Graphs, Tessellations, and Perfect Codes on Flat ... - Semantic Scholar

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 10, OCTOBER 2004

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Graphs, Tessellations, and Perfect Codes on Flat Tori Sueli I. R. Costa, Marcelo Muniz, Edson Agustini, and Reginaldo Palazzo, Senior Member, IEEE

Abstract—Quadrature amplitude modulation (QAM)-like signal sets are considered in this paper as coset constellations placed on regular graphs on surfaces known as flat tori. Such signal sets can be related to spherical, block, and trellis codes and may be viewed as geometrically uniform (GU) in the graph metric in a sense that extends the concept introduced by Forney [13]. Homogeneous signal sets of any order can then be labeled by a cyclic group, induced by translations on the Euclidean plane. We construct classes of perfect codes on square graphs including Lee spaces, and on hexagonal and triangular graphs, all on flat tori. Extension of this approach to higher dimensions is also considered. Index Terms—Codes on graphs, coset codes, flat torus, geometrically uniform (GU) codes, perfect codes, spherical codes.

I. INTRODUCTION

T

HIS paper is devoted to the design of quadrature amplitude modulation (QAM)-like signal sets, i.e., finite sets of points taken from a plane lattice, considered as coset codes with the graph distance induced by the Euclidean metric, and to their extension to higher dimensions. Such signal sets can be related to spherical, block, and trellis codes. Our approach is essentially geometric and will rely on the concept of geometrically uniform (GU) codes, introduced by Forney [13] which, although initially concerned with spherical, lattice, and coset codes, can be extended to any metric space . A set is a GU code there exists an isometry in if and only if for every such that and . A GU code is highly homogeneous in the sense that it has the same distance profile for each codeword, and the Voronoi regions are congruent. We are particularly interested in the case where can be isometrically labeled, i.e., labeled by a group of symmetries. The codes considered here are complete sets of coset repreof a lattice in . We focus spesentative of a sublattice and of the cially on sublattices of the integer lattice of , which also provides geometric hexagonal lattice insights into results concerning lattices in higher dimensions. Manuscript received November 9, 2001; revised May 12, 2004. This work was supported by FAPESP-Brazil 02/07473-7. The work of S. I. R. Costa was supported in part by CNPq 304573/2002-7. The work of M. Muniz was supported in part by FAPESP 97/12270-8. The work of E. Agustini was supported in part by FAPESP 97/12269-0. The work of R. Palazzo was supported in part by PROCAD/CAPES 0121/01-0 and CNPq 301416/85-0. S. I. R. Costa is with the Instituto de Matemática, UNICAMP. CEP 13081970, Campinas, SP, Brazil (e-mail: [email protected]). M. Muniz is with the Centro Politécnico, Universidade Federal do Paraná, CEP 81531-990, Curitiba, PR, Brazil (e-mail: [email protected]). E. Agustini is with the Faculdade de Matemática, Universidade Federal de Uberlândia, CEP 38408-100, Uberlândia, MG, Brazil (e-mail: agustini@ ufu.br). R. Palazzo is with the Departamento de Telemática, UNICAMP. CEP 13081970, Campinas, SP, Brazil (e-mail: [email protected]). Communicated by G. Battail, Associate Editor At Large. Digital Object Identifier 10.1109/TIT.2004.834754

