Codes over Graphs Derived from Quotient Rings of the Quaternion Orders

Abstract

We propose the construction of signal space codes over the quaternion orders 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.

1. Introduction

In the study of two-dimensional lattice codes, it is known that the lattice ℤ² is associated with a type of digital modulation known as quadrature amplitude modulation, QAM modulation, denoted by 𝑥_𝑖(𝑡)=𝛼_𝑖cos𝑤₀𝑡+𝛽_𝑖sin𝑤₀𝑡, where 𝛼_𝑖 and 𝛽_𝑖 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 𝑦_𝑖(𝑡)=𝐴cos(𝑤₀𝑡+𝜙_𝑖), where 𝜙_𝑖 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 group Γ₈. 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 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 [9–13].

2.1. 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 𝑖²=𝑎, 𝑗²=𝑏, 𝑖𝑗=−𝑗𝑖=𝑘, 𝑎,𝑏∈𝕂−{0}, and denoted by 𝒜=(𝑎,𝑏)_𝕂.

Let 𝛼∈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 Nrd_𝒜(𝛼), or simply Nrd(𝛼) when there is no confusion, is defined by Nrd(𝛼)=𝛼⋅𝛼⁻=𝑎²⁰−𝑎𝑎²¹−𝑏𝑎²²+𝑎𝑏𝑎²³,(2.1) and the reduced trace of 𝛼 by Trd(𝛼)=𝛼+𝛼⁻=2𝑎₀.(2.2)

Notice that the reduced norm is a quadratic form such that Nrd∶𝒜𝛼⟼⟶𝑎𝕂₂₀,−𝑎𝑎²¹−𝑏𝑎²²+𝑎𝑏𝑎²³,(2.3) which may also be denoted by its normal form ⟨1,−𝑎,−𝑏,𝑎𝑏⟩.

Let 𝒜=(𝑎,𝑏)_𝕂 be a quaternion algebra over a field 𝕂 and 𝜑∶𝕂→𝔽 a field homomorphism. Define 𝒜^𝜑=(𝜑(𝑎),𝜑(𝑏))_𝜑(𝕂),𝒜^𝜑⊗𝔽=(𝜑(𝑎),𝜑(𝑏))_𝔽,(2.4) where 𝒜^𝜑⊗𝔽 denotes the tensor product of the algebra 𝒜^𝜑 by the field 𝔽, [9]. Each homomorphism 𝜑 in the algebra 𝒜^𝜑=(𝜑(𝑎),𝜑(𝑏))_𝜑(𝕂) is called place of the quaternion algebra 𝒜.

Let 𝕂 be a totally real algebraic number field of degree 𝑛. This means that the 𝑛 monomorphisms 𝜑_𝑖, 𝑖=1,…,𝑛 are all real, that is, 𝜑_𝑖(𝕂)⊂ℝ. Therefore, the 𝑛 distinct places are defined by ℝ-isomorphisms 𝜌₁∶𝒜^𝜑1⊗ℝ⟶𝑀₂(ℝ),𝜌_𝑖∶𝒜^𝜑𝑖⊗ℝ⟶ℋ,(2.5) where 𝜑₁ is the identity, 𝜑_𝑖 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 𝜑_𝑖, for 2≤𝑖≤𝑛.

Let Nrd_ℋ and Trd_ℋ be the reduced norm and the reduced trace in ℋ, respectively. Given 𝛼∈𝒜, it is easy to verify that Nrd_ℋ(𝛼)=det𝜌₁(𝛼),Trd_ℋ(𝛼)=tr𝜌₁(𝛼).(2.6)

Now, from the identification of 𝛼_𝑖 with 𝜑_𝑖(𝛼_𝑖), for 𝑖=0,1,2,3, it follows that for every 2≤𝑖≤𝑛, 𝜑_𝑖Nrd_ℋ(𝛼)=Nrd_ℋ𝜌_𝑖(𝛼),𝜑_𝑖Trd_ℋ(𝛼)=Trd_ℋ𝜌_𝑖(𝛼).(2.7)

Furthermore, as the reduced norm of an element is given by the determinant of the isomorphism 𝜌₁, one may verify that Nrd_ℋ(𝛼⋅𝛽)=Nrd(𝛼)_ℋ⋅Nrd_ℋ(𝛽),(2.8) for any 𝛼,𝛽∈𝒜.

