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ISRN Algebra
VolumeΒ 2012Β (2012), Article IDΒ 956017, 14 pages
http://dx.doi.org/10.5402/2012/956017
Research Article

Codes over Graphs Derived from Quotient Rings of the Quaternion Orders

1Departamento de MatemΓ‘tica, ICEx, UNIFAL, R. Gabriel Monteiro da Silva, 700 Centro, 37130-000 Alfenas, MG, Brazil
2Departamento 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 Ξ“8. 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 β„€2 is associated with a type of digital modulation known as quadrature amplitude modulation, QAM modulation, denoted by π‘₯𝑖(𝑑)=𝛼𝑖cos𝑀0𝑑+𝛽𝑖sin𝑀0𝑑, 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(𝑑)=𝐴cos(𝑀0𝑑+πœ™π‘–), where πœ™π‘– 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 𝑔β‰₯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 β„€2, 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 Ξ“8. 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 π’œ over 𝕂 is a 𝕂-vector space of dimension 4 with a 𝕂-base 𝔅={1,𝑖,𝑗,π‘˜}, where 𝑖2=π‘Ž, 𝑗2=𝑏, 𝑖𝑗=βˆ’π‘—π‘–=π‘˜, π‘Ž,π‘βˆˆπ•‚βˆ’{0}, and denoted by π’œ=(π‘Ž,𝑏)𝕂.

Let π›Όβˆˆπ’œ be given by 𝛼=π‘Ž0+π‘Ž1𝑖+π‘Ž2𝑗+π‘Ž3𝑖𝑗, where π‘Ž0,π‘Ž1,π‘Ž2,π‘Ž3βˆˆπ•‚. The conjugate of 𝛼, denoted by βˆ’π›Ό, is defined by βˆ’π›Ό=π‘Ž0βˆ’π‘Ž1π‘–βˆ’π‘Ž2π‘—βˆ’π‘Ž3𝑖𝑗. Thus, the reduced norm of π›Όβˆˆπ’œ, denoted by Nrdπ’œ(𝛼), or simply Nrd(𝛼) when there is no confusion, is defined by Nrd(𝛼)=π›Όβ‹…βˆ’π›Ό=π‘Ž20βˆ’π‘Žπ‘Ž21βˆ’π‘π‘Ž22+π‘Žπ‘π‘Ž23,(2.1) and the reduced trace of 𝛼 by Trd(𝛼)=𝛼+βˆ’π›Ό=2π‘Ž0.(2.2)

Notice that the reduced norm is a quadratic form such that NrdβˆΆπ’œβŸΆπ•‚,π›ΌβŸΌπ‘Ž20βˆ’π‘Žπ‘Ž21βˆ’π‘π‘Ž22+π‘Žπ‘π‘Ž23,(2.3) which may also be denoted by its normal form ⟨1,βˆ’π‘Ž,βˆ’π‘,π‘Žπ‘βŸ©.

Let π’œ=(π‘Ž,𝑏)𝕂 be a quaternion algebra over a field 𝕂 and πœ‘βˆΆπ•‚β†’π”½ a field homomorphism. Define π’œπœ‘=(πœ‘(π‘Ž),πœ‘(𝑏))πœ‘(𝕂),π’œπœ‘βŠ—π”½=(πœ‘(π‘Ž),πœ‘(𝑏))𝔽,(2.4) where π’œπœ‘βŠ—π”½ denotes the tensor product of the algebra π’œπœ‘ by the field 𝔽, [9]. Each homomorphism πœ‘ in the algebra π’œπœ‘=(πœ‘(π‘Ž),πœ‘(𝑏))πœ‘(𝕂) is called place of the quaternion algebra π’œ.

Let 𝕂 be a totally real algebraic number field of degree 𝑛. This means that the 𝑛 monomorphisms πœ‘π‘–, 𝑖=1,…,𝑛 are all real, that is, πœ‘π‘–(𝕂)βŠ‚β„. Therefore, the 𝑛 distinct places are defined by ℝ-isomorphisms 𝜌1βˆΆπ’œπœ‘1βŠ—β„βŸΆπ‘€2(ℝ),πœŒπ‘–βˆΆπ’œπœ‘π‘–βŠ—β„βŸΆβ„‹,(2.5) where πœ‘1 is the identity, πœ‘π‘– is an embedding of 𝕂 on ℝ, for 𝑖=1,…,𝑛, and β„‹ is a division subalgebra of 𝑀2√(𝕂(π‘Ž)). Hence, π’œ is not ramified at the place πœ‘1 and ramified at the places πœ‘π‘–, for 2≀𝑖≀𝑛.

Let Nrdβ„‹ and Trdβ„‹ be the reduced norm and the reduced trace in β„‹, respectively. Given π›Όβˆˆπ’œ, it is easy to verify that Nrdβ„‹ξ€·πœŒ(𝛼)=det1ξ€Έ(𝛼),Trdβ„‹ξ€·πœŒ(𝛼)=tr1ξ€Έ(𝛼).(2.6)

Now, from the identification of 𝛼𝑖 with πœ‘π‘–(𝛼𝑖), for 𝑖=0,1,2,3, it follows that for every 2≀𝑖≀𝑛, πœ‘π‘–ξ€·Nrdβ„‹ξ€Έ(𝛼)=Nrdβ„‹ξ€·πœŒπ‘–ξ€Έ(𝛼),πœ‘π‘–ξ€·Trdβ„‹ξ€Έ(𝛼)=Trdβ„‹ξ€·πœŒπ‘–ξ€Έ(𝛼).(2.7)

Furthermore, as the reduced norm of an element is given by the determinant of the isomorphism 𝜌1, one may verify that Nrdβ„‹(𝛼⋅𝛽)=Nrd(𝛼)β„‹β‹…Nrdβ„‹(𝛽),(2.8) for any 𝛼,π›½βˆˆπ’œ.

