ + ib₂ what they are encoded. Then is full rate.

(3) If N ∈ ⊆ _4g(2), then N is given by

where a₁ + ic₁, i₂ + ic₂, d₁ + ib₁, d₂ + ib₂∈ [i]. Now, if H ∈ _4g(2), where H = then

Therefore

with x, y, z, w ∈ [ θ], where θ = if M ∈ ₈(2) and θ = 3 + 2 if M ∈ ₁₂(2). As consequence of Equation (4), it followsthat det(H) = det(f^-1(H)) ∈ _4g(2). Thus, det(H) ≠ 0 and therefore is fully diversity.

6 Conclusion

In this work, we constructed a new class of STBCs from symmetric groups(in this case arithmetic Fuchsian groups) associated with the regular polygon octogon and dodecagon of self-dual tessellation {8,8} and {12,12}, respectively.

However, we known there are infinitely possibilities of tessellations of hyperbolic plane by regular polygons. This fact suggest another possibilities of identifications of arithmetic Fuchsian groups by quaterion orders. Therefore, its open another possibilities to constructions STBCs that using this theory.

Received: 23/III/10.

Accepted: 16/III/11.

#CAM-196/10.

• [1] S.M. Alamouti, A simple transmit diversity technique for wireless communications, IEEE Journal on Selected Areas in Communications 16(8), (1998), 1451-1458.
• [2] V. Tarokh, N. Seshadri and A.R. Calderbank, Space-time codes for high datarate wireless communications: performance criterion and code construction IEEE Trans. Inform. Theory, 44 (1998), 744-765.
• [3] M.O. Damen, A. Tewfik and J.-C. Belfiore, A construction of a space-time code based on the theory numbers IEEE Trans. Inform. Theory, 48(3) (2002), 753-760.
• [4] B. Hassibi and B.M. Hochwald, High-rate codes that are linear in space andtime IEEE Trans. Inform. Theory, 48 (2002), 1804-1824.
• [5] F. Oggier, G. Rekaya, J-C. Belfiore and E. Viterbo, Perfect space-time blockcodes IEEE Trans. Inform Theory, 52(9) (2006), 3885-3902.
• [6] B.A. Sethuraman and B.S. Rajan, Full-diversity, hight-rate space-time block codes from division algebras IEEE Trans. Inform. Theory, 9(10) (2003), 2596-2616.
• [7] J.C. Belfiore, G. Rekaya and E. Viterbo, The golden code: a 2 ×2 full rate space time code with non vanishing determinants IEEE Trans. Inform. Theory, 51(4) (2005), 1432-1436.
• [8] R. Vehkalahti, C. Hollanti, J. Lahtonen and K. Ranto, On the densest MIMO lattices from cyclic division algebras IEEE Trans. Inform. Theory, 55(8) (2009),3751-3779.
• [9] S. Katok. Fuchsian groups The University of Chicago Press (1992).
• [10] K. Takeuchi, A characterization of arithmetic Fuchsian groups J. Math. Soc.Japan, 27(4) (1975), 600-612.
• [11] O.T. O'Meara, Maximal orders, Academic Press, London (1975).
• [12] A. Beardon, The geometry of discret groups Springer-Verlag, New York (1983).
• [13] M.L. Macasieb, Derived arithmetic Fuchsian groups of genus two Experimental Mathematics, 17(3) (2008), 347-369.

*

Acknowledgment to FAPESP by financial support 2007/56052-8.

Publication Dates

• Publication in this collection
  06 Jan 2012
• Date of issue
  2011

History

• Received
  23 Mar 2010
• Accepted
  16 Mar 2011