Proposition 2.1 (see [13]). Let 𝒜=(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 𝑅=𝛼𝑟^𝑚∶𝛼∈𝐼_𝕂and𝑚∈ℕ,(2.9) where 𝐼_𝕂 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 that 𝑅 is a subring of 𝕂 containing 𝐼_𝕂 and that 𝒪 is an 𝑅-module. On the other hand, if 𝛽∈𝕂, then there exists 𝑐∈ℤ−{0} such that 𝑐∈𝐼_𝕂. Therefore, for any 𝑥₀,𝑥₁,𝑥₂,𝑥₃∈𝕂, there exists 𝑐_𝑙∈ℤ−{0} such that 𝑐_𝑙𝑥_𝑙=𝛼𝑙∈𝐼_𝕂, 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. Let ℋ=(−1,−1)_ℝ be the Hamilton quaternion algebra and ℋ¹={𝛼∈ℋ∶Nrd_ℝ(𝛼)=1}. Hence, given 𝛼=𝑎₀+𝑎₁𝑖+𝑎₂𝑗+𝑎₃𝑘∈ℋ¹, from (2.1), we have Nrd_ℝ(𝛼)=𝑎²⁰−𝑎𝑎²¹−𝑏𝑎²²+𝑎𝑏𝑎²³𝑘=𝑎²⁰+𝑎²¹+𝑎²²+𝑎²³=1, which implies that 𝑎²⁰=1−𝑎²¹−𝑎²²−𝑎²³ and so |𝑎₀|≤1. Now, from (2.2), it follows that Trd_ℝ(𝛼)=2𝑎₀, and so Trd_ℝ(𝛼)=2𝑎₀∈[−2,2]. Therefore, Trd_ℝ(ℋ¹)=[−2,2].

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∈𝐼_𝕂−{0}, then 𝒪={𝑎₀+𝑎₁𝑖+𝑎₂𝑗+𝑎₃𝑖𝑗∶𝑎₀,𝑎₁,𝑎₂,𝑎₃∈𝐼_𝕂} is an order in 𝒜 denoted by 𝒪=(𝑎,𝑏)_𝐼𝕂.

Example 2.3. Given ℋ=(−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

Let 𝒜=(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 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 {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 order 𝒪 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(𝑥_𝑖,𝑥⁻_𝑗))∶0≤𝑖,𝑗≤3}.

Example 2.4. Let 𝒜=(𝑎,𝑏)_𝕂, and let 𝐼_𝕂 be the ring of integers of 𝕂, where 𝑎, 𝑏∈𝐼^∗𝕂=𝐼_𝕂−{0}. Then, [16], 𝒪={𝑥₀+𝑥₁𝑖+𝑥₂𝑗+𝑥₃𝑘∶𝑥₀,𝑥₁,𝑥₂,𝑥₃∈𝐼_𝕂} is an order in 𝒜 denoted by 𝒪=(𝑎,𝑏)_𝐼𝕂. The discriminant of 𝒪 is the principal ideal 𝑅⋅det(𝑇𝑟(𝑥_𝑖,𝑥⁻_𝑗)), where {𝑥₀,𝑥₁,𝑥₂,𝑥₃}={1,𝑖,𝑗,𝑘}, [14]. On the other hand, it is not difficult to see that Tr(𝑥𝑥_𝑖−𝑗) is the following diagonal matrix: ⎛⎜⎜⎜⎜⎜⎜⎝2000−0002𝑎−0002𝑏2000𝑎𝑏⎞⎟⎟⎟⎟⎟⎟⎠.(2.10) Therefore, 𝑑(𝒪)=4𝑎𝑏.(2.11)

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, [14].