Proposition 2.1 (see [13]). Let π’œ=(π‘Ž,𝑏)𝕂 be a quaternion algebra with a basis {1,𝑖,𝑗,π‘˜},π‘Ÿβˆˆβ„•βˆ—, with π‘Ÿ fixed, and let 𝑅 be the set 𝛼𝑅=π‘Ÿπ‘šβˆΆπ›ΌβˆˆπΌπ•‚ξ‚‡,andπ‘šβˆˆβ„•(2.9) where 𝐼𝕂 is the ring of integers of 𝕂. Then π’ͺ={π‘₯=π‘₯0+π‘₯1𝑖+π‘₯2𝑗+π‘₯3π‘˜βˆΆπ‘₯0,π‘₯1,π‘₯2,π‘₯3βˆˆπ‘…} is an order in π’œ.

Proof. We have that 𝑅 is a subring of 𝕂 containing 𝐼𝕂 and that π’ͺ is an 𝑅-module. On the other hand, if π›½βˆˆπ•‚, then there exists π‘βˆˆβ„€βˆ’{0} such that π‘βˆˆπΌπ•‚. Therefore, for any π‘₯0,π‘₯1,π‘₯2,π‘₯3βˆˆπ•‚, there exists π‘π‘™βˆˆβ„€βˆ’{0} such that 𝑐𝑙π‘₯𝑙=π›Όπ‘™βˆˆπΌπ•‚, 𝑙=0,1,2,3. Thus, given π‘₯=π‘₯0+π‘₯1𝑖+π‘₯2𝑗+π‘₯3π‘˜βˆˆπ’œ, there exists π›Ύβˆˆπ•‚ such that π‘₯=𝛾π‘₯ξ…ž, with π‘₯ξ…žβˆˆπ’ͺ. Therefore, π’œ=𝕂π’ͺ, which shows that π’ͺ is an order in π’œ.

Example 2.2. Let β„‹=(βˆ’1,βˆ’1)ℝ be the Hamilton quaternion algebra and β„‹1={π›Όβˆˆβ„‹βˆΆNrdℝ(𝛼)=1}. Hence, given 𝛼=π‘Ž0+π‘Ž1𝑖+π‘Ž2𝑗+π‘Ž3π‘˜βˆˆβ„‹1, from (2.1), we have Nrdℝ(𝛼)=π‘Ž20βˆ’π‘Žπ‘Ž21βˆ’π‘π‘Ž22+π‘Žπ‘π‘Ž23π‘˜=π‘Ž20+π‘Ž21+π‘Ž22+π‘Ž23=1, which implies that π‘Ž20=1βˆ’π‘Ž21βˆ’π‘Ž22βˆ’π‘Ž23 and so |π‘Ž0|≀1. Now, from (2.2), it follows that Trdℝ(𝛼)=2π‘Ž0, and so Trdℝ(𝛼)=2π‘Ž0∈[βˆ’2,2]. Therefore, Trdℝ(β„‹1)=[βˆ’2,2].

Given π’œ, a quaternion algebra over 𝕂, and 𝑅, a ring of 𝕂, an 𝑅-order π’ͺ in π’œ is a subring with unity of π’œ which is a finitely generated 𝑅-module such that π’œ=𝕂π’ͺ. Hence, if π’œ=(π‘Ž,𝑏)𝕂 and 𝐼𝕂, the integer ring of 𝕂, where π‘Ž,π‘βˆˆπΌπ•‚βˆ’{0}, then π’ͺ={π‘Ž0+π‘Ž1𝑖+π‘Ž2𝑗+π‘Ž3π‘–π‘—βˆΆπ‘Ž0,π‘Ž1,π‘Ž2,π‘Ž3βˆˆπΌπ•‚} is an order in π’œ denoted by π’ͺ=(π‘Ž,𝑏)𝐼𝕂.

Example 2.3. Given β„‹=(βˆ’1,βˆ’1)ℝ the Hamilton quaternion algebra, the integer ring of ℝ is β„€, and the quaternion order β„‹[β„€]={π‘Ž0+π‘Ž1𝑖+π‘Ž2𝑗+π‘Ž3π‘–π‘—βˆΆπ‘Ž0,π‘Ž1,π‘Ž2,π‘Ž3βˆˆβ„€} is called the ring of integral Hamiltonian quaternions, or the Lipschitz integers.

2.2. Hyperbolic Lattices

Let π’œ=(π‘Ž,𝑏)𝕂 be a quaternion algebra over 𝕂, let 𝑅 be a ring of 𝕂, and let be π’ͺ an 𝑅-order in π’œ. We also call π’ͺ a hyperbolic lattice due to its identification with an arithmetic Fuchsian group.

The lattices π’ͺ are used as the basic entity in generating the signals of a signal constellation in the hyperbolic plane. Since π’ͺ is an order in π’œ, then there exists a basis {𝑒1,𝑒2,𝑒3,𝑒4} of π’œ and 𝑅-ideal π”ž such that π’ͺ=π”žπ‘’1βŠ•π‘…π‘’2βŠ•π‘…π‘’3βŠ•π‘…π‘’4, where βŠ• denotes direct sum. Note that by definition, given π‘₯,π‘¦βˆˆπ’ͺ, we have π‘₯β‹…π‘¦βˆˆπ’ͺ. Furthermore, since every π‘₯∈π’ͺ is integral over 𝑅, [14], it follows that Nrd(π‘₯), Trd(π‘₯)βˆˆπ‘…, [15].

An invariant of an order π’ͺ is its discriminant, 𝑑(π’ͺ). For that, let {π‘₯0,π‘₯1,π‘₯2,π‘₯3} be a set consisting of the generators of π’ͺ over 𝑅. The discriminant of π’ͺ is defined as the square root of the 𝑅-ideal generated by the set {det(Tr(π‘₯𝑖,βˆ’π‘₯𝑗))∢0≀𝑖,𝑗≀3}.

