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

Authors: Cátia R. de O. Quilles Queiroz; Reginaldo Palazzo Júnior
Journal: ISRN Algebra
 (2012)

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## Abstract

We propose the construction of signal space codes over the quaternion orders from a graph associated with the arithmetic Fuchsian groupΓ8. 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.


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## Body

## 1. Introduction

In the study of two-dimensional lattice codes, it is known that the latticeℤ2 is associated with a type of digital modulation known as quadrature amplitude modulation, QAM modulation, denoted by xi(t)=αicosw0t+βisinw0t, where αi and βi 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 yi(t)=Acos(w0t+ϕi), where ϕi takes values on a finite set, for the same average energy. The PSK modulation is associated with the nth 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 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]. Here, we consider only the case g=2.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 ℝ2 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 ℤ2, whereas for the Eisenstein-Jacobi integer ring the Voronoi regions of the signal constellation are hexagons and may be represented by the lattice A2.In this paper, we propose the construction of signal space codes over the quaternion orders from graphs associated with the arithmetic Fuchsian groupΓ8. This Fuchsian group consists of the edge-pairing isometries of the regular hyperbolic polygon (fundamental region) P8 (8 edges) which tessellates the hyperbolic plane 𝔻2. The tessellation is the self-dual tessellation {8,8}, [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 Section2, 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 [9–13].


### 2.1. Quaternion Algebras

Let𝕂 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 i2=a, j2=b, ij=-ji=k, a,b∈𝕂-{0}, and denoted by 𝒜=(a,b)𝕂.Letα∈𝒜 be given by α=a0+a1i+a2j+a3ij, where a0,a1,a2,a3∈𝕂. The conjugate of α, denoted by α-, is defined by α-=a0-a1i-a2j-a3ij. Thus, the reduced norm of α∈𝒜, denoted by Nrd𝒜(α), or simply Nrd(α) when there is no confusion, is defined by
(2.1)Nrd(α)=α⋅α-=a02-aa12-ba22+aba32,
and the reduced trace of α by
(2.2)Trd(α)=α+α-=2a0.Notice that the reduced norm is a quadratic form such that(2.3)Nrd:A⟶K,α⟼a02-aa12-ba22+aba32,
which may also be denoted by its normal form 〈1,-a,-b,ab〉.Let𝒜=(a,b)𝕂 be a quaternion algebra over a field 𝕂 and φ:𝕂→𝔽 a field homomorphism. Define
(2.4)Aφ=(φ(a),φ(b))φ(K),Aφ⊗F=(φ(a),φ(b))F,
where 𝒜φ⊗𝔽 denotes the tensor product of the algebra 𝒜φ by the field 𝔽, [9]. Each homomorphism φ in the algebra 𝒜φ=(φ(a),φ(b))φ(𝕂) is called place of the quaternion  algebra 𝒜.Let𝕂 be a totally real algebraic number field of degree n. This means that the n monomorphisms φi, i=1,…,n are all real, that is, φi(𝕂)⊂ℝ. Therefore, the n distinct places are defined by ℝ-isomorphisms
(2.5)ρ1:Aφ1⊗R⟶M2(R),ρi:Aφi⊗R⟶H,
where φ1 is the identity, φi is an embedding of 𝕂 on ℝ, for i=1,…,n, and ℋ is a division subalgebra of M2(𝕂(a)). Hence, 𝒜 is not ramified at the place φ1 and ramified at the places φi, for 2≤i≤n.LetNrdℋ and Trdℋ be the reduced norm and the reduced trace in ℋ, respectively. Given α∈𝒜, it is easy to verify that
(2.6)NrdH(α)=det⁡(ρ1(α)),TrdH(α)=tr⁡(ρ1(α)).Now, from the identification ofαi with φi(αi), for i=0,1,2,3, it follows that for every 2≤i≤n,
(2.7)φi(NrdH(α))=NrdH(ρi(α)),φi(TrdH(α))=TrdH(ρi(α)).Furthermore, as the reduced norm of an element is given by the determinant of the isomorphismρ1, one may verify that
(2.8)NrdH(α⋅β)=Nrd(α)H⋅NrdH(β),
for any α,β∈𝒜.Proposition 2.1 (see [13]).
Let𝒜=(a,b)𝕂 be a quaternion algebra with a basis {1,i,j,k},r∈ℕ*, with r fixed, and let R be the set
(2.9)R={αrm:α∈IKandm∈N},
where I𝕂 is the ring of integers of 𝕂. Then 𝒪={x=x0+x1i+x2j+x3k:x0,x1,x2,x3∈R} is an order in 𝒜.Proof.
We have thatR is a subring of 𝕂 containing I𝕂 and that 𝒪 is an R-module. On the other hand, if β∈𝕂, then there exists c∈ℤ-{0} such that c∈I𝕂. Therefore, for any x0,x1,x2,x3∈𝕂, there exists cl∈ℤ-{0} such that clxl=αl∈I𝕂, l=0,1,2,3. Thus, given x=x0+x1i+x2j+x3k∈𝒜, there exists γ∈𝕂 such that x=γx′, with x′∈𝒪. Therefore, 𝒜=𝕂𝒪, which shows that 𝒪 is an order in 𝒜.Example 2.2.
Letℋ=(-1,-1)ℝ be the Hamilton quaternion algebra and ℋ1={α∈ℋ:Nrdℝ(α)=1}. Hence, given α=a0+a1i+a2j+a3k∈ℋ1, from (2.1), we have Nrdℝ(α)=a02-aa12-ba22+aba32k=a02+a12+a22+a32=1, which implies that a02=1-a12-a22-a32 and so |a0|≤1. Now, from (2.2), it follows that Trdℝ(α)=2a0, and so Trdℝ(α)=2a0∈[-2,2]. Therefore, Trdℝ(ℋ1)=[-2,2].Given𝒜, 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 I𝕂, the integer ring of 𝕂, where a,b∈I𝕂-{0}, then 𝒪={a0+a1i+a2j+a3ij:a0,a1,a2,a3∈I𝕂} is an order in 𝒜 denoted by 𝒪=(a,b)I𝕂.Example 2.3.
Givenℋ=(-1,-1)ℝ the Hamilton quaternion algebra, the integer ring of ℝ is ℤ, and the quaternion order ℋ[ℤ]={a0+a1i+a2j+a3ij:a0,a1,a2,a3∈ℤ} is called the ring of integral Hamiltonian quaternions, or the Lipschitz integers.