If ℳ 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. Let 𝒜 be an algebra 𝒜=(√2,−1)_√ℚ(2) with a basis {1,𝑖,𝑗,𝑘} satisfying 𝑖=√⁴2,  𝑗=Im, and 𝑘=√⁴2Im 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/2^𝑛∶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 plane ℍ²={𝑧∈ℂ∶Im(𝑧)>0} endowed with the Riemannian metric 𝑑𝑠=√𝑑𝑥²+𝑑𝑦²𝑦,(2.12) 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 𝑧⟶𝑐𝑎𝑧𝑧++𝑑𝑏∶𝑎,𝑏,𝑐,𝑑∈ℝ,𝑎𝑑−𝑏𝑐=1.(2.13) 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 of ℂ 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, PSL(2,ℝ)≈SL(2,ℝ)±𝐼₂.(2.14)

A Fuchsian group Γ 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 disc 𝔻²={𝑧∈ℂ∶|𝑧|<1} endowed with the Riemannian metric 𝑑𝑠²=4𝑑𝑥²+𝑑𝑦²1−𝑥²+𝑦²²,(2.15) where 𝑧=𝑥+𝑦Im. Analogously, the discrete group Γ_𝑝 of orientation-preserving isometries 𝑇∶𝔻²→𝔻² is also a Fuchsian group, given by the transformations 𝑇_𝑝∈Γ_𝑝<PSL(2,ℂ) such that 𝑇_𝑝(𝑧)=𝑎𝑧+𝑐₋𝑐𝑧+𝑎⁻,𝑎,𝑏∈ℂ,|𝑎|²−|𝑐|²=1.(2.16) Furthermore, we may write 𝑇_𝑝=𝑓∘𝑇∘𝑓⁻¹, where 𝑇∈PSL(2,ℝ), and 𝑓∶ℍ²→𝔻² is an isometry given by 𝑓(𝑧)=𝑧Im+1𝑧+Im.(2.17) 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 order 𝒪 in 𝒜, consider 𝒪¹ as the set 𝒪¹={𝛼∈𝒪∶Nrd_ℋ(𝛼)=1}. Note that 𝒪¹ is a multiplicative group.

Now, note that the Fuchsian groups may be obtained by the isomorphism 𝜌₁ in (2.5). In fact, from (2.6), we have Nrd_ℋ(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 Γ(𝒜,𝒪)=𝜌¹𝒪¹±𝐼𝑑₂<SL±(𝐼2𝑑,₂ℝ)≅PSL(2,ℝ).(2.18)

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 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.

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(Γ)={±T𝑟(𝑇)∶𝑇∈Γ}.

Theorem 2.7 (see [11, 16]). Let Γ 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)if 𝕂₁=ℚ(Tr(𝑇)∶𝑇∈Γ), then 𝕂₁ is an algebraic number field of finite degree, and Tr(Γ) is contained in 𝐼_𝕂1, the ring of integers of 𝕂₁;(2)if 𝜑 is an embedding of 𝕂₁ in ℂ such that 𝜑≠𝐼𝑑, then 𝜑(Tr(Γ)) is bounded in ℂ.

3. Identification of Γ₈ in Γ(𝒜,𝒪),𝒪⊂𝒜

In this section, we identify the arithmetic Fuchsian group Γ₈ 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 𝐼_𝕂 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 at 𝜑₁, and it is ramified at the remaining places.

Consider the Fuchsian group Γ₈, given a quaternion algebra 𝒜=(√2,−1)_𝕂, and the elements of 𝑇∈Γ are given by 𝑇=12^𝑠⎛⎜⎜⎝−𝑥𝑧_𝑙𝑙+𝑦_𝑙√2𝑧_𝑙+𝑤_𝑙√2+𝑤_𝑙√2𝑥_𝑙−𝑦_𝑙√2⎞⎟⎟⎠,(3.1) where 𝑠∈ℕ, 𝑥_𝑙,𝑦_𝑙,𝑧_𝑙,𝑤_𝑙∈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 𝜑_𝑖, except at 𝜑₁.

On the other hand, the order 𝒪=(√2,−1)_𝐼𝕂 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 [13], 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)_𝐼𝕂. Therefore, this is the order we are taking into consideration in the case of interest.

