Codes over Graphs Derived from Quotient Rings of the Quaternion Orders

Resumo: In this paper we propose the construction of signal space code over the quaternion order from a graph associated with the arithmetic Fuchsian group Γ8. This Fuchsian group consists of the edge-pairing isometries of the regular hyperbolic polygon (fundamental region) P8, which tessellates the hyperbolic plane D2. Knowing the generators of the quaternion orders which realize the edge-pairings of the polygon, the signal points of the signal constellation (geometrically uniform code) derived from the graph associated with the quotient ring of the quaternion order are determined.


Introduction
In the study of two dimensional lattice codes, it is known that the lattice Z 2 is associated with a QAM modulation whose performance under the (bit) error probability criterion is better than the P SK modulation for the same average energy.The question that emerges is why a QAM signal constellation achieves better performance in terms of the error probability?Topologically, the fundamental region of the P SK signal constellation is a polygon with two edges oriented in the same direction whereas the fundamental region of the QAM signal constellation is a square with opposite edges oriented in the same direction.The edge-pairing of each one of these fundamental regions leads to oriented compact surfaces with genus g = 0 (sphere) and g = 1 (torus), respectively.We infer that the topological invariant associated with the performance of the signal constellation is the genus of the surface which is obtained by pairing the edges of the fundamental region associated with the signal code.In the quest for the signal code with the best performance, we construct signal codes associated with surfaces with genus g ≥ 2. Such surfaces may be obtained by the quotient of Fuchsian groups of the first kind, [1].Due to space limitation we consider only the case g = 2.
The concept of geometrically uniform codes (GU codes)was proposed in [4] and generalized in [7].In [3] these GU codes are summarized for any specific metric space and in [8] new metrics are derived from graphs associated with quotient rings.Such codes have highly desirable symmetry properties, such as: every Voronoi region are congruent; the distance profile is the same for any codeword; the codewords have the same error probability; and the generator group is isomorphic to a permutation group acting transitively on the codewords.In [9], [10] geometrically uniform codes are constructed in R 2 from graphs associated with Gaussian and Einsenstein-Jacobi integer rings.For the Gaussian integer rings, the Voronoi regions of the signal constellation are squares and may be represented by the lattice Z 2 , whereas for the Einsenstein-Jacobi integer ring the Voronoi regions of the signal constellation are hexagons and may be represented by the lattice A 2 .
In this paper we propose the construction of signal space codes over the quaternion orders from graphs associated with the arithmetic Fuchsian group Γ 8 .This Fuchsian group consists of the edge-pairing isometries of the regular hyperbolic polygon (fundamental region) P 8 (8 edges) which tessellates the hyperbolic plane D 2 .The tessellation is the self-dual tessellation {8, 8}, where the first number denotes the number of edges of the regular hyperbolic polygon, and the second one denotes the number of such polygons which cover each vertex.
This paper is organized as follows.In Section 2, basic concepts on quaternion orders and arithmetic Fuchsian groups are presented.In Section 3 the identification of the arithmetic Fuchsian group derived from the octagon is realized by the associated quaternion order.In Section 4, quotient ring of the quaternion order is constructed and some new results are obtained.In Section 5 some concepts related to graphs and codes over graphs are presented.Finally, in Section 6 an example of a GU code derived from a graph over the quotient ring of the quaternion order is established.

Preliminary Results
In this section some basic and important concepts regarding quaternion algebras, quaternion orders, and arithmetic Fuchsian groups with respect to the development of this paper are presented.For a detailed description of these concepts we refer the reader to [13]- [12].

Quaternion algebras
Let K be a field.A quaternion algebra A over K is a K-vector space of dimension 4 with a K−base B = {1, i, j, k}, where Furthermore, for any α, β ∈ A, it is easy to verify that N rd H (α.β) = N rd H (α).N rd H (β).
Proposition 2.1 Let A = (a, b) K be a quaternion algebra with a basis {1, i, j, k} , r ∈ N * , with r fixed, and R be the set R = α r m : α ∈ I K and m ∈ N , where I K is the ring of integers of Given A, a quaternion algebra over K, and R a ring of K, an R-order O in A is a subring with unity of A which is a finitely generated R-module such that A = KO.Hence, if A = (a, b) K and I K , the integer ring of K, where a, b Example 2.1 Given H = (−1, −1) R the Hamilton quaternion algebra, the integer ring of R is Z and the quaternion order H[Z] = {a 0 + a 1 i + a 2 j + a 3 ij : a 0 , a 1 , a 2 , a 3 ∈ Z}, is called integer ring of the Hamilton quaternion, or the Lipschitz integers.