### 2.2. Hyperbolic Lattices

Let𝒜=(a,b)𝕂 be a quaternion algebra over 𝕂, let R be a ring of 𝕂, and let be 𝒪 an R-order in 𝒜. We also call 𝒪 a hyperbolic lattice due to its identification with an arithmetic Fuchsian group.The lattices𝒪 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 {e1,e2,e3,e4} of 𝒜  and R-ideal 𝔞 such that  𝒪=𝔞e1⊕Re2⊕Re3⊕Re4, 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, [14], it follows that Nrd(x), Trd(x)∈R, [15].An invariant of an order𝒪 is its discriminant, d(𝒪). For that, let {x0,x1,x2,x3} 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(Tr(xi,xj-)):0≤i,j≤3}.Example 2.4.
Let𝒜=(a,b)𝕂, and let I𝕂 be the ring of integers of 𝕂, where a, b∈I𝕂*=I𝕂-{0}. Then, [16], 𝒪={x0+x1i+x2j+x3k:x0,x1,x2,x3∈I𝕂} is an order in 𝒜 denoted by 𝒪=(a,b)I𝕂. The discriminant of 𝒪 is the principal ideal R·det(Tr(xi,xj-)), where {x0,x1,x2,x3}={1,i,j,k}, [14]. On the other hand, it is not difficult to see that Tr(xixj-) is the following diagonal matrix:
(2.10)(20000-2a0000-2b00002ab).
Therefore,
(2.11)d(O)=4ab.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 bemaximal. An order ℳ in a quaternion algebra 𝒜 is called maximal if ℳ is not contained in any other order in 𝒜, [14].Ifℳ is a maximal order in 𝒜 containing another order 𝒪, then the discriminant satisfies, [15], d(𝒪)=d(ℳ)·[ℳ:𝒪],  d(ℳ)=d(𝒜). Conversely, if d(𝒪)=d(𝒜), then 𝒪 is a maximal order in 𝒜.Example 2.5.
Let𝒜 be an algebra 𝒜=(2,-1)ℚ(2) with a basis {1,i,j,k} satisfying i=24,  j=Im, and k=24Im where Im denotes an imaginary unit, Im2=-1. Let us also consider the following order (Proposition 2.1 considers a more general case for 𝒪) in 𝒜, 𝒪={x=x0+x1i+x2j+x3k:x0,x1,x2,x3∈R}, that is, 𝒪=(2,-1)R, where R={x/2n:x∈ℤ[2] and n∈ℕ}. Thus, by (2.11), d(𝒪)=-2. Furthermore, d(𝒜)=-2, [15]. Hence, 𝒪  is a maximal order in  𝒜.


### 2.3. Arithmetic Fuchsian Groups

Consider the upper-half planeℍ2={z∈ℂ:Im(z)>0}  endowed with the Riemannian metric
(2.12)ds=dx2+dy2y,
where z=x+yIm. With this metric ℍ2 is the model of the hyperbolic plane or the Lobachevski plane. Let PSL(2,ℝ) be the set of all the Möbius transformations of ℂ over itself as
(2.13){z⟶az+bcz+d:a,b,c,d∈R,ad-bc=1}.
Consider the group of real matrices g=(abcd) with det(g)=ad-bc=1, and Tr(g)=a+d is the trace of the matrix g. This group is called unimodular, and it is denoted by SL(2,ℝ).The set of linear fractional Möbius transformations ofℂ 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 T is represented by a pair of matrices ±g∈SL(2,ℝ). Thus, the group of all transformations (2.13), called PSL(2,ℝ), is isomorphic to SL(2,ℝ)/{±I2}, where I2 is the 2×2 identity matrix, that is,
(2.14)PSL(2,R)≈SL(2,R){±I2}.A Fuchsian groupΓ is a discrete subgroup of PSL(2,ℝ), that is, Γ consists of the orientation-preserving isometries T:ℍ2→ℍ2, acting on ℍ2 by homeomorphisms.Another Euclidean model of the hyperbolic plane is given by the Poincaré disc𝔻2={z∈ℂ:|z|<1} endowed with the Riemannian metric
(2.15)ds2=4(dx2+dy2)[1-(x2+y2)]2,
where z=x+yIm. Analogously, the discrete group Γp of orientation-preserving isometries T:𝔻2→𝔻2 is also a Fuchsian group, given by the transformations Tp∈Γp<PSL(2,ℂ) such that
(2.16)Tp(z)=az+cc-z+a-,a,b∈C,|a|2-|c|2=1.
Furthermore, we may write Tp=f∘T∘f-1, where T∈PSL(2,ℝ), and f:ℍ2→𝔻2 is an isometry given by
(2.17)f(z)=zIm⁡+1z+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.For each order𝒪 in 𝒜, consider 𝒪1 as the set 𝒪1={α∈𝒪:Nrdℋ(α)=1}. Note that 𝒪1 is a multiplicative group.Now, note that the Fuchsian groups may be obtained by the isomorphismρ1 in (2.5). In fact, from (2.6), we have Nrdℋ(α)=det(ρ1(α)). Furthermore, we know that 𝒪1 is a multiplicative group, and so ρ1(𝒪1) is a subgroup of SL(2,ℝ), that is, ρ1(𝒪1)<SL(2,ℝ). Therefore, the group derived from a quaternion algebra 𝒜=(a,b)𝕂 and whose order is 𝒪, denoted by Γ(𝒜,𝒪), is given by
(2.18)Γ(A,O)=ρ1(O1){±Id2}<SL(2,R){±Id2}≅PSL(2,R).As a consequence, consider the following.Theorem 2.6 (see [11]).
Γ
(
𝒜
,
𝒪
) 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 algebra𝒜 and an order 𝒪⊂𝒜 such that Γ has finite index in Γ(𝒜,𝒪). The group Γ is called an arithmetic Fuchsian group.Theorem2.7 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(Γ)={±Tr(T):T∈Γ}.Theorem 2.7 (see [11, 16]).
LetΓ be a Fuchsian group where the fundamental region has finite area, that is, μ(ℍ2/Γ)<∞. Then Γ is derived from a quaternion algebra 𝒜 over a totally real number field 𝕂 if and only if the following conditions are satisfied by Γ:  (1)
if𝕂1=ℚ(Tr(T):T∈Γ), then 𝕂1 is an algebraic number field of finite degree, and Tr(Γ) is contained in I𝕂1, the ring of integers of 𝕂1;(2)
ifφ is an embedding of 𝕂1 in ℂ such that φ≠Id, then φ(Tr(Γ)) is bounded in ℂ.