in can be viewed as vertices of The coset classes of a graph with the graph distance between two of these classes taken as the smallest distance between any two of their representatives. This leads us to identify the borders of a fundamental enclosing the code (a parallelogram in the plane, region of ). Such an identification occurs in the or parallelotopes in usual treatment of QAM codes: although the Voronoi regions of the boundary points are not congruent with those of the interior points, this “boundary effect” can be ignored if the number of points is great enough. Hence, QAM codes and other codes taken from lattices may be considered as GU codes—but this works only locally because they are not GU in the plane. Moreover, binary labelings of QAM sets are not normally associated with isometries. For these reasons, we propose the use of a geometrical model endowing QAM-like signal sets with a true GU structure, with the associate natural isometric labelings (e.g., labelings induced by translations in a Euclidean space). This model is based on an -dimensional surface known as a flat torus [3] which has zero . curvature in the sense that it can be locally “flattened” in , our approach consists of considering When restricted to planar codes as belonging to a two-dimensional flat torus. A two-dimensional flat torus is a surface generated by the identification of the opposite sides of a parallelogram [19]. It can be algebraically defined as the quotient of the Euclidean plane by the group of translations generated by two linearly independent vectors. If the flat torus is generated by a rectangle (i.e., if the vectors defining the translations are orthogonal) it can be embedded in the surface of a hypersphere of the four-dimensional Euclidean space. Thus, a signal set of this flat torus actually defines a spherical code. To illustrate this, first consider a rectangle with just one pair of opposite sides identified. This geometric object can be isometrically mapped (i.e., without “stretching”) onto a cylinder in the three-dimensional space. This cylinder can be considered flat in the sense that it can be flattened and any measure of length or area of a figure on it can be taken after it has been “opened” into the rectangle from which it was constructed. In this same way, a rectangle with the two pairs of opposite sides identified can be isometrically mapped onto a surface in the four-dimensional space which is contained in a hypersphere. This surface can be viewed (topologically) as the standard torus of the three-dimensional space, although differing from it because, like a cylinder, it can also be flattened. For example, the length of any curve on the flat torus can be measured without distortion, considering its plane counterpart in the rectangle. Signal sets on it, which are spherical codes in , can also be analyzed as plane constellations, if we consider the proper identification of the rectangle boundaries. Plane isometries and tessellations can then be used to design and label spherical codes in . We introduced this approach

0018-9448/04$20.00 © 2004 IEEE

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Fig. 1. On the top, a topological view of the flat torus as the standard three-dimensional torus is obtained by identification of the opposite sides of a parallelogram in two steps. On the bottom, the distance d on the flat torus is viewed as the Euclidean distance d in : d (a ; b) = d(a; b ) but d (a ; c ) = d(a ; c ).

in [8], [9], where we considered a class of spherical codes in which are cyclically labeled and discussed its minimum-distance performance. Higher dimensional flat tori are built as the quotient space by a group of translations, in an analogous way. If these of translations are mutually orthogonal, signal sets of the -dimensional flat torus generated by some -dimensional lattice will define -dimensional spherical codes. Flat tori thus appear as intermediates in the design of spherical codes generated by lattices of the Euclidean space, a feature which provides a strong motivation for studying codes on flat tori. As a special application, let us consider a torus constructed . from an -dimensional cube mapped into a sphere of are thus mapped onto highly homogenous Straight lines of . The resulting “stretching,” together spherical curves of with the possibility of “flattening ” these curves, were used in [21] to design a coding/decoding system for a continuous-alphabet, discrete-time source in the presence of additive Gaussian noise. The present paper is organized as follows. In Section II, we introduce the construction of an -dimensional flat torus and show how it can be embedded in a higher dimensional sphere. In Section III, we focus the analysis on dimension two and vertex signal sets generated by a regular tessellation of a flat torus by squares, deducing their main characteristics, namely labelings, expressions for the graph metric, and distance profiles. Special , partition-based GU codes examples including the Lee space [13], coset codes [12], [7], and codes over the Gaussian integers with the Mannheim distance [15] are viewed in this context. New constructions of perfect codes on a class of flat tori graphs are proposed. Section IV is dewhich include the Lee spaces voted to triangular and hexagonal tessellation graphs on flat tori. We deal with codes of Shikhande-type and triangular–hexagonal graphs, stating their main characteristics and presenting classes of perfect codes on them. In Sections V and VI, extensions to higher dimensions and block codes related to the approach proposed in [15] and [16] are developed. Finally, in Section VII, concluding remarks are given. II. THE FLAT TORUS Given a basis of , the flat torus is , where algebraically defined as the quotient space is the lattice generated by .