Example 2.4. Let π’œ=(π‘Ž,𝑏)𝕂, and let 𝐼𝕂 be the ring of integers of 𝕂, where π‘Ž, π‘βˆˆπΌβˆ—π•‚=πΌπ•‚βˆ’{0}. Then, [16], π’ͺ={π‘₯0+π‘₯1𝑖+π‘₯2𝑗+π‘₯3π‘˜βˆΆπ‘₯0,π‘₯1,π‘₯2,π‘₯3βˆˆπΌπ•‚} is an order in π’œ denoted by π’ͺ=(π‘Ž,𝑏)𝐼𝕂. The discriminant of π’ͺ is the principal ideal 𝑅⋅det(π‘‡π‘Ÿ(π‘₯𝑖,βˆ’π‘₯𝑗)), where {π‘₯0,π‘₯1,π‘₯2,π‘₯3}={1,𝑖,𝑗,π‘˜}, [14]. On the other hand, it is not difficult to see that Tr(π‘₯π‘–βˆ’π‘₯𝑗) is the following diagonal matrix: βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ 20000βˆ’2π‘Ž0000βˆ’2𝑏00002π‘Žπ‘.(2.10) Therefore, 𝑑(π’ͺ)=4π‘Žπ‘.(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 β„³ in a quaternion algebra π’œ is called maximal if β„³ is not contained in any other order in π’œ, [14].

If β„³ is a maximal order in π’œ containing another order π’ͺ, then the discriminant satisfies, [15], 𝑑(π’ͺ)=𝑑(β„³)β‹…[β„³βˆΆπ’ͺ],  𝑑(β„³)=𝑑(π’œ). Conversely, if 𝑑(π’ͺ)=𝑑(π’œ), then π’ͺ is a maximal order in π’œ.

Example 2.5. Let π’œ be an algebra βˆšπ’œ=(2,βˆ’1)βˆšβ„š(2) with a basis {1,𝑖,𝑗,π‘˜} satisfying 𝑖=4√2,  𝑗=Im, and π‘˜=4√2Im where Im denotes an imaginary unit, Im2=βˆ’1. Let us also consider the following order (Proposition 2.1 considers a more general case for π’ͺ) in π’œ, π’ͺ={π‘₯=π‘₯0+π‘₯1𝑖+π‘₯2𝑗+π‘₯3π‘˜βˆΆπ‘₯0,π‘₯1,π‘₯2,π‘₯3βˆˆπ‘…}, that is, √π’ͺ=(2,βˆ’1)𝑅, where 𝑅={π‘₯/2π‘›βˆšβˆΆπ‘₯βˆˆβ„€[2] and π‘›βˆˆβ„•}. Thus, by (2.11), βˆšπ‘‘(π’ͺ)=βˆ’2. Furthermore, βˆšπ‘‘(π’œ)=βˆ’2, [15]. Hence, π’ͺ is a maximal order in π’œ.

2.3. Arithmetic Fuchsian Groups

Consider the upper-half plane ℍ2={π‘§βˆˆβ„‚βˆΆIm(𝑧)>0} endowed with the Riemannian metric βˆšπ‘‘π‘ =𝑑π‘₯2+𝑑𝑦2𝑦,(2.12) where 𝑧=π‘₯+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 MΓΆbius transformations of β„‚ over itself as ξ‚†π‘§βŸΆπ‘Žπ‘§+𝑏.𝑐𝑧+π‘‘βˆΆπ‘Ž,𝑏,𝑐,π‘‘βˆˆβ„,π‘Žπ‘‘βˆ’π‘π‘=1(2.13) Consider the group of real matrices 𝑔=π‘Žπ‘π‘π‘‘ξ€Έ with det(𝑔)=π‘Žπ‘‘βˆ’π‘π‘=1, and Tr(𝑔)=π‘Ž+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 MΓΆbius transformations of β„‚ 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 Β±π‘”βˆˆSL(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 2Γ—2 identity matrix, that is, PSL(2,ℝ)β‰ˆSL(2,ℝ)±𝐼2ξ€Ύ.(2.14)

A Fuchsian group Ξ“ is a discrete subgroup of PSL(2,ℝ), that is, Ξ“ consists of the orientation-preserving isometries π‘‡βˆΆβ„2→ℍ2, acting on ℍ2 by homeomorphisms.

Another Euclidean model of the hyperbolic plane is given by the PoincarΓ© disc 𝔻2={π‘§βˆˆβ„‚βˆΆ|𝑧|<1} endowed with the Riemannian metric 𝑑𝑠2=4𝑑π‘₯2+𝑑𝑦2ξ€Έξ€Ίξ€·π‘₯1βˆ’2+𝑦2ξ€Έξ€»2,(2.15) where 𝑧=π‘₯+𝑦Im. Analogously, the discrete group Γ𝑝 of orientation-preserving isometries π‘‡βˆΆπ”»2→𝔻2 is also a Fuchsian group, given by the transformations π‘‡π‘βˆˆΞ“π‘<PSL(2,β„‚) such that 𝑇𝑝(𝑧)=π‘Žπ‘§+π‘βˆ’π‘π‘§+βˆ’π‘Ž,π‘Ž,π‘βˆˆβ„‚,|π‘Ž|2βˆ’|𝑐|2=1.(2.16) Furthermore, we may write 𝑇𝑝=π‘“βˆ˜π‘‡βˆ˜π‘“βˆ’1, where π‘‡βˆˆPSL(2,ℝ), and π‘“βˆΆβ„2→𝔻2 is an isometry given by 𝑓(𝑧)=𝑧Im+1.𝑧+Im(2.17) 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 π’ͺ in π’œ, consider π’ͺ1 as the set π’ͺ1={π›Όβˆˆπ’ͺ∢Nrdβ„‹(𝛼)=1}. Note that π’ͺ1 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 Nrdβ„‹(𝛼)=det(𝜌1(𝛼)). Furthermore, we know that π’ͺ1 is a multiplicative group, and so 𝜌1(π’ͺ1) is a subgroup of SL(2,ℝ), that is, 𝜌1(π’ͺ1)<SL(2,ℝ). Therefore, the group derived from a quaternion algebra π’œ=(π‘Ž,𝑏)𝕂 and whose order is π’ͺ, denoted by Ξ“(π’œ,π’ͺ), is given by πœŒΞ“(π’œ,π’ͺ)=1ξ€·π’ͺ1±𝐼𝑑2ξ€Ύ<SL(2,ℝ)±𝐼𝑑2ξ€Ύβ‰…PSL(2,ℝ).(2.18)

As a consequence, consider the following.