## 2.1. Quaternion Algebras

Let𝕂 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 i2=a, j2=b, ij=-ji=k, a,b∈𝕂-{0}, and denoted by 𝒜=(a,b)𝕂.Letα∈𝒜 be given by α=a0+a1i+a2j+a3ij, where a0,a1,a2,a3∈𝕂. The conjugate of α, denoted by α-, is defined by α-=a0-a1i-a2j-a3ij. Thus, the reduced norm of α∈𝒜, denoted by Nrd𝒜(α), or simply Nrd(α) when there is no confusion, is defined by
(2.1)Nrd(α)=α⋅α-=a02-aa12-ba22+aba32,
and the reduced trace of α by
(2.2)Trd(α)=α+α-=2a0.Notice that the reduced norm is a quadratic form such that(2.3)Nrd:A⟶K,α⟼a02-aa12-ba22+aba32,
which may also be denoted by its normal form 〈1,-a,-b,ab〉.Let𝒜=(a,b)𝕂 be a quaternion algebra over a field 𝕂 and φ:𝕂→𝔽 a field homomorphism. Define
(2.4)Aφ=(φ(a),φ(b))φ(K),Aφ⊗F=(φ(a),φ(b))F,
where 𝒜φ⊗𝔽 denotes the tensor product of the algebra 𝒜φ by the field 𝔽, [9]. Each homomorphism φ in the algebra 𝒜φ=(φ(a),φ(b))φ(𝕂) is called place of the quaternion  algebra 𝒜.Let𝕂 be a totally real algebraic number field of degree n. This means that the n monomorphisms φi, i=1,…,n are all real, that is, φi(𝕂)⊂ℝ. Therefore, the n distinct places are defined by ℝ-isomorphisms
(2.5)ρ1:Aφ1⊗R⟶M2(R),ρi:Aφi⊗R⟶H,
where φ1 is the identity, φi is an embedding of 𝕂 on ℝ, for i=1,…,n, and ℋ is a division subalgebra of M2(𝕂(a)). Hence, 𝒜 is not ramified at the place φ1 and ramified at the places φi, for 2≤i≤n.LetNrdℋ and Trdℋ be the reduced norm and the reduced trace in ℋ, respectively. Given α∈𝒜, it is easy to verify that
(2.6)NrdH(α)=det⁡(ρ1(α)),TrdH(α)=tr⁡(ρ1(α)).Now, from the identification ofαi with φi(αi), for i=0,1,2,3, it follows that for every 2≤i≤n,
(2.7)φi(NrdH(α))=NrdH(ρi(α)),φi(TrdH(α))=TrdH(ρi(α)).Furthermore, as the reduced norm of an element is given by the determinant of the isomorphismρ1, one may verify that
(2.8)NrdH(α⋅β)=Nrd(α)H⋅NrdH(β),
for any α,β∈𝒜.Proposition 2.1 (see [13]).
Let𝒜=(a,b)𝕂 be a quaternion algebra with a basis {1,i,j,k},r∈ℕ*, with r fixed, and let R be the set
(2.9)R={αrm:α∈IKandm∈N},
where I𝕂 is the ring of integers of 𝕂. Then 𝒪={x=x0+x1i+x2j+x3k:x0,x1,x2,x3∈R} is an order in 𝒜.Proof.
We have thatR is a subring of 𝕂 containing I𝕂 and that 𝒪 is an R-module. On the other hand, if β∈𝕂, then there exists c∈ℤ-{0} such that c∈I𝕂. Therefore, for any x0,x1,x2,x3∈𝕂, there exists cl∈ℤ-{0} such that clxl=αl∈I𝕂, l=0,1,2,3. Thus, given x=x0+x1i+x2j+x3k∈𝒜, there exists γ∈𝕂 such that x=γx′, with x′∈𝒪. Therefore, 𝒜=𝕂𝒪, which shows that 𝒪 is an order in 𝒜.Example 2.2.
Letℋ=(-1,-1)ℝ be the Hamilton quaternion algebra and ℋ1={α∈ℋ:Nrdℝ(α)=1}. Hence, given α=a0+a1i+a2j+a3k∈ℋ1, from (2.1), we have Nrdℝ(α)=a02-aa12-ba22+aba32k=a02+a12+a22+a32=1, which implies that a02=1-a12-a22-a32 and so |a0|≤1. Now, from (2.2), it follows that Trdℝ(α)=2a0, and so Trdℝ(α)=2a0∈[-2,2]. Therefore, Trdℝ(ℋ1)=[-2,2].Given𝒜, 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 I𝕂, the integer ring of 𝕂, where a,b∈I𝕂-{0}, then 𝒪={a0+a1i+a2j+a3ij:a0,a1,a2,a3∈I𝕂} is an order in 𝒜 denoted by 𝒪=(a,b)I𝕂.Example 2.3.
Givenℋ=(-1,-1)ℝ the Hamilton quaternion algebra, the integer ring of ℝ is ℤ, and the quaternion order ℋ[ℤ]={a0+a1i+a2j+a3ij:a0,a1,a2,a3∈ℤ} is called the ring of integral Hamiltonian quaternions, or the Lipschitz integers.


## 2.2. Hyperbolic Lattices

Let𝒜=(a,b)𝕂 be a quaternion algebra over 𝕂, let R be a ring of 𝕂, and let be 𝒪 an R-order in 𝒜. We also call 𝒪 a hyperbolic lattice due to its identification with an arithmetic Fuchsian group.The lattices𝒪 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 {e1,e2,e3,e4} of 𝒜  and R-ideal 𝔞 such that  𝒪=𝔞e1⊕Re2⊕Re3⊕Re4, 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, [14], it follows that Nrd(x), Trd(x)∈R, [15].An invariant of an order𝒪 is its discriminant, d(𝒪). For that, let {x0,x1,x2,x3} 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(Tr(xi,xj-)):0≤i,j≤3}.Example 2.4.
Let𝒜=(a,b)𝕂, and let I𝕂 be the ring of integers of 𝕂, where a, b∈I𝕂*=I𝕂-{0}. Then, [16], 𝒪={x0+x1i+x2j+x3k:x0,x1,x2,x3∈I𝕂} is an order in 𝒜 denoted by 𝒪=(a,b)I𝕂. The discriminant of 𝒪 is the principal ideal R·det(Tr(xi,xj-)), where {x0,x1,x2,x3}={1,i,j,k}, [14]. On the other hand, it is not difficult to see that Tr(xixj-) is the following diagonal matrix:
(2.10)(20000-2a0000-2b00002ab).
Therefore,
(2.11)d(O)=4ab.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 bemaximal. An order ℳ in a quaternion algebra 𝒜 is called maximal if ℳ is not contained in any other order in 𝒜, [14].Ifℳ is a maximal order in 𝒜 containing another order 𝒪, then the discriminant satisfies, [15], d(𝒪)=d(ℳ)·[ℳ:𝒪],  d(ℳ)=d(𝒜). Conversely, if d(𝒪)=d(𝒜), then 𝒪 is a maximal order in 𝒜.Example 2.5.
Let𝒜 be an algebra 𝒜=(2,-1)ℚ(2) with a basis {1,i,j,k} satisfying i=24,  j=Im, and k=24Im where Im denotes an imaginary unit, Im2=-1. Let us also consider the following order (Proposition 2.1 considers a more general case for 𝒪) in 𝒜, 𝒪={x=x0+x1i+x2j+x3k:x0,x1,x2,x3∈R}, that is, 𝒪=(2,-1)R, where R={x/2n:x∈ℤ[2] and n∈ℕ}. Thus, by (2.11), d(𝒪)=-2. Furthermore, d(𝒜)=-2, [15]. Hence, 𝒪  is a maximal order in  𝒜.