It may be defined through a modulus function (1) where and denotes the floor of , i.e., the greatest integer less than or equal to . Two vectors and of are thus in the same coset if and only if (i.e., , which means

The Euclidean distance in induces a distance on the in a natural way [19], [14]. The distance measured flat torus on the flat torus between two cosets and of and , with , is (2) where is the Euclidean vector norm in . can be characterized as the Geometrically, the flat torus by the group of translations generated by , also quotient of . For and this quotient denoted by can be viewed as the parallelogram generated by and with the opposite sides identified (this parallelogram contains all the coset representatives with redundancy in the border). and shows the distances Fig. 1 illustrates a flat torus for and , where , , and are the cosets of , , . and , respectively, A. Embedding a Flat Torus on a Sphere , the flat torus can also be viewed as the standard For torus surface in the three-dimensional Euclidean space (Fig. 1). However, it can be distinguished from this last one as being like a cylinder in : it is perfectly homogeneous (no point can be distinguished from another one) and can be cut and flattened into a rectangle (i.e., it is locally isometric to the plane). Differently from the cylinder, which is obtained from the identification of just two sides of a rectangle and can be isometrically realized in the usual space, the flat torus, being completely closed, can , . In only be mapped isometrically as a surface in [9] and [8], we have considered the identification of special flat

COSTA et al.: GRAPHS, TESSELLATIONS AND PERFECT CODES ON FLAT TORI

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tori with two-dimensional surfaces contained in a three-dimensional sphere to construct some Slepian-type group codes in . We describe next how a flat torus constructed from a rectangle can be mapped into . Our intention is to illustrate how some results obtained here can possibly be read in the -dimensional Euclidean space. , , If we consider the plane vectors , and the adjacent rectangular region , the can be realized starting from the map flat torus

supported on , with its paras the fundamental polytope in allel boundary faces identified. If is an orthogonal basis there into a sphere of which can be easily is an embedding of described in terms of the basis . For

We first note that

if, and only if, ; . Therefore, and precisely identifies the opposite sides of the rectangle. Formally, induces a one-to-one correspondence we can say that between and the surface in . This correspondence is, in fact, an isometry, that is, a distance-preserving map when we consider the geodesic distance . This means that if and , then in is precisely the length of the shortest path on connecting and . This last assertion can be deduced from the relations , that is,

and

([19]). The flat torus generated by a rectangle supported on plane and , , , can be vectors placed in through a local isometry which is the , , composition of with a rotation of that is,

where

We remark that the surface

is contained in a

. That is, as it is three-dimensional sphere of radius usual to say in Riemannian Geometry, the flat torus can be isometrically embedded in a hypersphere of . , the torus is generated by a square and we have: If

We deal in this paper with flat tori and study signal sets placed on them considering their labelings through plane isometries, lattices, and tessellations in their “flattened” counterparts. The construction described above shows that those signal sets and associated results can, in some cases, be used to design spherical codes with special labelings and decoding procedures, as was done in [21]. B. Tessellations and Graphs on Flat Tori Tessellations of by regular polytopes and associated graphs given by their edges may, under certain conditions, induce graphs and tessellations on flat tori. Let us consider a generated by a basis , the associated quotient flat torus induced by as defined by (1) map

in terms of the polytope in supported on . A crucial fact to be used is that is injective and is a local isometry when restricted to the region inside this polytope. This implies that any length, area, or -dimensional volume induced by (2) on on its flattened counterpart the flat torus can be measured in inside . It is why we say that the flat torus, like a cylinder, can be flattened. we may ask under what conditions Given a tessellation of induces a tessellation in . For , we will the map consider tessellations of the plane by squares (lattice ), regular hexagons and triangles and discuss the possibility that they induce regular graphs and tessellations on a two-dimensional flat torus. We start by summarizing some definitions and stating general results on tessellations. be a discrete group of isometries of a Definition 1: Let is called a fundamental remetric space . A subset of gion associated with if and only if: a) b)

(3)

( is defined as the largest open set contained in ); c) . given by copies of (referred to as tiles) The covering of associated under the action of is called a tessellation of with , or -tessellation.