Theorem 2.6 (see [11]). Ξ“(π’œ,π’ͺ) 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 π’œ and an order π’ͺβŠ‚π’œ such that Ξ“ has finite index in Ξ“(π’œ,π’ͺ). The group Ξ“ 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(Ξ“)={Β±Tπ‘Ÿ(𝑇)βˆΆπ‘‡βˆˆΞ“}.

Theorem 2.7 (see [11, 16]). Let Ξ“ be a Fuchsian group where the fundamental region has finite area, that is, πœ‡(ℍ2/Ξ“)<∞. Then Ξ“ is derived from a quaternion algebra π’œ over a totally real number field 𝕂 if and only if the following conditions are satisfied by Ξ“: (1)if 𝕂1=β„š(Tr(𝑇)βˆΆπ‘‡βˆˆΞ“), then 𝕂1 is an algebraic number field of finite degree, and Tr(Ξ“) is contained in 𝐼𝕂1, the ring of integers of 𝕂1;(2)if πœ‘ is an embedding of 𝕂1 in β„‚ such that πœ‘β‰ πΌπ‘‘, then πœ‘(Tr(Ξ“)) is bounded in β„‚.

3. Identification of Ξ“8 in Ξ“(π’œ,π’ͺ),π’ͺβŠ‚π’œ

In this section, we identify the arithmetic Fuchsian group Ξ“8 derived from a quaternion algebra π’œ over a number field 𝕂, for [π•‚βˆΆβ„š]=2, where [π•‚βˆΆβ„š] denotes the degree of the field extension, and 𝑔=2 denotes the genus of the surface 𝔻2/Ξ“8 in a quaternion order.

From [17], if 𝑔=2, the arithmetic Fuchsian group Ξ“8 is derived from a quaternion algebra π’œ over a totally real number field βˆšπ•‚=β„š(2). The elements of the Fuchsian group Ξ“8 are identified, by an isomorphism, with the elements of the order √π’ͺ=(2,βˆ’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 at πœ‘1, and it is ramified at the remaining places.

Consider the Fuchsian group Ξ“8, given a quaternion algebra βˆšπ’œ=(2,βˆ’1)𝕂, and the elements of π‘‡βˆˆΞ“ are given by 1𝑇=2π‘ βŽ›βŽœβŽœβŽπ‘₯𝑙+π‘¦π‘™βˆš2𝑧𝑙+π‘€π‘™βˆš2βˆ’π‘§π‘™+π‘€π‘™βˆš2π‘₯π‘™βˆ’π‘¦π‘™βˆš2⎞⎟⎟⎠,(3.1) where π‘ βˆˆβ„•, π‘₯𝑙,𝑦𝑙,𝑧𝑙,π‘€π‘™βˆšβˆˆβ„€[2]. Since πœ‘1 is the identity, it follows that π’œβ‰ƒπ‘€2(𝕂) is not ramified at πœ‘1. Now, observe that √2 is square-free for βˆšπ•‚=β„š(2), that is, there is no π‘‘βˆˆπ•‚βˆ’{0} such that 𝑑2=√2. Therefore, π’œ is ramified at all places πœ‘π‘–, except at πœ‘1.

On the other hand, the order √π’ͺ=(2,βˆ’1)𝐼𝕂 is not a maximal order in the quaternion algebra βˆšπ’œ=(2,βˆ’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 π’ͺ in π’œ and that it is maximal. From [13], we have that √π’ͺ=(2,βˆ’1)𝑅, where 𝑅={𝛼/2π‘šβˆΆπ›ΌβˆˆI𝕂,mβˆˆβ„•} is a maximal order that contains √π’ͺ=(2,βˆ’1)𝐼𝕂. Therefore, this is the order we are taking into consideration in the case of interest.

4. Quotient Rings of the Quaternion Order √π’ͺ=(2,βˆ’1)𝑅 Where 𝑅={𝛼/2π‘šβˆΆπ›ΌβˆˆπΌπ•‚,π‘šβˆˆβ„•}

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 Ξ“8 is derived from a quaternion algebra over βˆšπ•‚=β„š(2), with the identification of the generators by the order √π’ͺ=(2,βˆ’1)𝐼𝕂. Thus, let βˆšπ•‚=β„š(2) and {1,𝑖,𝑗,π‘˜}={1,βˆšξ”2,Im,√2Im} be a basis of the quaternion algebra βˆšπ’œ=(2,βˆ’1)𝐼𝕂, where 𝑖2=√2, 𝑗2=βˆ’1,π‘˜=𝑖𝑗=√2Im.

The ring of integers of βˆšπ•‚=β„š(2) is βˆšβ„€[2]; hence, π’ͺ={π‘Ž0+π‘Ž1𝑖+π‘Ž2𝑗+π‘Ž3π‘˜βˆΆπ‘Ž0,π‘Ž1,π‘Ž2,π‘Ž3βˆšβˆˆβ„€[2]} is in fact an order in π’œ. Due to the simplicity of this order, we start with it and gradually extend it to the order √π’ͺ=(2,βˆ’1)𝑅, where 𝑅={𝛼/2π‘šβˆšβˆΆπ›Όβˆˆβ„€[2],π‘šβˆˆβ„•} which realizes the complete labeling.