## 2.3. Arithmetic Fuchsian Groups

Consider the upper-half planeℍ2={z∈ℂ:Im(z)>0}  endowed with the Riemannian metric
(2.12)ds=dx2+dy2y,
where z=x+yIm. With this metric ℍ2 is the model of the hyperbolic plane or the Lobachevski plane. Let PSL(2,ℝ) be the set of all the Möbius transformations of ℂ over itself as
(2.13){z⟶az+bcz+d:a,b,c,d∈R,ad-bc=1}.
Consider the group of real matrices g=(abcd) with det(g)=ad-bc=1, and Tr(g)=a+d is the trace of the matrix g. This group is called unimodular, and it is denoted by SL(2,ℝ).The set of linear fractional Möbius transformations ofℂ 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 T is represented by a pair of matrices ±g∈SL(2,ℝ). Thus, the group of all transformations (2.13), called PSL(2,ℝ), is isomorphic to SL(2,ℝ)/{±I2}, where I2 is the 2×2 identity matrix, that is,
(2.14)PSL(2,R)≈SL(2,R){±I2}.A Fuchsian groupΓ is a discrete subgroup of PSL(2,ℝ), that is, Γ consists of the orientation-preserving isometries T:ℍ2→ℍ2, acting on ℍ2 by homeomorphisms.Another Euclidean model of the hyperbolic plane is given by the Poincaré disc𝔻2={z∈ℂ:|z|<1} endowed with the Riemannian metric
(2.15)ds2=4(dx2+dy2)[1-(x2+y2)]2,
where z=x+yIm. Analogously, the discrete group Γp of orientation-preserving isometries T:𝔻2→𝔻2 is also a Fuchsian group, given by the transformations Tp∈Γp<PSL(2,ℂ) such that
(2.16)Tp(z)=az+cc-z+a-,a,b∈C,|a|2-|c|2=1.
Furthermore, we may write Tp=f∘T∘f-1, where T∈PSL(2,ℝ), and f:ℍ2→𝔻2 is an isometry given by
(2.17)f(z)=zIm⁡+1z+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.For each order𝒪 in 𝒜, consider 𝒪1 as the set 𝒪1={α∈𝒪:Nrdℋ(α)=1}. Note that 𝒪1 is a multiplicative group.Now, note that the Fuchsian groups may be obtained by the isomorphismρ1 in (2.5). In fact, from (2.6), we have Nrdℋ(α)=det(ρ1(α)). Furthermore, we know that 𝒪1 is a multiplicative group, and so ρ1(𝒪1) is a subgroup of SL(2,ℝ), that is, ρ1(𝒪1)<SL(2,ℝ). Therefore, the group derived from a quaternion algebra 𝒜=(a,b)𝕂 and whose order is 𝒪, denoted by Γ(𝒜,𝒪), is given by
(2.18)Γ(A,O)=ρ1(O1){±Id2}<SL(2,R){±Id2}≅PSL(2,R).As a consequence, consider the following.Theorem 2.6 (see [11]).
Γ
(
𝒜
,
𝒪
) 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 algebra𝒜 and an order 𝒪⊂𝒜 such that Γ has finite index in Γ(𝒜,𝒪). The group Γ is called an arithmetic Fuchsian group.Theorem2.7 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(Γ)={±Tr(T):T∈Γ}.Theorem 2.7 (see [11, 16]).
LetΓ be a Fuchsian group where the fundamental region has finite area, that is, μ(ℍ2/Γ)<∞. Then Γ is derived from a quaternion algebra 𝒜 over a totally real number field 𝕂 if and only if the following conditions are satisfied by Γ:  (1)
if𝕂1=ℚ(Tr(T):T∈Γ), then 𝕂1 is an algebraic number field of finite degree, and Tr(Γ) is contained in I𝕂1, the ring of integers of 𝕂1;(2)
ifφ is an embedding of 𝕂1 in ℂ such that φ≠Id, then φ(Tr(Γ)) is bounded in ℂ.


## 3. Identification ofΓ8 in Γ(𝒜,𝒪),𝒪⊂𝒜

In this section, we identify the arithmetic Fuchsian groupΓ8 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, and g=2 denotes the genus of the surface 𝔻2/Γ8 in a quaternion order.From [17], if g=2, the arithmetic Fuchsian group Γ8 is derived from a quaternion algebra 𝒜 over a totally real number field 𝕂=ℚ(2). The elements of the Fuchsian group Γ8 are identified, by an isomorphism, with the elements of the order 𝒪=(2,-1)I𝕂, where I𝕂 denotes the integer ring of 𝕂.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 atφ1, and it is ramified at the remaining places.Consider the Fuchsian groupΓ8, given a quaternion algebra 𝒜=(2,-1)𝕂, and the elements of T∈Γ are given by
(3.1)T=12s(xl+yl2zl+wl2-zl+wl2xl-yl2),
where s∈ℕ, xl,yl,zl,wl∈ℤ[2]. Since φ1 is the identity, it follows that 𝒜≃M2(𝕂) is not ramified at φ1. Now, observe that 2 is square-free for 𝕂=ℚ(2), that is, there is no t∈𝕂-{0} such that t2=2. Therefore, 𝒜 is ramified at all places φi, except at φ1.On the other hand, the order𝒪=(2,-1)I𝕂 is not a maximal order in the quaternion algebra 𝒜=(2,-1)𝕂 for the discriminant is not 42. 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 [13], we have that 𝒪=(2,-1)R, where R={α/2m:α∈I𝕂,m∈ℕ} is a maximal order that contains 𝒪=(2,-1)I𝕂. Therefore, this is the order we are taking into consideration in the case of interest.