Hyperbolic lattices
Let A = (a, b) K be a quaternion algebra over K and R be a ring of K.An R-order O in A is a subring of A containing 1, equivalently, it is a finitely generated R-module such that A = KO.We also call an R-order O a hyperbolic lattice due to its identification with an arithmetic Fuchsian group.
The lattices O are used as the basic entity in generating the signals of a signal constellation in the hyperbolic plane.Since O is an order in A, then there exists a basis {e 1 , e 2 , e 3 , e 4 } of A and an R-ideal a such that, O = ae 1 ⊕ Re 2 ⊕ Re 3 ⊕ Re 4 , where ⊕ denotes direct sum.Note that by definition, given x, y ∈ O, we have x • y ∈ O. Furthermore, since every x ∈ O is integral over R, [11], it follows that N rd(x) ∈ R, [5].
An invariant of an order O is its discriminant, d(O).For that, let {x 0 , x 1 , x 2 , x 3 } be a set consisting of the generators of O over R. The discriminant of O, is defined as the square root of the R-ideal generated by the set {det(T r(x i x j )) : 0 ≤ i, j ≤ 3}.Then, [6], O = {x 0 + x 1 i + x 2 j + x 3 k : x 0 , x 1 , x 2 , x 3 ∈ I K }, is an order in A denoted by O = (a, b) I K .The discriminant of O is the principal ideal R • det(T r(x i x j )), where {x 0 , x 1 , x 2 , x 3 } = {1, i, j, k}, [11].On the other hand, it is not difficult to see that T r(x i x j ) is the following diagonal matrix One of the main objectives of this paper is to identify the arithmetic Fuchsian group in a quaternion order.Once this identification is realized, then the next step is to show the codewords of a code over graphs or the signals of a signal constellation (quotient of an order by a proper ideal).However, for the algebraic labeling to be complete, it is necessary that the corresponding order be maximal.An order M in a quaternion algebra A is called maximal if M is not contained in any other order in A, [11].If M is a maximal order in A containing another order O, then the discriminant satisfies, [5], ) with a basis {1, i, j, k} satisfying i = 4 . Hence, O is a maximal order in A.

Arithmetic Fuchsian groups
Consider the upper-half plane H 2 = {z ∈ C : Im(z) > 0} endowed with the Riemannian metric The set of linear fractional Mobius transformations of C over itself as in ( 3) is a group such that the product of two transformations corresponds to the product of the corresponding matrices and the inverse transformation corresponds to the inverse matrix.Each transformation T is represented by a pair of matrices ±g ∈ SL(2, R).Thus, the group of all transformations (3), called P SL(2, R), is isomorphic to SL(2, R)/{±I 2 }, where I 2 is the 2 × 2 identity matrix, that is, A Fuchsian group Γ is a discrete subgroup of P SL(2, R), that is, Γ consists of the orientation preserving isometries T : H 2 → H 2 , acting on H 2 by homeomorphisms.
Another Euclidean model of the hyperbolic plane is given by the Poincaré disc D 2 = {z ∈ C : |z| < 1} endowed with the Riemannian metric where z = x + y Im.Analogously, the discrete group Γ p of orientation preserving isometries T : D 2 → D 2 is also a Fuchsian group, given by the transformations T p ∈ Γ p < P SL(2, C) such that T p (z) = az+c cz+ā , a, b ∈ C, |a| 2 − |c| 2 = 1.Furthermore, we may write T p = f • T • f −1 , where T ∈ P SL(2, R) and f : H 2 → D 2 is an isometry given by f (z) = z Im +1 z+Im .Therefore, the Euclidean models of the hyperbolic plane such as the Poincaré disc and the upper-half plane are isomorphic and they will be used according to the need.Notice that the Poincaré disc model is useful for the visualization whereas the upper-half plane is useful for the algebraic manipulations.Now, we know that the group derived from a quaternion algebra A = (a, b) K and whose order is O, denoted by Γ(A, O), is given by As a consequence, These previous concepts and results lead to the concept of arithmetic Fuchsian groups.Since every Fuchsian group may be obtained in this way, we say that a Fuchsian group is derived from a quaternion algebra if there exists a quaternion algebra A and an order O ⊂ A such that Γ has finite index in Γ(A, O).The group Γ is called an arithmetic Fuchsian group.

Identification of Γ 8 in Γ(A, O), O ⊂ A
In this section we identify the arithmetic Fuchsian group Γ 8 derived from a quaternion algebra A over a number field K, for [K : Q] = 2, where [K : Q] denotes the degree of the field extension.From [2] if g = 2 the arithmetic Fuchsian group Γ 8 is derived from a quaternion algebra A over a totally real number field K = Q( √ 2).The elements of the Fuchsian group Γ 8 are identified, by an isomorphism, with the elements of the order O = ( √ 2, −1) I K , where I K denotes the integer ring of K. To verify if a Fuchsian group associated with an order is in fact arithmetic is very simple.
Consider the Fuchsian group Γ 8 , given a quaternion algebra A = ( √ 2, −1) K , the elements of T ∈ Γ are given by: T ]. On the other hand, the order O = ( √ 2, −1) I K is not a maximal order in the quaternion algebra A = ( √ 2, −1) K for the discriminant is not 4 √ 2. Since we are interested in realizing a complete algebraic labeling, we have to find an order that contains the order O in A and that it is maximal.From [14] we have that O = ( √ 2, −1) R , where R = { α 2 m : α ∈ I K , m ∈ N} is a maximal order that contains O = ( √ 2, −1) I K .Therefore, this is the order we are taking into consideration in the case of interest.
4 Quotient Rings of the Quaternion Order O Given the genus g = 2. Consider the self-dual tessellation {8, 8} having an octagon as the fundamental region.We know from the previous sections that the arithmetic Fuchsian

Example 2 . 2
Let A = (a, b) K and I K the ring of integers of K, where a, b ∈ I * K = I K − {0}.

√ 2 ,
j = Im, and k = 4 √ 2 Im, where Im denotes an imaginary unit, Im 2 = −1.Let us also consider the following order (Proposition 2.1 considers a more general case for O) in A, O

3 )
where z = x+y Im.With this metric H 2 is the model of the hyperbolic plane or the Lobachevski plane.Let P SL(2, R) be the set of all the Mobius transformations of C over itself as {z → az + b cz + d : a, b, c, d ∈ R, ad − bc = 1}.(Consider the group of real matrices g = a b c d with det(g) = ad − bc = 1, and T r(g) = a + d the trace of the matrix g.This group is called unimodular and it is denoted by SL(2, R).