Observe that π’ͺ={π‘Ž0+π‘Ž1𝑖+π‘Ž2𝑗+π‘Ž3π‘˜βˆΆπ‘Žπ‘–βˆšβˆˆβ„€[2]} is an extension of βˆšβ„€[2] of dimension 4, for it has {1,𝑖,𝑗,π‘˜} as its basis, and we have that π’ͺ is a subring of π’œ containing 1 and which is a finitely generated βˆšβ„€[2]-module. Now, if we look at the order π’ͺ as an extension of β„€, the dimension increases to 8, and the basis of π’ͺ over β„€ is given by {1,√√2,𝑖,2𝑖,𝑗,βˆšξ”2𝑗,π‘˜,√2π‘˜}. In this case, the order will be denoted by π’ͺβ„€. We may still verify that according to the definition of order, π’ͺβ„€ is a free β„€-module with rank 4𝑛=8, where βˆšπ‘›=[π•‚βˆΆβ„š]=[β„š(2)βˆΆβ„š]=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 Ξ“8 is derived from a quaternion algebra π’œ over a totally real number field βˆšπ•‚=β„š(2), and the elements of Ξ“8 are identified, via an isomorphism, with the elements of √π’ͺ=(2,βˆ’1)βˆšβ„€[2]. Hence, given βˆšπ•‚=β„š(2) and √π’ͺ=(2,βˆ’1)βˆšβ„€[2] such that π’ͺ={π‘Ž0+π‘Ž1𝑖+π‘Ž2𝑗+π‘Ž3π‘˜βˆΆπ‘Žπ‘–βˆšβˆˆβ„€[2],𝑖2=√2,𝑗2=βˆ’1,π‘˜2√=βˆ’2}, the reduced norm of an element 𝛼=π‘Ž0+π‘Ž1𝑖+π‘Ž2𝑗+π‘Ž3π‘˜βˆˆπ’ͺ is given by Nrdβˆšβ„€[2](𝛼)=π›Όβˆ’π›Ό=π‘Ž20βˆ’βˆš2π‘Ž21+π‘Ž22βˆ’βˆš2π‘Ž23ξ‚ƒβˆšβˆˆβ„€2ξ‚„,(4.1) and it satisfies Nrdβˆšβ„€[2](𝛼)βˆˆπΌπ•‚βˆš=β„€[2]. Next, we verify in which cases this norm is an element belonging to β„€.

Proposition 4.1. Given 𝛼=π‘Ž0+π‘Ž1𝑖+π‘Ž2𝑗+π‘Ž3π‘˜βˆˆπ’ͺ, where π‘Žπ‘–=π‘₯𝑖+π‘¦π‘–βˆš2, where π‘₯𝑖,π‘¦π‘–βˆˆβ„€, then Nrdβˆšβ„€[2](𝛼)βˆˆβ„€ if and only if 2π‘₯0𝑦0βˆ’π‘₯21βˆ’2𝑦21+2π‘₯2𝑦2βˆ’π‘₯23βˆ’2𝑦23=0. In this case, the norm is given by Nrdβˆšβ„€[2](𝛼)=π‘₯20+2𝑦20βˆ’4π‘₯1𝑦1+π‘₯22+2𝑦22βˆ’4π‘₯3𝑦3.

Proof. From (4.1), we have Nrdβˆšβ„€[2](𝛼)=π›Όβˆ’π›Ό=π‘Ž20βˆ’βˆš2π‘Ž21+π‘Ž22βˆ’βˆš2π‘Ž23. Since π‘Žπ‘–βˆšβˆˆβ„€[2], it may be written as π‘Žπ‘–=π‘₯𝑖+π‘¦π‘–βˆš2, where π‘₯𝑖,π‘¦π‘–βˆˆβ„€. Thus, π‘Ž2𝑖=π‘₯2𝑖+2𝑦2π‘–βˆš+22π‘₯𝑖𝑦𝑖, and from this, it follows that Nrdβˆšβ„€[2](𝛼)=π‘Ž20βˆ’βˆš2π‘Ž21+π‘Ž22βˆ’βˆš2π‘Ž23=π‘₯20+2𝑦20√+22π‘₯0𝑦0βˆ’βˆš2ξ‚€π‘₯21+2𝑦21√+22π‘₯1𝑦1+π‘₯22+2𝑦22√+22π‘₯2𝑦2βˆ’βˆš2ξ‚€π‘₯23+2𝑦23√+22π‘₯3𝑦3=π‘₯20+2𝑦20βˆ’4π‘₯1𝑦1+π‘₯22+2𝑦22βˆ’4π‘₯3𝑦3+√2ξ€·2π‘₯0𝑦0βˆ’π‘₯21βˆ’2𝑦21+2π‘₯2𝑦2βˆ’π‘₯23βˆ’2𝑦23ξ€Έ.(4.2)
Hence, Nrdβˆšβ„€[2](𝛼)βˆˆβ„€ if and only if 2π‘₯0𝑦0βˆ’π‘₯21βˆ’2𝑦21+2π‘₯2𝑦2βˆ’π‘₯23βˆ’2𝑦23=0, from which it follows that Nrdβˆšβ„€[2](𝛼)=π‘₯20+2𝑦20βˆ’4π‘₯1𝑦1+π‘₯22+2𝑦22βˆ’4π‘₯3𝑦3βˆˆβ„€.(4.3)

Now, considering the order π’ͺ as an extension of β„€, denoted by π’ͺβ„€, the reduced norm of an element 𝛼=π‘Ž0+π‘Ž1𝑖+π‘Ž2𝑗+π‘Ž3π‘˜βˆˆπ’ͺβ„€ is given by Nrdβ„€(𝛼)=π›Όβˆ’π›Ό=π‘Ž0βˆ’π‘Ž0βˆ’βˆš2π‘Ž1βˆ’π‘Ž1+π‘Ž2βˆ’π‘Ž2βˆ’βˆš2π‘Ž3βˆ’π‘Ž3ξ‚ƒβˆšβˆˆβ„€2ξ‚„,(4.4) where βˆ’π‘Žπ‘– denotes the conjugate of π‘Žπ‘–.