## 4. Quotient Rings of the Quaternion Order𝒪=(2,-1)R Where R={α/2m:α∈I𝕂,m∈ℕ}

Consider the self-dual tessellation{8,8} having an octagon as the fundamental region. We know from the previous sections that the arithmetic Fuchsian group Γ8 is derived from a quaternion algebra over 𝕂=ℚ(2), with the identification of the generators by the order 𝒪=(2,-1)I𝕂. Thus, let 𝕂=ℚ(2) and {1,i,j,k}={1,2,Im,2Im} be a basis of the quaternion algebra 𝒜=(2,-1)I𝕂, where i2=2, j2=-1,k=ij=2Im.The ring of integers of𝕂=ℚ(2) is ℤ[2]; hence, 𝒪={a0+a1i+a2j+a3k:a0,a1,a2,a3∈ℤ[2]} is in fact an order in 𝒜. Due to the simplicity of this order, we start with it and gradually extend it to the order 𝒪=(2,-1)R, where R={α/2m:α∈ℤ[2],m∈ℕ} which realizes the complete labeling.Observe that𝒪={a0+a1i+a2j+a3k:ai∈ℤ[2]} is an extension of ℤ[2] of dimension 4, for it has {1,i,j,k} as its basis, and we have that 𝒪 is a subring of 𝒜 containing 1 and which is a finitely generated ℤ[2]-module. Now, if we look at the order 𝒪 as an extension of ℤ, the dimension increases to 8, and the basis of 𝒪 over ℤ is given by {1,2,i,2i,j,2j,k,2k}. In this case, the order will be denoted by 𝒪ℤ. We may still verify that according to the definition of order, 𝒪ℤ is a free ℤ-module with rank 4n=8, where n=[𝕂:ℚ]=[ℚ(2):ℚ]=2, 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. Caseg=2