This procedure can be generalized to higher dimensions. The -dimensional flat torus , is then viewed

All tessellations we consider here are induced by lattices in . A given lattice induces two tessellations of in a natural way. The discrete group is the same in both cases,

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, the group of translations generated by , but we may consider as the tile either the polytope supported on the basis or the Voronoi region of a vertex. As an example, for , , , and , generates either a lozenge or a hexagonal tessellation of the plane (see Fig. 10). We deal with both in Section IV. For and , induces two tessellations of by squares which are congruent. We state next conditions to obtain a tessellation on a flat torus induced by the tessellation associated with a lattice in . The proof is included in the Appendix. Proposition 2: Let and be bases of , and let be the -tessellation of which has as fundamental region the polytope supported on . If is a sublattice of , and is the quotient map on the flat torus, we can assert the following. a) induces a -tessellation on with fundamental , and . region induces a homogeneous graph on underlining , b) , the vertices and edges of which are induced by . Remark: In Proposition 2 one may consider different tiles in the tessellation induced by , For example, the fundamental . region can be the Voronoi region of a vertex of For the lattice , the result stated by the last induces a tessellation on the flat torus proposition says that (by unit squares) if , where and with , , , integers. Besides, the graph associated induces through the quotient map a graph and a with tessellation by squares on the flat torus . This tessellation can supported on tessellated be viewed as the parallelogram by with the opposite sides glued together. Fig. 4 illustrates and : we have 13 vertices and this for . In general, 13 squares on the tessellation of the flat torus and the the vertex set of the induced graph is given by number of vertices and squares in the flat torus is . This result is proved in a wider context in the next proposition. In the th what follows, we denote by . vector of the standard basis of Proposition 3: Let be a basis of and the associated flat torus. , the standard basis of , the lattice a) For induces (through the quotient map ) and a tessellation of the flat torus , a regular graph by unit hypercubes, if have integer coordinates. In this case: are the vertices of . b) c)

integer, is the union of the edges. d) integer, are the hypercubic tiles.

e) The number of vertices, , and the number of hypercubic are both equal to . tiles, , of Proof: Parts a)–d) are deduced from Proposition 2 speassociated with the discrete group of cialized to the lattice is the standard basis of isometries , where , the fundamental region is the unit hypercube in and is a sublattice of . The proof of e) is based on the homogeneity of the induced is obtained through a quotient of tessellation. Since the graph isometries, one cannot distinguish in any way a vertex from anwill inherit all the homogeneity other one. Besides, the graph of the tessellation associated with (which can be viewed as of the polytope with its boundaries the tessellation by inside the polyidentified). If we have a vertex of tope , all vertices of will have the same characteristics as (e.g., hypercubes meet at a vertex in the standard lattice one of those vertices). To clarify the geometrical arguments to be used, we start with low-dimensional cases. , and , , , , integers, Let and the parallelogram generated by and . and restricted to the interior of is Since a local isometry and is injective (one-to-one), it preserves area, equals that of , which is that is, the area of . On the other hand, since the tessellation of is by unit squares, this implies we have precisely squares in this tessellation. To see that this is also the number of vertices, we point out that the tessellation in the flat torus is perfectly homogeneous since the parallelogram boundaries are restricted to a tile is injective, each squared face identified. If vertices and each vertex on the flat torus will have also belongs to four faces as we have in . This implies that . Proceeding with the same reasoning for , we of unit cubes for the tessellation, get a number since the volume of the flat torus is the volume of the prism . The number of vertices of each generated by but again each vertex belongs cube on this tessellation is . to eight cubes. This implies that The proof for general follows in the same way. The flat torus is homogeneously tessellated by hypercubes and is a local isometry which is injective when restricted to the interior which implies that the -dimensional volume of is of equal to the -dimensional volume of , which is equal to the number of hypercubes of the tessellation. Supposing i) that is injective inside a region containing a hypercube of , has vertices and each and since each hypercube of in belongs to hypercubes, we get vertex of the graph . Next we show that the hypothesis i) can be dropped. is not injective on the hypercubic tiles of (some part If of the boundary of a tile is identified) we may consider a new which is a refinement of lattice

By Proposition 2, induces a new tessellation in with a of tiles equal to . Now we number have injectivity of the quotient map on each tile of the refined is equal to . tessellation, and hence its number of vertices


Fig. 2. A cyclic labeling for the graph 0 ,

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= fu; v g, u = (4; 3), and v = (05; 2).