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

Proposition 4.2. Given 𝛼=π‘Ž0+π‘Ž1𝑖+π‘Ž2𝑗+π‘Ž3π‘˜βˆˆπ’ͺβ„€, where π‘Žπ‘–=π‘₯𝑖+π‘¦π‘–βˆš2, where π‘₯𝑖,π‘¦π‘–βˆˆβ„€, then Nrdβ„€(𝛼)βˆˆβ„€ if and only if π‘₯21+π‘₯23βˆ’2(𝑦21+𝑦23)=0. In this case, the norm is given by Nrdβ„€(𝛼)=π‘₯20βˆ’2𝑦20+π‘₯22βˆ’2𝑦22βˆˆβ„€.

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 0β‰ π›Όβˆˆπ’ͺ. If Nrd(𝛼)βˆˆβ„€, then π’ͺ/βŸ¨π›ΌβŸ© has Nrd(𝛼)4 elements.

Proof. Let 0β‰ π›Όβˆˆπ’ͺ and Nrd(𝛼)=π‘βˆˆβ„€. First, we show that π’ͺ/βŸ¨π‘βŸ© has 𝑁8 elements. As π‘βˆˆβ„€, let us consider π’ͺ over β„€. However, [π’ͺβˆΆβ„€]=8, and the basis of π’ͺ over β„€ is √{1,√2,𝑖,π‘–βˆš2,𝑗,π‘—βˆš2,π‘˜,π‘˜2}. Thus, π›Όβˆˆπ’ͺ is of the form 𝛼=π‘Ž0+π‘Ž1√2+π‘Ž2𝑖+π‘Ž3π‘–βˆš2+π‘Ž4𝑗+a5√2𝑗+π‘Ž6π‘˜+π‘Ž7π‘˜βˆš2.
Now, given two elements 𝛽,π›½β€²βˆˆπ’ͺ, 𝛽=𝑏0+𝑏1√2+𝑏2𝑖+𝑏3π‘–βˆš2+𝑏4𝑗+𝑏5√2𝑗+𝑏6π‘˜+𝑏7π‘˜βˆš2,π‘π‘–π›½βˆˆβ„€,ξ…ž=π‘ξ…ž0+π‘ξ…ž1√2+π‘ξ…ž2𝑖+π‘ξ…ž3π‘–βˆš2+π‘ξ…ž4𝑗+π‘ξ…ž5√2𝑗+π‘ξ…ž6π‘˜+π‘ξ…ž7π‘˜βˆš2,π‘ξ…žπ‘–βˆˆβ„€,(4.5) we say that 𝛽 and 𝛽′ are congruent modulo 𝑁 if there exists π›½ξ…žξ…ž=𝑏0ξ…žξ…ž+𝑏1ξ…žξ…žβˆš2+𝑏2ξ…žξ…žπ‘–+𝑏3ξ…žξ…žπ‘–βˆš2+𝑏4ξ…žξ…žπ‘—+𝑏5ξ…žξ…žβˆš2𝑗+𝑏6ξ…žξ…žπ‘˜+𝑏7ξ…žξ…žπ‘˜βˆš2,π‘π‘–ξ…žξ…žβˆˆβ„€,(4.6) such that π›½βˆ’π›½β€²=π›½ξ…žξ…žπ‘. Thus, π‘π‘–βˆ’π‘π‘–β€²=π‘π‘–ξ…žξ…žπ‘, 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, since Nrd(𝛼)=π›Όβˆ’π›Ό, we have the following chain of ideals: ⟨Nrd(𝛼)⟩=βŸ¨βˆ’π›Όπ›ΌβŸ©βŠ†βŸ¨π›ΌβŸ©. From the third isomorphism theorem for 𝐴-modules, [10], we have the following sequence of left 𝐴-module: 0βŸΆβŸ¨π›ΌβŸ©βŸ¨π›Όβˆ’βŸΆπ΄π›ΌβŸ©βŸ¨π›Όβˆ’βŸΆπ΄π›ΌβŸ©βŸ¨π›ΌβŸ©βŸΆ0.(4.7)
We denote the number of elements of 𝐴/βŸ¨π›ΌβŸ© by 𝑛 and the number of elements of βŸ¨π›ΌβŸ©/βŸ¨π›Όβˆ’π›ΌβŸ© by π‘š. 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=π‘›π‘š. If we prove that 𝑛=π‘š, we may finally conclude that 𝑛=Nrd(𝛼)4. Now, observe that the function π΄π‘“βˆΆβŸ¨βˆ’βŸΆπ›ΌβŸ©βŸ¨π›ΌβŸ©βŸ¨π›Όβˆ’,π›ΌβŸ©(4.8) defined by 𝑓(𝛽+βŸ¨βˆ’π›ΌβŸ©)=𝛽𝛼+βŸ¨π›Όβˆ’π›ΌβŸ©, is well defined, and it is an isomorphism of the left 𝐴-module. Therefore, π‘š is exactly the number of elements of 𝐴/βŸ¨βˆ’π›ΌβŸ©.
Finally, the quaternion conjugation is an antiautomorphism, which implies that 𝐴/βŸ¨βˆ’π›ΌβŸ© and 𝐴/βŸ¨π›ΌβŸ© have the same cardinality, that is, 𝑛=π‘š.

Example 4.5. Let βˆšπ›Ό=1+π‘—βˆˆπ’ͺ=(2,βˆ’1)βˆšβ„€[2], then Nrd(𝛼)=2. From Theorem 4.4, π’ͺ/βŸ¨π›ΌβŸ© has 16 elements, obtained by the quotient of the order π’ͺ and the ideal ⟨1+π‘—βŸ©, that is, we take the elements of π’ͺ and reduce them modulo (1+𝑗), obtaining π’ͺ=ξ‚†βˆšβŸ¨1+π‘—βŸ©0,1,√2,1+√2,𝑖,1+𝑖,√2+𝑖,ξ‚€βˆš2𝑖,1+ξ‚βˆš2+𝑖,1+√2𝑖,√2+ξ‚€βˆš2𝑖,1+2+βˆšξ‚€βˆš2𝑖,1+2ξ‚ξ‚€βˆšπ‘–,1+1+2ξ‚βˆšπ‘–,ξ‚€βˆš2+1+2ξ‚ξ‚€βˆšπ‘–,1+2+ξ‚€βˆš1+2𝑖.(4.9)