4. Quotient Rings of the Quaternion Order 𝒪=(√2,−1)_𝑅 Where 𝑅={𝛼/2^𝑚∶𝛼∈𝐼_𝕂,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 of 𝕂=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/2^𝑚∶𝛼∈ℤ[√2],𝑚∈ℕ} which realizes the complete labeling.

Observe that 𝒪={𝑎₀+𝑎₁𝑖+𝑎₂𝑗+𝑎₃𝑘∶𝑎_𝑖∈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 𝒪_ℤ. We may still verify that according to the definition of order, 𝒪_ℤ 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. Case 𝑔=2

Given the genus 𝑔=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∶𝑎_𝑖∈ℤ[√2],𝑖²=√2,𝑗²=−1,𝑘²=−√2}, the reduced norm of an element 𝛼=𝑎₀+𝑎₁𝑖+𝑎₂𝑗+𝑎₃𝑘∈𝒪 is given by Nrd_√ℤ[2](𝛼)=𝛼𝛼⁻=𝑎²⁰−√2𝑎²¹+𝑎²²−√2𝑎²³∈ℤ√2,(4.1) and it satisfies Nrd_√ℤ[2](𝛼)∈𝐼_𝕂=ℤ[√2]. Next, we verify in which cases this norm is an element belonging to ℤ.

Proposition 4.1. Given 𝛼=𝑎₀+𝑎₁𝑖+𝑎₂𝑗+𝑎₃𝑘∈𝒪, where 𝑎_𝑖=𝑥_𝑖+𝑦_𝑖√2, where 𝑥_𝑖,𝑦_𝑖∈ℤ, then Nrd_√ℤ[2](𝛼)∈ℤ if and only if 2𝑥₀𝑦₀−𝑥²¹−2𝑦²¹+2𝑥₂𝑦₂−𝑥²³−2𝑦²³=0. In this case, the norm is given by Nrd_√ℤ[2](𝛼)=𝑥²⁰+2𝑦²⁰−4𝑥₁𝑦₁+𝑥²²+2𝑦²²−4𝑥₃𝑦₃.

Proof. From (4.1), we have Nrd_√ℤ[2](𝛼)=𝛼𝛼⁻=𝑎²⁰−√2𝑎²¹+𝑎²²−√2𝑎²³. Since 𝑎_𝑖∈ℤ[√2], it may be written as 𝑎_𝑖=𝑥_𝑖+𝑦_𝑖√2, where 𝑥_𝑖,𝑦_𝑖∈ℤ. Thus, 𝑎^2𝑖=𝑥^2𝑖+2𝑦^2𝑖+2√2𝑥_𝑖𝑦_𝑖, and from this, it follows that Nrd_√ℤ[2](𝛼)=𝑎+²⁰−𝑥₂₂√2𝑎²¹+𝑎²²−√2𝑎²³=𝑥²⁰+2𝑦²⁰+2√2𝑥₀𝑦₀−√2𝑥²¹+2𝑦²¹+2√2𝑥₁𝑦₁+2𝑦²²+2√2𝑥₂𝑦₂−√2𝑥²³+2𝑦²³+2√2𝑥₃𝑦₃=𝑥²⁰+2𝑦²⁰−4𝑥₁𝑦₁+𝑥²²+2𝑦²²−4𝑥₃𝑦₃+√22𝑥₀𝑦₀−𝑥²¹−2𝑦²¹+2𝑥₂𝑦₂−𝑥²³−2𝑦²³.(4.2)
Hence, Nrd_√ℤ[2](𝛼)∈ℤ if and only if 2𝑥₀𝑦₀−𝑥²¹−2𝑦²¹+2𝑥₂𝑦₂−𝑥²³−2𝑦²³=0, from which it follows that Nrd_√ℤ[2](𝛼)=𝑥²⁰+2𝑦²⁰−4𝑥₁𝑦₁+𝑥²²+2𝑦²²−4𝑥₃𝑦₃∈ℤ.(4.3)

Now, considering the order 𝒪 as an extension of ℤ, denoted by 𝒪_ℤ, the reduced norm of an element 𝛼=𝑎₀+𝑎₁𝑖+𝑎₂𝑗+𝑎₃𝑘∈𝒪_ℤ is given by Nrd_ℤ(𝛼)=𝛼𝛼⁻=𝑎𝑎₀₋₀−√2𝑎𝑎₁₋₁+𝑎𝑎₂₋₂−√2𝑎𝑎₃₋₃∈ℤ√2,(4.4) where 𝑎⁻_𝑖 denotes the conjugate of 𝑎_𝑖.

