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ISRN Algebra

VolumeΒ 2012Β (2012), Article IDΒ 956017, 14 pages

http://dx.doi.org/10.5402/2012/956017

## Codes over Graphs Derived from Quotient Rings of the Quaternion Orders

^{1}Departamento de MatemΓ‘tica, ICEx, UNIFAL, R. Gabriel Monteiro da Silva, 700 Centro, 37130-000 Alfenas, MG, Brazil^{2}Departamento de TelemΓ‘tica, FEEC, UNICAMP, Avenida Albert Einstein 400, Cidade Universtaria Zeferino Vaz, 13083-852 Campinas, SP, Brazil

Received 13 February 2012; Accepted 6 March 2012

Academic Editors: H.Β Airault, A.Β Milas, and H.Β You

Copyright Β© 2012 CΓ‘tia R. de O. Quilles Queiroz and Reginaldo Palazzo JΓΊnior. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose the construction of signal space codes over the quaternion orders from a graph associated with the arithmetic Fuchsian group . This Fuchsian group consists of the edge-pairing isometries of the regular hyperbolic polygon (fundamental region) *P*_{8}, which tessellates the hyperbolic plane . 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.

#### 1. Introduction

In the study of two-dimensional lattice codes, it is known that the lattice is associated with a type of digital modulation known as quadrature amplitude modulation, QAM modulation, denoted by , where and take values on a finite integer set, whose performance under the (bit) error probability criterion is better than that of the phase-shift keying modulation, PSK modulation, denoted by , where takes values on a finite set, for the same average energy. The PSK modulation is associated with the th roots of unity. 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 PSK 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 (sphere) and (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 . Such surfaces may be obtained by the quotient of Fuchsian groups of the first kind, [1]. Here, we consider only the case .

The concept of geometrically uniform codes (GU codes) was proposed in [2] and generalized in [3]. In [4], these GU codes are summarized for any specific metric space, and in [5], new metrics are derived from graphs associated with quotient rings. Such codes have highly desirable symmetry properties, such as the following: every Voronoi region is congruent; the distance profile is the same for any codeword; the codewords have the same error probability; the generator group is isomorphic to a permutation group acting transitively on the codewords. In [6, 7], geometrically uniform codes are constructed in from graphs associated with Gaussian and Eisenstein-Jacobi integer rings. For the Gaussian integer rings, the Voronoi regions of the signal constellation are squares and may be represented by the lattice , whereas for the Eisenstein-Jacobi integer ring the Voronoi regions of the signal constellation are hexagons and may be represented by the lattice .

In this paper, we propose the construction of signal space codes over the quaternion orders from graphs associated with the arithmetic Fuchsian group . This Fuchsian group consists of the edge-pairing isometries of the regular hyperbolic polygon (fundamental region) (8 edges) which tessellates the hyperbolic plane . The tessellation is the self-dual tessellation , [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 we show that the cardinality of this quotient ring is given by the norm to the fourth power. 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.

#### 2. 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 [9β13].

##### 2.1. Quaternion Algebras

Let be a field. A quaternion algebra over is a -vector space of dimension 4 with a -base , where , , , , and denoted by .

Let be given by , where . The conjugate of , denoted by , is defined by . Thus, the reduced norm of , denoted by , or simply when there is no confusion, is defined by and the reduced trace of by

Notice that the reduced norm is a quadratic form such that which may also be denoted by its normal form .

Let be a quaternion algebra over a field and a field homomorphism. Define
where denotes the tensor product of the algebra by the field , [9]. Each homomorphism in the algebra is called *place* of the quaternion algebra .

Let be a totally real algebraic number field of degree . This means that the monomorphisms , are all real, that is, . Therefore, the distinct places are defined by -isomorphisms where is the identity, is an embedding of on , for , and is a division subalgebra of . Hence, is not ramified at the place and ramified at the places , for .

Let and be the reduced norm and the reduced trace in , respectively. Given , it is easy to verify that

Now, from the identification of with , for , it follows that for every ,

Furthermore, as the reduced norm of an element is given by the determinant of the isomorphism , one may verify that for any .