Given the genusg=2, the arithmetic Fuchsian group Γ8 is derived from a quaternion algebra 𝒜 over a totally real number field 𝕂=ℚ(2), and the elements of Γ8 are identified, via an isomorphism, with the elements of 𝒪=(2,-1)ℤ[2]. Hence, given 𝕂=ℚ(2) and 𝒪=(2,-1)ℤ[2] such that 𝒪={a0+a1i+a2j+a3k:ai∈ℤ[2],i2=2,j2=-1,k2=-2}, the reduced norm of an element α=a0+a1i+a2j+a3k∈𝒪 is given by
(4.1)NrdZ[2](α)=αα-=a02-2a12+a22-2a32∈Z[2],
and it satisfies Nrdℤ[2](α)∈I𝕂=ℤ[2]. Next, we verify in which cases this norm is an element belonging to ℤ.Proposition 4.1.
Givenα=a0+a1i+a2j+a3k∈𝒪, where ai=xi+yi2, where xi,yi∈ℤ, then  Nrdℤ[2](α)∈ℤ if and only if  2x0y0-x12-2y12+2x2y2-x32-2y32=0. In this case, the norm is given by Nrdℤ[2](α)=x02+2y02-4x1y1+x22+2y22-4x3y3.Proof.
From (4.1), we have Nrdℤ[2](α)=αα-=a02-2a12+a22-2a32. Since ai∈ℤ[2], it may be written as ai=xi+yi2, where xi,yi∈ℤ. Thus, ai2=xi2+2yi2+22xiyi, and from this, it follows that
(4.2)NrdZ[2](α)=a02-2a12+a22-2a32=x02+2y02+22x0y0-2(x12+2y12+22x1y1)+x22+2y22+22x2y2-2(x32+2y32+22x3y3)=x02+2y02-4x1y1+x22+2y22-4x3y3+2(2x0y0-x12-2y12+2x2y2-x32-2y32).
Hence,Nrdℤ[2](α)∈ℤ if and only if  2x0y0-x12-2y12+2x2y2-x32-2y32=0, from which it follows that
(4.3)NrdZ[2](α)=x02+2y02-4x1y1+x22+2y22-4x3y3∈Z.Now, considering the order𝒪 as an extension of ℤ, denoted by 𝒪ℤ, the reduced norm of an element α=a0+a1i+a2j+a3k∈𝒪ℤ is given by
(4.4)NrdZ(α)=αα-=a0a0--2a1a1-+a2a2--2a3a3-∈Z[2],
where ai- denotes the conjugate of ai.Since the proof of the next result is similar to the proof of Proposition4.1, we omit it.Proposition 4.2.
Givenα=a0+a1i+a2j+a3k∈𝒪ℤ, where ai=xi+yi2, where xi,yi∈ℤ, then  Nrdℤ(α)∈ℤ if and only if  x12+x32-2(y12+y32)=0. In this case, the norm is given by Nrdℤ(α)=x02-2y02+x22-2y22∈ℤ.Remark 4.3.
When there is no confusion in the notation being used, we will denote for simplicity the reduced norm ofα by Nrd(α).Theorem 4.4.
Let0≠α∈𝒪. If Nrd(α)∈ℤ, then 𝒪/〈α〉 has Nrd(α)4 elements.Proof.
Let0≠α∈𝒪 and Nrd(α)=N∈ℤ. First, we show that 𝒪/〈N〉 has N8 elements. As N∈ℤ, let us consider 𝒪 over ℤ. However, [𝒪:ℤ]=8, and the basis of 𝒪 over ℤ is {1,2,i,i2,j,j2,k,k2}. Thus, α∈𝒪 is of the form α=a0+a12+a2i+a3i2+a4j+a52j+a6k+a7k2.
Now, given two elementsβ,β′∈𝒪,
(4.5)β=b0+b12+b2i+b3i2+b4j+b52j+b6k+b7k2,bi∈ℤ,β′=b0′+b1′2+b2′i+b3′i2+b4′j+b5′2j+b6′k+b7′k2,bi′∈Z,
we say that β and β′ are congruent modulo N if there exists
(4.6)β′′=b0′′+b1′′2+b2′′i+b3′′i2+b4′′j+b5′′2j+b6′′k+b7′′k2,bi′′∈Z,
such that β-β′=β′′N. Thus, bi-bi′=bi′′N, for i=0,1,…,7, that is, bi≡bi′(modN) which implies that there exist N possibilities for each bi, and thus, N8 different equivalence classes modulo N.
Now, sinceNrd(α)=αα-, we have the following chain of ideals: 〈Nrd(α)〉=〈α-α〉⊆〈α〉. From the third isomorphism theorem for A-modules, [10], we have the following sequence of left A-module:
(4.7)0⟶〈α〉〈αα-〉⟶A〈αα-〉⟶A〈α〉⟶0.
We denote the number of elements ofA/〈α〉 by n and the number of elements of 〈α〉/〈αα-〉 by m. 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 Nrd(α)8=nm. If we prove that n=m, we may finally conclude that n=Nrd(α)4. Now, observe that the function
(4.8)f:A〈α-〉⟶〈α〉〈αα-〉,
defined by f(β+〈α-〉)=βα+〈αα-〉, is well defined, and it is an isomorphism of the left A-module. Therefore, m is exactly the number of elements of A/〈α-〉.
Finally, the quaternion conjugation is an antiautomorphism, which implies thatA/〈α-〉 and A/〈α〉 have the same cardinality, that is, n=m.Example 4.5.
Letα=1+j∈𝒪=(2,-1)ℤ[2], then Nrd(α)=2. From Theorem 4.4, 𝒪/〈α〉 has 16 elements, obtained by the quotient of the order 𝒪 and the ideal 〈1+j〉, that is, we take the elements of 𝒪 and reduce them modulo (1+j), obtaining
(4.9)O〈1+j〉={0,1,2,1+2,i,1+i,2+i,2i,(1+2+i),1+2i,2+2i,(1+2)+2i,(1+2)i,1+(1+2)i,2+(1+2)i,(1+2)+(1+2)i}.Example 4.6.
Givenα=2+2∈𝒪=(2,-1)ℤ[2], from Proposition 4.1, we have Nrdℤ[2](α)=6+42∉ℤ. However, taking the order as an extension of ℤ, that is, 𝒪ℤ=(2,-1)ℤ from Proposition 4.2, we have Nrdℤ(α)=2, and 𝒪/〈α〉 has 16 elements, given by
(4.10)O〈2+2〉={0,1,i,j,k,1+i,1+j,1+k,i+j,i+k,j+k,1+i+j,1+i+k,1+j+k,i+j+k,1+i+j+k}.Remark 4.7.
We are not interested in orders such as𝒪ℤ, for when the order 𝒪 is extended to the order 𝒪ℤ, 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 I𝕂 is not an element in ℤ.Corollary 4.8.
Ifβ∈𝒪=(2,-1)ℤ[2] is a right divisor of α and Nrd(β)∈ℤ, then the left ideal generated by β, 〈β〉⊆𝒪 has Nrd(α)4/Nrd(β)4 elements.Note from Corollary4.8 that 〈β〉 generates a code with Nrd(α)4/Nrd(β)4 codewords, therefore, a subcode of 𝒪/〈α〉, whose minimum distance Dβ(η,τ)>Dα(η,τ).Example 4.9.
Givenα=1+22j, from Proposition 4.1, we have Nrdℤ[2](α)=(1)2+(22)2=9. Now, α=1+22j may be written as 1+22j=(2+j)2; hence, β is a right divisor of α and Nrdℤ[2](β)=3∈ℤ, then the left ideal generated by β, 〈β〉⊆𝒪 has Nrd(α)4/Nrd(β)4=94/34=81 elements.As can be seen in Example2.5, for the proof see [13], the order 𝒪=(2,-1)ℤ[2] is not a maximal order. Therefore, we have to consider the order over the ring R={α/2m:α∈ℤ[2],m∈ℕ}, which makes it maximal, hence, given 𝕂=ℚ(2) and 𝒪=(2,-1)R, where R={α/2m:α∈ℤ[2],m∈ℕ} such that
(4.11)O={a0+a12i+a22j+a32k:ai∈Z[2],i2=2,j2=-1,k2=-2},
we have that the reduced norm of an element α∈𝒪 is given by
(4.12)NrdR(α)=a02-122a12+12a22-142a32∈Z[2].
Now, for the maximal order, the cardinality of the quotient ring satisfy the following results:Theorem 4.10.
Letα∈𝒪=(2,-1)R, where R={α/2m:α∈ℤ[2],m∈ℕ}. If NrdR(α)=2n, then 𝒪/〈α〉 has just one element.Proof.
Letγ∈𝒪/〈α〉. We have to show that γ≡0(modα). To show that γ≡0(modα) is equivalent to proving that γ=xα, where x∈𝒪. As NrdR(α)=2n, we have that α-α=2n;  hence, γ may be written as
(4.13)γ=γ2nα-α,
that is, γ≡0(modα). In particular, one may verify that 1≡0(modα), for 1=(1/2n)α-α.Theorem 4.11.
Letα∈𝒪=(2,-1)R, where R={α/2m:α∈ℤ[2],m∈ℕ}. If NrdR(α)≠2n, then 𝒪/〈α〉 has NrdR(α)4 elements.Proof.
Let0≠α∈𝒪 and NrdR(α)≠2n, NrdR(α)=N∈ℤ. We have to show that the left 𝒪-module 𝒪/〈N〉 has N8 elements. As N∈ℤ, let us take 𝒪 over ℤ. However, [𝒪:ℤ]=8 and the basis of 𝒪 over ℤ is {1/2n,2/2n,i/2n,i2/2n,j/2n,j2/2n,k/2n,k2/2n}. Hence, the proof is analogous to the proof of Theorem 4.4.Example 4.12.
Letα=2∈𝒪R. Hence, from (4.12), we have that NrdR(α)=4, and by Theorem 4.10, it follows that 𝒪/〈α〉 has just one element {0}.Example 4.13.
Letα=2+(2/2)j∈𝒪R. Hence, from (4.12), we have that NrdR(α)=3 and by Theorem 4.11, it follows that 𝒪/〈α〉 has 81 elements.