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

Proposition 4.2. Given 𝛼=𝑎₀+𝑎₁𝑖+𝑎₂𝑗+𝑎₃𝑘∈𝒪_ℤ, where 𝑎_𝑖=𝑥_𝑖+𝑦_𝑖√2, where 𝑥_𝑖,𝑦_𝑖∈ℤ, then Nrd_ℤ(𝛼)∈ℤ if and only if 𝑥²¹+𝑥²³−2(𝑦²¹+𝑦²³)=0. In this case, the norm is given by Nrd_ℤ(𝛼)=𝑥²⁰−2𝑦²⁰+𝑥²²−2𝑦²²∈ℤ.

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

Theorem 4.4. Let 0≠𝛼∈𝒪. If Nrd(𝛼)∈ℤ, then 𝒪/⟨𝛼⟩ has Nrd(𝛼)⁴ elements.

Proof. Let 0≠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,𝑖√2,𝑗,𝑗√2,𝑘,𝑘√2}. Thus, 𝛼∈𝒪 is of the form 𝛼=𝑎₀+𝑎₁√2+𝑎₂𝑖+𝑎₃𝑖√2+𝑎₄𝑗+a₅√2𝑗+𝑎₆𝑘+𝑎₇𝑘√2.
Now, given two elements 𝛽,𝛽′∈𝒪, 𝛽𝛽=𝑏₀+𝑏₁√2+𝑏₂𝑖+𝑏₃𝑖√2+𝑏₄𝑗+𝑏₅√2𝑗+𝑏₆𝑘+𝑏₇𝑘√2,𝑏_𝑖∈ℤ,_=𝑏^0+𝑏^1√2+𝑏^2𝑖+𝑏^3𝑖√2+𝑏^4𝑗+𝑏^5√2𝑗+𝑏^6𝑘+𝑏^7𝑘√2,𝑏^𝑖∈ℤ,(4.5) we say that 𝛽 and 𝛽′ are congruent modulo 𝑁 if there exists 𝛽^=𝑏^0+𝑏^1√2+𝑏^2𝑖+𝑏^3𝑖√2+𝑏^4𝑗+𝑏^5√2𝑗+𝑏^6𝑘+𝑏^7𝑘√2,𝑏^𝑖∈ℤ,(4.6) such that 𝛽−𝛽′=𝛽^𝑁. Thus, 𝑏_𝑖−𝑏_𝑖′=𝑏^𝑖𝑁, for 𝑖=0,1,…,7, that is, 𝑏_𝑖≡𝑏_𝑖′(mod𝑁) which implies that there exist 𝑁 possibilities for each 𝑏_𝑖, and thus, 𝑁⁸ different equivalence classes modulo 𝑁.
Now, since Nrd(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: 0⟶⟨𝛼⟩⟨𝛼𝛼⁻⟩⟶⟨𝛼𝐴𝛼⁻⟩⟶⟨𝐴𝛼⟩⟶0.(4.7)
We denote the number of elements of 𝐴/⟨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 𝑓∶𝐴⟨𝛼⁻⟩⟶⟨𝛼⟨𝛼𝛼⁻⟩⟩,(4.8) 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 that 𝐴/⟨𝛼⁻⟩ and 𝐴/⟨𝛼⟩ have the same cardinality, that is, 𝑛=𝑚.