Proposition 2.1 (see [13]). *Let be a quaternion algebra with a basis , with fixed, and let be the set
**
where is the ring of integers of . Then is an order in .*

*Proof. *We have that is a subring of containing and that is an -module. On the other hand, if , then there exists such that . Therefore, for any , there exists such that , . Thus, given , there exists such that , with . Therefore, , which shows that is an order in .

*Example 2.2. *Let be the Hamilton quaternion algebra and . Hence, given , from (2.1), we have , which implies that and so . Now, from (2.2), it follows that , and so . Therefore, .

Given , a quaternion algebra over , and , a ring of , an -order in is a subring with unity of which is a finitely generated -module such that . Hence, if and , the integer ring of , where , then is an order in denoted by .

*Example 2.3. *Given the Hamilton quaternion algebra, the integer ring of is , and the quaternion order is called the ring of integral Hamiltonian quaternions, or the Lipschitz integers.

##### 2.2. Hyperbolic Lattices

Let be a quaternion algebra over , let be a ring of , and let be an -order in . We also call a hyperbolic lattice due to its identification with an arithmetic Fuchsian group.

The lattices are used as the basic entity in generating the signals of a signal constellation in the hyperbolic plane. Since is an order in , then there exists a basis of and -ideal such that , where denotes direct sum. Note that by definition, given , we have . Furthermore, since every is integral over , [14], it follows that , , [15].

An invariant of an order is its discriminant, . For that, let be a set consisting of the generators of over . The discriminant of is defined as the square root of the -ideal generated by the set .

*Example 2.4. *Let , and let be the ring of integers of , where , . Then, [16], is an order in denoted by . The discriminant of is the principal ideal , where , [14]. On the other hand, it is not difficult to see that is the following diagonal matrix:
Therefore,

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 in a quaternion algebra is called *maximal* if is not contained in any other order in , [14].

If is a maximal order in containing another order , then the discriminant satisfies, [15], ,β . Conversely, if , then is a maximal order in .

*Example 2.5. *Let be an algebra with a basis satisfying ,ββ, and where Im denotes an imaginary unit, . Let us also consider the following order (Proposition 2.1 considers a more general case for in , , that is, , where and . Thus, by (2.11), . Furthermore, , [15]. Hence, is a maximal order in .

##### 2.3. Arithmetic Fuchsian Groups

Consider the upper-half plane endowed with the Riemannian metric where . With this metric is the model of the hyperbolic plane or the Lobachevski plane. Let PSL be the set of all the MΓΆbius transformations of over itself as Consider the group of real matrices with , and is the trace of the matrix . This group is called unimodular, and it is denoted by .

The set of linear fractional MΓΆbius transformations of over itself as in (2.13) 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 is represented by a pair of matrices . Thus, the group of all transformations (2.13), called , is isomorphic to , where is the identity matrix, that is,

A Fuchsian group is a discrete subgroup of PSL, that is, consists of the orientation-preserving isometries , acting on by homeomorphisms.

Another Euclidean model of the hyperbolic plane is given by the PoincarΓ© disc endowed with the Riemannian metric where . Analogously, the discrete group of orientation-preserving isometries is also a Fuchsian group, given by the transformations such that Furthermore, we may write , where , and is an isometry given by Therefore, the Euclidean models of the hyperbolic plane such as the PoincarΓ© disc and the upper-half plane are isomorphic, and they will be used according to the need. Notice that the PoincarΓ© disc model is useful for the visualization, whereas the upper-half plane is useful for the algebraic manipulations.

For each order in , consider as the set . Note that is a multiplicative group.

Now, note that the Fuchsian groups may be obtained by the isomorphism in (2.5). In fact, from (2.6), we have . Furthermore, we know that is a multiplicative group, and so is a subgroup of , that is, . Therefore, the group derived from a quaternion algebra and whose order is , denoted by , is given by

As a consequence, consider the following.

Theorem 2.6 (see [11]). * is a Fuchsian group. *

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 and an order such that has finite index in . The group is called an *arithmetic Fuchsian group*.

Theorem 2.7 establishes the necessary and sufficient conditions for arithmetic of Fuchsian groups, and its characterization makes use of the set consisting of the traces of its elements, that is, Tr.