## 4.1. Caseg=2

Given the genusg=2, the arithmetic Fuchsian group Γ8 is derived from a quaternion algebra 𝒜 over a totally real number field 𝕂=ℚ(2), and the elements of Γ8 are identified, via an isomorphism, with the elements of 𝒪=(2,-1)ℤ[2]. Hence, given 𝕂=ℚ(2) and 𝒪=(2,-1)ℤ[2] such that 𝒪={a0+a1i+a2j+a3k:ai∈ℤ[2],i2=2,j2=-1,k2=-2}, the reduced norm of an element α=a0+a1i+a2j+a3k∈𝒪 is given by
(4.1)NrdZ[2](α)=αα-=a02-2a12+a22-2a32∈Z[2],
and it satisfies Nrdℤ[2](α)∈I𝕂=ℤ[2]. Next, we verify in which cases this norm is an element belonging to ℤ.Proposition 4.1.
Givenα=a0+a1i+a2j+a3k∈𝒪, where ai=xi+yi2, where xi,yi∈ℤ, then  Nrdℤ[2](α)∈ℤ if and only if  2x0y0-x12-2y12+2x2y2-x32-2y32=0. In this case, the norm is given by Nrdℤ[2](α)=x02+2y02-4x1y1+x22+2y22-4x3y3.Proof.
From (4.1), we have Nrdℤ[2](α)=αα-=a02-2a12+a22-2a32. Since ai∈ℤ[2], it may be written as ai=xi+yi2, where xi,yi∈ℤ. Thus, ai2=xi2+2yi2+22xiyi, and from this, it follows that
(4.2)NrdZ[2](α)=a02-2a12+a22-2a32=x02+2y02+22x0y0-2(x12+2y12+22x1y1)+x22+2y22+22x2y2-2(x32+2y32+22x3y3)=x02+2y02-4x1y1+x22+2y22-4x3y3+2(2x0y0-x12-2y12+2x2y2-x32-2y32).
Hence,Nrdℤ[2](α)∈ℤ if and only if  2x0y0-x12-2y12+2x2y2-x32-2y32=0, from which it follows that
(4.3)NrdZ[2](α)=x02+2y02-4x1y1+x22+2y22-4x3y3∈Z.Now, considering the order𝒪 as an extension of ℤ, denoted by 𝒪ℤ, the reduced norm of an element α=a0+a1i+a2j+a3k∈𝒪ℤ is given by
(4.4)NrdZ(α)=αα-=a0a0--2a1a1-+a2a2--2a3a3-∈Z[2],
where ai- denotes the conjugate of ai.Since the proof of the next result is similar to the proof of Proposition4.1, we omit it.Proposition 4.2.
Givenα=a0+a1i+a2j+a3k∈𝒪ℤ, where ai=xi+yi2, where xi,yi∈ℤ, then  Nrdℤ(α)∈ℤ if and only if  x12+x32-2(y12+y32)=0. In this case, the norm is given by Nrdℤ(α)=x02-2y02+x22-2y22∈ℤ.Remark 4.3.
When there is no confusion in the notation being used, we will denote for simplicity the reduced norm ofα by Nrd(α).Theorem 4.4.
Let0≠α∈𝒪. If Nrd(α)∈ℤ, then 𝒪/〈α〉 has Nrd(α)4 elements.Proof.
Let0≠α∈𝒪 and Nrd(α)=N∈ℤ. First, we show that 𝒪/〈N〉 has N8 elements. As N∈ℤ, let us consider 𝒪 over ℤ. However, [𝒪:ℤ]=8, and the basis of 𝒪 over ℤ is {1,2,i,i2,j,j2,k,k2}. Thus, α∈𝒪 is of the form α=a0+a12+a2i+a3i2+a4j+a52j+a6k+a7k2.
Now, given two elementsβ,β′∈𝒪,
(4.5)β=b0+b12+b2i+b3i2+b4j+b52j+b6k+b7k2,bi∈ℤ,β′=b0′+b1′2+b2′i+b3′i2+b4′j+b5′2j+b6′k+b7′k2,bi′∈Z,
we say that β and β′ are congruent modulo N if there exists
(4.6)β′′=b0′′+b1′′2+b2′′i+b3′′i2+b4′′j+b5′′2j+b6′′k+b7′′k2,bi′′∈Z,
such that β-β′=β′′N. Thus, bi-bi′=bi′′N, for i=0,1,…,7, that is, bi≡bi′(modN) which implies that there exist N possibilities for each bi, and thus, N8 different equivalence classes modulo N.
Now, sinceNrd(α)=αα-, we have the following chain of ideals: 〈Nrd(α)〉=〈α-α〉⊆〈α〉. From the third isomorphism theorem for A-modules, [10], we have the following sequence of left A-module:
(4.7)0⟶〈α〉〈αα-〉⟶A〈αα-〉⟶A〈α〉⟶0.
We denote the number of elements ofA/〈α〉 by n and the number of elements of 〈α〉/〈αα-〉 by m. 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 Nrd(α)8=nm. If we prove that n=m, we may finally conclude that n=Nrd(α)4. Now, observe that the function
(4.8)f:A〈α-〉⟶〈α〉〈αα-〉,
defined by f(β+〈α-〉)=βα+〈αα-〉, is well defined, and it is an isomorphism of the left A-module. Therefore, m is exactly the number of elements of A/〈α-〉.
Finally, the quaternion conjugation is an antiautomorphism, which implies thatA/〈α-〉 and A/〈α〉 have the same cardinality, that is, n=m.Example 4.5.
Letα=1+j∈𝒪=(2,-1)ℤ[2], then Nrd(α)=2. From Theorem 4.4, 𝒪/〈α〉 has 16 elements, obtained by the quotient of the order 𝒪 and the ideal 〈1+j〉, that is, we take the elements of 𝒪 and reduce them modulo (1+j), obtaining
(4.9)O〈1+j〉={0,1,2,1+2,i,1+i,2+i,2i,(1+2+i),1+2i,2+2i,(1+2)+2i,(1+2)i,1+(1+2)i,2+(1+2)i,(1+2)+(1+2)i}.Example 4.6.
Givenα=2+2∈𝒪=(2,-1)ℤ[2], from Proposition 4.1, we have Nrdℤ[2](α)=6+42∉ℤ. However, taking the order as an extension of ℤ, that is, 𝒪ℤ=(2,-1)ℤ from Proposition 4.2, we have Nrdℤ(α)=2, and 𝒪/〈α〉 has 16 elements, given by
(4.10)O〈2+2〉={0,1,i,j,k,1+i,1+j,1+k,i+j,i+k,j+k,1+i+j,1+i+k,1+j+k,i+j+k,1+i+j+k}.Remark 4.7.
We are not interested in orders such as𝒪ℤ, for when the order 𝒪 is extended to the order 𝒪ℤ, 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 I𝕂 is not an element in ℤ.Corollary 4.8.
Ifβ∈𝒪=(2,-1)ℤ[2] is a right divisor of α and Nrd(β)∈ℤ, then the left ideal generated by β, 〈β〉⊆𝒪 has Nrd(α)4/Nrd(β)4 elements.Note from Corollary4.8 that 〈β〉 generates a code with Nrd(α)4/Nrd(β)4 codewords, therefore, a subcode of 𝒪/〈α〉, whose minimum distance Dβ(η,τ)>Dα(η,τ).Example 4.9.