Example 4.5. Let 𝛼=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 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.(4.9)

Example 4.6. Given 𝛼=2+√2∈𝒪=(√2,−1)_√ℤ[2], from Proposition 4.1, we have Nrd_√ℤ[2](𝛼)=6+4√2∉ℤ. However, taking the order as an extension of ℤ, that is, 𝒪_ℤ=(√2,−1)_ℤ from Proposition 4.2, we have Nrd_ℤ(𝛼)=2, and 𝒪/⟨𝛼⟩ has 16 elements, given by 𝒪2+√2={+0,𝑘1,,1𝑖,+𝑗,𝑗𝑘+,𝑘1,+𝑖+𝑖,𝑗1++𝑘𝑗,,11++𝑖𝑘+,𝑖𝑗++𝑗𝑘,}𝑖+.𝑘,𝑗+𝑘,1+𝑖+𝑗,1+𝑖(4.10)

Remark 4.7. We are not interested in orders such as 𝒪_ℤ, for when the order 𝒪 is extended to the order 𝒪_ℤ, 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 𝐼_𝕂 is not an element in ℤ.

Corollary 4.8. If 𝛽∈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 Corollary 4.8 that ⟨𝛽⟩ generates a code with Nrd(𝛼)⁴/Nrd(𝛽)⁴ codewords, therefore, a subcode of 𝒪/⟨𝛼⟩, whose minimum distance 𝐷_𝛽(𝜂,𝜏)>𝐷_𝛼(𝜂,𝜏).

Example 4.9. Given 𝛼=1+2√2𝑗, from Proposition 4.1, we have Nrd_√ℤ[2](𝛼)=(1)²+(2√2)²=9. Now, 𝛼=1+2√2𝑗 may be written as 1+2√2𝑗=(√2+𝑗)²; hence, 𝛽 is a right divisor of 𝛼 and Nrd_√ℤ[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 Example 2.5, 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/2^𝑚∶𝛼∈ℤ[√2],𝑚∈ℕ}, which makes it maximal, hence, given 𝕂=ℚ(√2) and 𝒪=(√2,−1)_𝑅, where 𝑅={𝛼/2^𝑚∶𝛼∈ℤ[√2],𝑚∈ℕ} such that 𝒪=𝑎₀+𝑎¹2𝑖+𝑎2²𝑗+𝑎2³𝑘∶𝑎^𝑖∈ℤ√2,𝑖²=√2,𝑗²=−1,𝑘²=−√2,(4.11) we have that the reduced norm of an element 𝛼∈𝒪 is given by Nrd_𝑅(𝛼)=𝑎²⁰−12√2𝑎²¹+12𝑎²²−14√2𝑎²³∈ℤ√2.(4.12) Now, for the maximal order, the cardinality of the quotient ring satisfy the following results:

Theorem 4.10. Let 𝛼∈𝒪=(√2,−1)_𝑅, where 𝑅={𝛼/2^𝑚∶𝛼∈ℤ[√2],𝑚∈ℕ}. If Nrd_𝑅(𝛼)=2^𝑛, then 𝒪/⟨𝛼⟩ has just one element.

Proof. Let 𝛾∈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 Nrd_𝑅(𝛼)=2^𝑛, we have that 𝛼⁻𝛼=2^𝑛; hence, 𝛾 may be written as 𝛾=𝛾2^𝑛−𝛼𝛼,(4.13) that is, 𝛾≡0(mod𝛼). In particular, one may verify that 1≡0(mod𝛼), for 1=(1/2^𝑛)𝛼⁻𝛼.

Theorem 4.11. Let 𝛼∈𝒪=(√2,−1)_𝑅, where 𝑅={𝛼/2^𝑚∶𝛼∈ℤ[√2],𝑚∈ℕ}. If Nrd_𝑅(𝛼)≠2^𝑛, then 𝒪/⟨𝛼⟩ has Nrd_𝑅(𝛼)⁴ elements.

Proof. Let 0≠𝛼∈𝒪 and Nrd_𝑅(𝛼)≠2^𝑛, Nrd_𝑅(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 {1/2^𝑛,√2/2^𝑛,𝑖/2^𝑛,𝑖√2/2^𝑛,𝑗/2^𝑛,𝑗√2/2^𝑛,𝑘/2^𝑛,𝑘√2/2^𝑛}. Hence, the proof is analogous to the proof of Theorem 4.4.