Under this subdivision, we see immediately that . To also see that , we point out that each vertex of is either a vertex of or is of the form , or . Therefore, for each vertex there are new where vertices in , and this concludes the proof. III. REGULAR TESSELLATIONS BY SQUARES AND THEIR LABELINGS We now specialize to . The aim in this section is to construct regular graphs on two-dimensional flat tori with labelings induced by plane isometries and use them to design codes. Some of the results presented here were summarized in [1]. , plane translations associated with We consider , the tile , , and , , , , integers. The graph on the flat torus and its tessellation given by Proposition 3 have then vertices and the same number of unit squares. We point out also that, in the notation used in the context of coset codes [7], the correspond to the cosets of where is the vertices of and lattice

A. Labelings Induced by Translations Horizontal and vertical translations by one in the plane induce natural labelings on flat tori. We start with two examples. and , we get a flat torus i) For squares, tessellated by as illustrated on the right of Fig. 2. Note that in this case, the vertical segments of the graph are connected when we identify the parallelogram opposite sides and they form a closed curve (which is a knot) on the flat torus. The plain vertical translation by one unit on the plane induces an action on this flat torus: if we start from any vertex of the graph and go north step by step, all vertices will be reached (they all lie on the torus knot). This means that we , which have a cyclic group of isometries,

labels the whole regular graph . Fig. 2 also illustrates a cyclic labeling of the graph by in this case. ii) For , , and , the -tessellation is parallel to the square which generates the flat torus, and unit vertical (horizontal) translations starting vertices, and from any vertex produce a cycle with there are such cycles. is then naturally labeled by . We point out that in this case is known as the Lee space, an the graph metric is the Lee metric. The and illustration of Fig. 7 refers to the Lee space for generated by unit vertical and horizontal translations . starting at point The “vertical” labeling proposed in example i) can be exand . From tended to the general case a geometric viewpoint, this labeling can be done as follows: if , the image through the quotient map of any ver, , is a simple closed curve on the flat tical line vertices of , . torus which contains all the The labeling by is done going cyclically on this curve. We start from a vertex of and then go north from there, step by step, until the fundamental parallelogram border is reached. Then, crossing the border, we restart from the equivalent point in the parallelogram and go north again. We repeat this procedure ,a until the last point of the curve is reached. If can be done in the same way through horcyclic labeling of , a labeling going north on izontal lines. If a vertical line will reach only vertices of . (An analogous and horizontal labelings.) statement holds for An algebraic approach of a labeling procedure on a flat torus generated by and can be considered in the following way: the graph is naturally labeled by the . The next proposition, which is a particquotient group ular case of Proposition 22 to be presented in Section V, states a condition which assures a cyclic labeling. and and Proposition 4: Let . If , then the group is isomorphic to the cyclic group , the unit vertical translation is one of its . Mutatis generators and can be used to label isometrically and the unit mutandis, the same result holds for horizontal translation.

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Under the conditions of the this proposition we will denote by

, . the modulus function Note that through this labeling the neighbors at a distance from are

represented by the labeling of

by the vertical translation.

B. Induced Graph Metric on the Label Set

and

or metric on is defined as usual The graph distance as the minimum number of edges connecting two vertices. This distance is also induced by the Euclidean distance on the lattice. and in the graph For two vertices

We now consider a metric in metric on the flat torus

where and . In this section, we present a systematic way of determining for and also of visualizing the graph distance profile in , , and or . Since in by a cyclic group , , this case we can label in this is equivalent to finding out what is the metric which translates the graph distance of on the flat torus. We is induced by start from an important remark: given that (vertical) translations in the Euclidean plane which are isomemust satisfy tries in the flat torus, this metric

for any , . In order to get expressions for the labeling and the graph disintroduced in (1) which tance, we use the modulus function the representative of the associates with a vector supported on same class which lies inside the parallelogram . Expressing in terms of the basis , , and solving in and by Cramer formulas, we get and so the modulus function is expressed as (4) where denotes the integer part (floor). is the fundaand, of course, mental parallelogram which defines the torus since is an integer combination of and . The vertical translation , which establishes the cyclic la, can be used to label the vertices of beling when . In this way, the label of the vertex is the same as that of vertex defined as