Example 4.6. Given βˆšπ›Ό=2+√2∈π’ͺ=(2,βˆ’1)βˆšβ„€[2], from Proposition 4.1, we have Nrdβˆšβ„€[2]√(𝛼)=6+42βˆ‰β„€. However, taking the order as an extension of β„€, that is, π’ͺβ„€βˆš=(2,βˆ’1)β„€ from Proposition 4.2, we have Nrdβ„€(𝛼)=2, and π’ͺ/βŸ¨π›ΌβŸ© has 16 elements, given by π’ͺξ‚¬βˆš2+2ξ‚­={0,1,𝑖,𝑗,π‘˜,1+𝑖,1+𝑗,1+π‘˜,𝑖+𝑗,𝑖+π‘˜,𝑗+π‘˜,1+𝑖+𝑗,1+𝑖+π‘˜,1+𝑗+π‘˜,𝑖+𝑗+π‘˜,1+𝑖+𝑗+π‘˜}.(4.10)

Remark 4.7. We are not interested in orders such as π’ͺβ„€, for when the order π’ͺ is extended to the order π’ͺβ„€, 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 β„€.

Corollary 4.8. If βˆšπ›½βˆˆπ’ͺ=(2,βˆ’1)βˆšβ„€[2] is a right divisor of 𝛼 and Nrd(𝛽)βˆˆβ„€, then the left ideal generated by 𝛽, βŸ¨π›½βŸ©βŠ†π’ͺ has Nrd(𝛼)4/Nrd(𝛽)4 elements.

Note from Corollary 4.8 that βŸ¨π›½βŸ© generates a code with Nrd(𝛼)4/Nrd(𝛽)4 codewords, therefore, a subcode of π’ͺ/βŸ¨π›ΌβŸ©, whose minimum distance 𝐷𝛽(πœ‚,𝜏)>𝐷𝛼(πœ‚,𝜏).

Example 4.9. Given βˆšπ›Ό=1+22𝑗, from Proposition 4.1, we have Nrdβˆšβ„€[2](𝛼)=(1)2√+(22)2=9. Now, βˆšπ›Ό=1+22𝑗 may be written as √1+2√2𝑗=(2+𝑗)2; hence, 𝛽 is a right divisor of 𝛼 and Nrdβˆšβ„€[2](𝛽)=3βˆˆβ„€, then the left ideal generated by 𝛽, βŸ¨π›½βŸ©βŠ†π’ͺ 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 √π’ͺ=(2,βˆ’1)βˆšβ„€[2] is not a maximal order. Therefore, we have to consider the order over the ring 𝑅={𝛼/2π‘šβˆšβˆΆπ›Όβˆˆβ„€[2],π‘šβˆˆβ„•}, which makes it maximal, hence, given βˆšπ•‚=β„š(2) and √π’ͺ=(2,βˆ’1)𝑅, where 𝑅={𝛼/2π‘šβˆšβˆΆπ›Όβˆˆβ„€[2],π‘šβˆˆβ„•} such that ξ‚†π‘Žπ’ͺ=0+π‘Ž12π‘Žπ‘–+22π‘Žπ‘—+32π‘˜βˆΆπ‘Žπ‘–ξ‚ƒβˆšβˆˆβ„€2ξ‚„,𝑖2=√2,𝑗2=βˆ’1,π‘˜2√=βˆ’2,(4.11) we have that the reduced norm of an element π›Όβˆˆπ’ͺ is given by Nrd𝑅(𝛼)=π‘Ž20βˆ’12√2π‘Ž21+12π‘Ž22βˆ’14√2π‘Ž23ξ‚ƒβˆšβˆˆβ„€2ξ‚„.(4.12) Now, for the maximal order, the cardinality of the quotient ring satisfy the following results:

Theorem 4.10. Let βˆšπ›Όβˆˆπ’ͺ=(2,βˆ’1)𝑅, where 𝑅={𝛼/2π‘šβˆšβˆΆπ›Όβˆˆβ„€[2],π‘šβˆˆβ„•}. If Nrd𝑅(𝛼)=2𝑛, then π’ͺ/βŸ¨π›ΌβŸ© has just one element.

Proof. Let π›Ύβˆˆπ’ͺ/βŸ¨π›ΌβŸ©. We have to show that 𝛾≑0(mod𝛼). To show that 𝛾≑0(mod𝛼) is equivalent to proving that 𝛾=π‘₯𝛼, where π‘₯∈π’ͺ. As Nrd𝑅(𝛼)=2𝑛, we have that βˆ’π›Όπ›Ό=2𝑛; hence, 𝛾 may be written as 𝛾𝛾=2π‘›βˆ’π›Όπ›Ό,(4.13) that is, 𝛾≑0(mod𝛼). In particular, one may verify that 1≑0(mod𝛼), for 1=(1/2𝑛)βˆ’π›Όπ›Ό.

Theorem 4.11. Let βˆšπ›Όβˆˆπ’ͺ=(2,βˆ’1)𝑅, where 𝑅={𝛼/2π‘šβˆšβˆΆπ›Όβˆˆβ„€[2],π‘šβˆˆβ„•}. If Nrd𝑅(𝛼)β‰ 2𝑛, then π’ͺ/βŸ¨π›ΌβŸ© has Nrd𝑅(𝛼)4 elements.

Proof. Let 0β‰ π›Όβˆˆπ’ͺ and Nrd𝑅(𝛼)β‰ 2𝑛, Nrd𝑅(𝛼)=π‘βˆˆβ„€. We have to show that the left π’ͺ-module π’ͺ/βŸ¨π‘βŸ© has 𝑁8 elements. As π‘βˆˆβ„€, let us take π’ͺ over β„€. However, [π’ͺβˆΆβ„€]=8 and the basis of π’ͺ over β„€ is {1/2𝑛,√2/2𝑛,𝑖/2π‘›βˆš,𝑖2/2𝑛,𝑗/2π‘›βˆš,𝑗2/2𝑛,π‘˜/2π‘›βˆš,π‘˜2/2𝑛}. Hence, the proof is analogous to the proof of Theorem 4.4.