Givenα=1+22j, from Proposition 4.1, we have Nrdℤ[2](α)=(1)2+(22)2=9. Now, α=1+22j may be written as 1+22j=(2+j)2; hence, β is a right divisor of α and Nrdℤ[2](β)=3∈ℤ, then the left ideal generated by β, 〈β〉⊆𝒪 has Nrd(α)4/Nrd(β)4=94/34=81 elements.As can be seen in Example2.5, for the proof see [13], the order 𝒪=(2,-1)ℤ[2] is not a maximal order. Therefore, we have to consider the order over the ring R={α/2m:α∈ℤ[2],m∈ℕ}, which makes it maximal, hence, given 𝕂=ℚ(2) and 𝒪=(2,-1)R, where R={α/2m:α∈ℤ[2],m∈ℕ} such that
(4.11)O={a0+a12i+a22j+a32k:ai∈Z[2],i2=2,j2=-1,k2=-2},
we have that the reduced norm of an element α∈𝒪 is given by
(4.12)NrdR(α)=a02-122a12+12a22-142a32∈Z[2].
Now, for the maximal order, the cardinality of the quotient ring satisfy the following results:Theorem 4.10.
Letα∈𝒪=(2,-1)R, where R={α/2m:α∈ℤ[2],m∈ℕ}. If NrdR(α)=2n, then 𝒪/〈α〉 has just one element.Proof.
Letγ∈𝒪/〈α〉. We have to show that γ≡0(modα). To show that γ≡0(modα) is equivalent to proving that γ=xα, where x∈𝒪. As NrdR(α)=2n, we have that α-α=2n;  hence, γ may be written as
(4.13)γ=γ2nα-α,
that is, γ≡0(modα). In particular, one may verify that 1≡0(modα), for 1=(1/2n)α-α.Theorem 4.11.
Letα∈𝒪=(2,-1)R, where R={α/2m:α∈ℤ[2],m∈ℕ}. If NrdR(α)≠2n, then 𝒪/〈α〉 has NrdR(α)4 elements.Proof.
Let0≠α∈𝒪 and NrdR(α)≠2n, NrdR(α)=N∈ℤ. We have to show that the left 𝒪-module 𝒪/〈N〉 has N8 elements. As N∈ℤ, let us take 𝒪 over ℤ. However, [𝒪:ℤ]=8 and the basis of 𝒪 over ℤ is {1/2n,2/2n,i/2n,i2/2n,j/2n,j2/2n,k/2n,k2/2n}. Hence, the proof is analogous to the proof of Theorem 4.4.Example 4.12.
Letα=2∈𝒪R. Hence, from (4.12), we have that NrdR(α)=4, and by Theorem 4.10, it follows that 𝒪/〈α〉 has just one element {0}.Example 4.13.
Letα=2+(2/2)j∈𝒪R. Hence, from (4.12), we have that NrdR(α)=3 and by Theorem 4.11, it follows that 𝒪/〈α〉 has 81 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.
Let0≠α∈𝒪=(θ,-1)I𝕂. The distance in 𝒪 is the distance induced by the graph Gα. Hence, if η,τ∈𝒪, then the distance is given by
(5.1)Dα(η,τ)=min⁡⁡{|x1|+|x2|+2|x3|+2|x4|-2|x2x3|},
such that τ-η≡x1+x2i+x3j+x4k(modα).Example 5.2.
ForV=𝒪/〈2+j〉. If τ=1 and η=i, then τ-η=1-i. Thus, Dα(η,τ)=2, if τ=(2/2)(1+i) and η=(2/2)(1-i), then τ-η=2i≡k(modα). Thus, Dα(η,τ)=2.Given the distanceDα, a graph generated by α∈𝒪 is defined as follows.Definition 5.3.
Let0≠α∈𝒪=(θ,-1)I𝕂. The graph generated by α is defined as Gα=(V,E), where (1)
V=𝒪/〈α〉 denotes the set of vertices;(2)
E={(η,τ)∈V×V:Dα(η,τ)=1} denotes the set of edges.Example 5.4.
Givenα=2+j∈𝒪=(2,-1)ℤ[2], from Proposition 4.1, the reduced norm is Nrdℤ[2](α)=3. The set of vertices is V=𝒪/〈2+j〉, and the set of edges satisfies E.Remark 5.5.
Note that the distance between two signal pointsη and τ in the graph is the least number of traversed edges connecting the signal point η to the signal point τ.Given a graphGα with a set of vertices V and distance Dα, a code in Gα is a nonempty subset 𝒞 of Gα. The Voronoi region Vη associated with η∈𝒞 is the subset consisting of the elements of V for which η is the closest signal point in 𝒞, that is, Vη={τ∈V;D(η,τ)=D(η,𝒞)}. The number t=max{D(η,𝒞);η∈V} is called covering radius of the code. The covering radius is the least number t such that each ball of radius t centered at the signal points of 𝒞, given by Bt(η)={τ∈V:D(η,τ)≤t}, covers V. The number δ=min{D(η,τ):η,τ∈𝒞,η≠τ} is the minimum distance of 𝒞, and δ≤2t+1; the equality holds when each ball of radius t centered at the signal points of 𝒞 forms a partition of V. A code satisfying this property is called perfect and corrects t errors. A code is called quasiperfect if the code is capable of correcting every error pattern up to t errors and some patterns with t+1 errors and no errors greater than t+1. 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 elementα∈𝒪R, we may generate a code over a graph by use of the quotient ring 𝒪R/〈α〉 as the vertices of the graph. Thus, by choosing β a divisor of α, 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.
Forg= 2, given α=1+22j, such that α∈𝒪ℤ(2)=(2,-1)ℤ(2), the reduced norm is Nrdℤ[2](α)=9. Thus, from Theorem 4.4, the cardinality of the set of vertices V is Nrdℤ[2](α)4=94=6561. Note that α may be written as 1+22j=(2+j)2, and so β is a right divisor of α and Nrdℤ[2](β)=3∈ℤ. Therefore, the code generated by β, 〈β〉⊆𝒪 has Nrdℤ[2](α)4/Nrdℤ[2](β)4=94/34=81 codewords. Note that the Voronoi region associated with each codeword consists of 81 elements. If τ=2+j and η=0, then τ-η=2+j=(1+22j)2-3j≡-3j(modα). Thus, Dα(η,τ)=6. The minimum distance of this code is Dα(η,τ)=6.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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*Source: 956017-2012-06-27.xml*