Theorem 2.7 (see [11, 16]). *Let be a Fuchsian group where the fundamental region has finite area, that is, . Then is derived from a quaternion algebra over a totally real number field if and only if the following conditions are satisfied by : *(1)*if , then is an algebraic number field of finite degree, and is contained in , the ring of integers of ;*(2)*if is an embedding of in such that , then is bounded in . *

#### 3. Identification of in

In this section, we identify the arithmetic Fuchsian group derived from a quaternion algebra over a number field , for , where denotes the degree of the field extension, and denotes the genus of the surface in a quaternion order.

From [17], if , the arithmetic Fuchsian group is derived from a quaternion algebra over a totally real number field . The elements of the Fuchsian group are identified, by an isomorphism, with the elements of the order , where denotes the integer ring of .

To verify if a Fuchsian group associated with an order as specified in the previous paragraph is in fact arithmetic, it suffices to show that the quaternion algebra is not ramified at , and it is ramified at the remaining places.

Consider the Fuchsian group , given a quaternion algebra , and the elements of are given by where , . Since is the identity, it follows that is not ramified at . Now, observe that is square-free for , that is, there is no such that . Therefore, is ramified at all places , except at .

On the other hand, the order is not a maximal order in the quaternion algebra for the discriminant is not . Since we are interested in realizing a complete algebraic labeling, we have to find an order that contains the order in and that it is maximal. From [13], we have that , where is a maximal order that contains . Therefore, this is the order we are taking into consideration in the case of interest.

#### 4. Quotient Rings of the Quaternion Order Where

Consider the self-dual tessellation having an octagon as the fundamental region. We know from the previous sections that the arithmetic Fuchsian group is derived from a quaternion algebra over , with the identification of the generators by the order . Thus, let and be a basis of the quaternion algebra , where , .

The ring of integers of is ; hence, is in fact an order in . Due to the simplicity of this order, we start with it and gradually extend it to the order , where which realizes the complete labeling.

Observe that is an extension of of dimension 4, for it has as its basis, and we have that is a subring of containing 1 and which is a finitely generated -module. Now, if we look at the order as an extension of , the dimension increases to 8, and the basis of over is given by . In this case, the order will be denoted by . We may still verify that according to the definition of order, is a free -module with rank , where , and in this way, we are not working with the quaternions anymore, but with the octonions, a set which besides being noncommutative is also nonassociative.

##### 4.1. Case

Given the genus , the arithmetic Fuchsian group is derived from a quaternion algebra over a totally real number field , and the elements of are identified, via an isomorphism, with the elements of . Hence, given and such that , the reduced norm of an element is given by and it satisfies . Next, we verify in which cases this norm is an element belonging to .

Proposition 4.1. *Given , where , where , then if and only if . In this case, the norm is given by .*

*Proof. *From (4.1), we have . Since , it may be written as , where . Thus, , and from this, it follows that

Hence, if and only if , from which it follows that

Now, considering the order as an extension of , denoted by , the reduced norm of an element is given by where denotes the conjugate of .

Since the proof of the next result is similar to the proof of Proposition 4.1, we omit it.

Proposition 4.2. *Given , where , where , then if and only if . In this case, the norm is given by .*

*Remark 4.3. *When there is no confusion in the notation being used, we will denote for simplicity the reduced norm of by .

Theorem 4.4. *Let . If , then has elements. *

*Proof. *Let and . First, we show that has elements. As , let us consider over . However, , and the basis of over is . Thus, is of the form .

Now, given two elements ,
we say that and are congruent modulo if there exists
such that . Thus, , for , that is, which implies that there exist possibilities for each , and thus, different equivalence classes modulo .

Now, since , we have the following chain of ideals: . From the third isomorphism theorem for -modules, [10], we have the following sequence of left -module:

We denote the number of elements of by and the number of elements of by . Then, as a consequence of the Lagrange theorem, [10], we may consider the previous exact sequence as a sequence of Abelian groups, thus leading to . If we prove that , we may finally conclude that . Now, observe that the function
defined by , is well defined, and it is an isomorphism of the left -module. Therefore, is exactly the number of elements of .

Finally, the quaternion conjugation is an antiautomorphism, which implies that and have the same cardinality, that is, .