Example 4.12. Let 𝛼=2∈π’ͺ𝑅. Hence, from (4.12), we have that Nrd𝑅(𝛼)=4, and by Theorem 4.10, it follows that π’ͺ/βŸ¨π›ΌβŸ© has just one element {0}.

Example 4.13. Let βˆšπ›Ό=√2+(2/2)π‘—βˆˆπ’ͺ𝑅. Hence, from (4.12), we have that Nrd𝑅(𝛼)=3 and by Theorem 4.11, it follows that π’ͺ/βŸ¨π›ΌβŸ© 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 0β‰ π›Όβˆˆπ’ͺ=(πœƒ,βˆ’1)𝐼𝕂. The distance in π’ͺ is the distance induced by the graph 𝐺𝛼. Hence, if πœ‚,𝜏∈π’ͺ, then the distance is given by 𝐷𝛼||π‘₯(πœ‚,𝜏)=min1||+||π‘₯2||||π‘₯+23||||π‘₯+24||||π‘₯βˆ’22π‘₯3||ξ€Ύ,(5.1) such that πœβˆ’πœ‚β‰‘π‘₯1+π‘₯2𝑖+π‘₯3𝑗+π‘₯4π‘˜(mod𝛼).

Example 5.2. For βˆšπ‘‰=π’ͺ/⟨2+π‘—βŸ©. If 𝜏=1 and πœ‚=𝑖, then πœβˆ’πœ‚=1βˆ’π‘–. Thus, 𝐷𝛼(πœ‚,𝜏)=2, if √𝜏=(2/2)(1+𝑖) and βˆšπœ‚=(2/2)(1βˆ’π‘–), then βˆšπœβˆ’πœ‚=2π‘–β‰‘π‘˜(mod𝛼). Thus, 𝐷𝛼(πœ‚,𝜏)=2.

Given the distance 𝐷𝛼, a graph generated by π›Όβˆˆπ’ͺ is defined as follows.

Definition 5.3. Let 0β‰ π›Όβˆˆπ’ͺ=(πœƒ,βˆ’1)𝐼𝕂. The graph generated by 𝛼 is defined as 𝐺𝛼=(𝑉,𝐸), where (1)𝑉=π’ͺ/βŸ¨π›ΌβŸ© denotes the set of vertices; (2)𝐸={(πœ‚,𝜏)βˆˆπ‘‰Γ—π‘‰βˆΆπ·π›Ό(πœ‚,𝜏)=1} denotes the set of edges.

Example 5.4. Given βˆšπ›Ό=√2+π‘—βˆˆπ’ͺ=(2,βˆ’1)βˆšβ„€[2], from Proposition 4.1, the reduced norm is Nrdβˆšβ„€[2](𝛼)=3. The set of vertices is βˆšπ‘‰=π’ͺ/⟨2+π‘—βŸ©, and the set of edges satisfies 𝐸.

Remark 5.5. Note that the distance between two signal points πœ‚ and 𝜏 in the graph is the least number of traversed edges connecting the signal point πœ‚ to the signal point 𝜏.

Given a graph 𝐺𝛼 with a set of vertices 𝑉 and distance 𝐷𝛼, a code in 𝐺𝛼 is a nonempty subset π’ž of 𝐺𝛼. The Voronoi region π‘‰πœ‚ associated with πœ‚βˆˆπ’ž is the subset consisting of the elements of 𝑉 for which πœ‚ is the closest signal point in π’ž, that is, π‘‰πœ‚={πœβˆˆπ‘‰;𝐷(πœ‚,𝜏)=𝐷(πœ‚,π’ž)}. The number 𝑑=max{𝐷(πœ‚,π’ž);πœ‚βˆˆπ‘‰} is called covering radius of the code. The covering radius is the least number 𝑑 such that each ball of radius 𝑑 centered at the signal points of π’ž, given by 𝐡𝑑(πœ‚)={πœβˆˆπ‘‰βˆΆπ·(πœ‚,𝜏)≀𝑑}, covers 𝑉. The number 𝛿=min{𝐷(πœ‚,𝜏)βˆΆπœ‚,πœβˆˆπ’ž,πœ‚β‰ πœ} is the minimum distance of π’ž, and 𝛿≀2𝑑+1; the equality holds when each ball of radius 𝑑 centered at the signal points of π’ž forms a partition of 𝑉. A code satisfying this property is called perfect and corrects 𝑑 errors. A code is called quasiperfect if the code is capable of correcting every error pattern up to 𝑑 errors and some patterns with 𝑑+1 errors and no errors greater than 𝑑+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 π›Όβˆˆπ’ͺ𝑅, we may generate a code over a graph by use of the quotient ring π’ͺ𝑅/βŸ¨π›ΌβŸ© 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 βˆšπ›Ό=1+22𝑗, such that π›Όβˆˆπ’ͺβˆšβ„€(2)√=(2,βˆ’1)βˆšβ„€(2), the reduced norm is Nrdβˆšβ„€[2](𝛼)=9. Thus, from Theorem 4.4, the cardinality of the set of vertices 𝑉 is Nrdβˆšβ„€[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 Nrdβˆšβ„€[2](𝛽)=3βˆˆβ„€. Therefore, the code generated by 𝛽, βŸ¨π›½βŸ©βŠ†π’ͺ has Nrdβˆšβ„€[2](𝛼)4/Nrdβˆšβ„€[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 πœ‚=0, then βˆšπœβˆ’πœ‚=√2+𝑗=(1+2√2𝑗)2βˆ’3π‘—β‰‘βˆ’3𝑗(mod𝛼). Thus, 𝐷𝛼(πœ‚,𝜏)=6. The minimum distance of this code is 𝐷𝛼(πœ‚,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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