Example 4.12. Let 𝛼=2∈𝒪_𝑅. Hence, from (4.12), we have that Nrd_𝑅(𝛼)=4, and by Theorem 4.10, it follows that 𝒪/⟨𝛼⟩ has just one element {0}.

Example 4.13. Let 𝛼=√2+(√2/2)𝑗∈𝒪_𝑅. Hence, from (4.12), we have that Nrd_𝑅(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. Let 0≠𝛼∈𝒪=(𝜃,−1)_𝐼𝕂. The distance in 𝒪 is the distance induced by the graph 𝐺_𝛼. Hence, if 𝜂,𝜏∈𝒪, then the distance is given by 𝐷_𝛼(𝜂,𝜏)=min||𝑥₁||+||𝑥₂||+2||𝑥₃||+2||𝑥₄||−2||𝑥₂𝑥₃||,(5.1) such that 𝜏−𝜂≡𝑥₁+𝑥₂𝑖+𝑥₃𝑗+𝑥₄𝑘(mod𝛼).

Example 5.2. For 𝑉=𝒪/⟨√2+𝑗⟩. If 𝜏=1 and 𝜂=𝑖, then 𝜏−𝜂=1−𝑖. Thus, 𝐷_𝛼(𝜂,𝜏)=2, if 𝜏=(√2/2)(1+𝑖) and 𝜂=(√2/2)(1−𝑖), then 𝜏−𝜂=√2𝑖≡𝑘(mod𝛼). Thus, 𝐷_𝛼(𝜂,𝜏)=2.

Given the distance 𝐷_𝛼, a graph generated by 𝛼∈𝒪 is defined as follows.

Definition 5.3. Let 0≠𝛼∈𝒪=(𝜃,−1)_𝐼𝕂. The graph generated by 𝛼 is defined as 𝐺_𝛼=(𝑉,𝐸), where (1)𝑉=𝒪/⟨𝛼⟩ denotes the set of vertices; (2)𝐸={(𝜂,𝜏)∈𝑉×𝑉∶𝐷_𝛼(𝜂,𝜏)=1} denotes the set of edges.

Example 5.4. Given 𝛼=√2+𝑗∈𝒪=(√2,−1)_√ℤ[2], from Proposition 4.1, the reduced norm is Nrd_√ℤ[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 points 𝜂 and 𝜏 in the graph is the least number of traversed edges connecting the signal point 𝜂 to the signal point 𝜏.

Given a graph 𝐺_𝛼 with a set of vertices 𝑉 and distance 𝐷_𝛼, a code in 𝐺_𝛼 is a nonempty subset 𝒞 of 𝐺_𝛼. The Voronoi region 𝑉_𝜂 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, 𝑉_𝜂={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 𝐵_𝑡(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 element 𝛼∈𝒪_𝑅, we may generate a code over a graph by use of the quotient ring 𝒪_𝑅/⟨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. For 𝑔= 2, given 𝛼=1+2√2𝑗, such that 𝛼∈𝒪_√ℤ(2)=(√2,−1)_√ℤ(2), the reduced norm is Nrd_√ℤ[2](𝛼)=9. Thus, from Theorem 4.4, the cardinality of the set of vertices 𝑉 is Nrd_√ℤ[2](𝛼)⁴=9⁴=6561. Note that 𝛼 may be written as 1+2√2𝑗=(√2+𝑗)², and so 𝛽 is a right divisor of 𝛼 and Nrd_√ℤ[2](𝛽)=3∈ℤ. Therefore, the code generated by 𝛽, ⟨𝛽⟩⊆𝒪 has Nrd_√ℤ[2](𝛼)⁴/Nrd_√ℤ[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+2√2𝑗)√2−3𝑗≡−3𝑗(mod𝛼). Thus, 𝐷_𝛼(𝜂,𝜏)=6. The minimum distance of this code is 𝐷_𝛼(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.