*Example 4.5. *Let , then . From Theorem 4.4, has 16 elements, obtained by the quotient of the order and the ideal , that is, we take the elements of and reduce them modulo , obtaining

*Example 4.6. *Given , from Proposition 4.1, we have . However, taking the order as an extension of , that is, from Proposition 4.2, we have , and has 16 elements, given by

*Remark 4.7. *We are not interested in orders such as , for when the order is extended to the order , it implies working with octonions; hence, some important properties are lost. Therefore, we consider such an extension when there is no other alternative, that is, when the norm over is not an element in .

Corollary 4.8. *If is a right divisor of and , then the left ideal generated by , has elements.*

Note from Corollary 4.8 that generates a code with codewords, therefore, a subcode of , whose minimum distance .

*Example 4.9. *Given , from Proposition 4.1, we have . Now, may be written as ; hence, is a right divisor of and , then the left ideal generated by , has elements.

As can be seen in Example 2.5, for the proof see [13], the order is not a maximal order. Therefore, we have to consider the order over the ring , which makes it maximal, hence, given and , where such that we have that the reduced norm of an element is given by Now, for the maximal order, the cardinality of the quotient ring satisfy the following results:

Theorem 4.10. * Let , where . If , then has just one element. *

*Proof. *Let . We have to show that . To show that is equivalent to proving that , where . As , we have that ; hence, may be written as
that is, . In particular, one may verify that , for .

Theorem 4.11. *Let , where . If , then has elements. *

*Proof. *Let and , . We have to show that the left -module has elements. As , let us take over . However, and the basis of over is . Hence, the proof is analogous to the proof of Theorem 4.4.

*Example 4.12. *Let . Hence, from (4.12), we have that , and by Theorem 4.10, it follows that has just one element .

*Example 4.13. * Let . Hence, from (4.12), we have that and by Theorem 4.11, it follows that has elements.

#### 5. Codes over Graphs

In this section some concepts of graphs and codes over graphs are considered which will be useful in the next section.

*Definition 5.1. *Let . The distance in is the distance induced by the graph . Hence, if , then the distance is given by
such that .

*Example 5.2. *For . If and , then . Thus, , if and , then . Thus, .

Given the distance , a graph generated by is defined as follows.

*Definition 5.3. *Let . The graph generated by is defined as , where (1) denotes the set of vertices; (2) denotes the set of edges.

*Example 5.4. *Given , from Proposition 4.1, the reduced norm is . The set of vertices is and the set of edges satisfies .

*Remark 5.5. *Note that the distance between two signal points and in the graph is the least number of traversed edges connecting the signal point to the signal point .

Given a graph with a set of vertices and distance , a code in is a nonempty subset of . The Voronoi region associated with is the subset consisting of the elements of for which is the closest signal point in , that is, . The number is called *covering radius* of the code. The covering radius is the least number such that each ball of radius centered at the signal points of , given by , covers . The number is the minimum distance of , and ; the equality holds when each ball of radius centered at the signal points of forms a partition of . A code satisfying this property is called *perfect* and corrects errors. A code is called *quasiperfect* if the code is capable of correcting every error pattern up to errors and some patterns with errors and no errors greater than . Perfect codes and quasiperfect codes are part of a more general class of codes called *geometrically uniform codes*.

#### 6. Example

A code derived from a graph is defined as geometrically uniform if for any two-code sequences, there exists an isometry that takes a code sequence into the other, while it leaves the code invariant. Hence, geometrically uniform codes partition a set of vertices of a graph by the Voronoi regions.

Given an element , we may generate a code over a graph by use of the quotient ring as the vertices of the graph. Thus, by choosing a divisor of , we obtain a geometrically uniform code, and the vertices of the graph are covered by the action of the isometries on the fundamental region as shown in Section 4.

*Example 6.1. *For 2, given , such that , the reduced norm is . Thus, from Theorem 4.4, the cardinality of the set of vertices is . Note that may be written as , and so is a right divisor of and . Therefore, the code generated by , has codewords. Note that the Voronoi region associated with each codeword consists of 81 elements. If and , then . Thus, . The minimum distance of this code is .

The procedures considered may be extended to surfaces with any genus once the associated quaternion order is known. This allows us to construct new geometrically uniform codes over different signal constellations.

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