in the fundamental parallelogram. In example i) illustrated in Fig. 2, we considered the cyclic labeling of the 23 vertices given by vertical translations, using

which translates the graph

Because of the homogeneity, we will conclude that, in order in that to deduce an analytical expression for the metric translates the graph metric

all we need is to know which are the neighbors of in . Besides and , we have to find and , . This is what is stated in the next proposition. , , . Proposition 5: Let is labeled by , , and for , the Then which translates the graph distance in , we metric in are , have that the four neighbors at distance of , , and , where is the least positive integer such that and for some . We call the “sum-modulus factor.” The solution can be obtained from any solution of the above system via Euclid’s algorithm. Proof: The first part of the assertion is given by Proposition 4. The neighbors at graph distance one from are , , , and , where . For any such that , we get the condition , , that is,

The first equation always has a solution, since A solution , can be determined via Euclid’s algorithm. in the second equation. It can In this way, we can obtain differ one from be easily checked that all the solutions another by multiples of . Since the labeling is defined in , we are interested in the smallest positive solution . This solution is the remainder of the division of by . We can visualize the graph setting a circular diagram for , nected with the vertices

through its cyclic labeling by , where each vertex is con, , and .


Fig. 3.

05 2).

Circular label diagram for the graph 0 , u = (4; 3), and v = (

;

The graph distance in this circular graph is precisely the distance induced by . This is assured by the homogeneity of the in flat torus and the isometries on it, induced by integer vertical translations. (Proposition 5), and In the example of Fig. 2, we have we may construct the labeling and the diagram illustrated in Fig. 3, where the vertices at distance are connected by either a line segment or an arc. In the next proposition we state the analytical expression for , which is simply deduced from taking the circular graph (with self-intersection) equivalent to . Proposition 6: Let , . Then the metric in is given by distance in

, , and which translates the graph

Proof: Let be a vertex of the graph. There are positive integers , such that

We have that

is the length of the shortest path connecting to in the counterclockwise direction, and that

is the length of the shortest path in the clockwise direction. Hence, we have

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Fig. 4.

A cyclic labeling for the graph 0 on the flat torus generated by u =

(3; 2) and v = (02; 3).

C. Tori Generated by Squares and Labelings of Spherical Codes If we consider a torus generated by a square, i.e., generated and , and integers, we have that by is a signal set of points. Fig. 4 illustrates the case , , and the vertical labeling of . Using the embedding of a torus generated by squares (Section II-A), we can place , which is a QAM-like signal set, naturally and homogeneously on a (hyper)sphere in the four-dimensional space. The Slepian- type spherical codes thus obtained have good performances when compared with -PSK -PSK four-dimensional sets [9]. Here we focus with the graph metric, sketching the data obtained for on specific subcases. Signal sets on flat tori generated by squares may have the same order but different performances. Consider, for instance, a Pythagorean triad , that is, , with , the vectors , , , , and the associated basis , . Then and have the same order although the later graph has a cyclic labeling whereas the former, which is the Lee space , does not (see [10]). (In fact, there is no Abelian labeling of the Lee spaces other than the trivial one [17]). , We point out also that in the special cases where prime, or , prime, the dealt with in this section reduce to the ones introsignal sets duced in [15] where the modulus relation is defined via complex , , , numbers: for where is the Gaussian integer obtained by rounding the real and imaginary parts of . In fact, we note that for and , the modulus function and the modulus relation above are equivalent. (Since in this last one we use “round” instead of floor function it is then natural to consider the fundamental square centered at the origin). It should be pointed out that in [15] and [16], the use of stronger algebraic structures are also powerful in the construction of block codes (Section VI). 1) Labelings of the Associated Spherical Codes: Embedprovide a class of cyclic labeled spherical dings of flat tori in codes of order , . Translations in the Euclidean plane correspond to rotations on the flat torus placed in a natural way. If we consider the labeling given by the in one step vertical translation in the plane it corresponds, through

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the isometry as defined by (3), to a rotation in the orthogonal given by the matrix group

(5) Since and are coprime, this matrix generates a cyclic . group of order An important consequence of the existence of four-dimensional isometries generated by horizontal and vertical translations in the plane is that the distance in four-dimensional space between two signals on a flat torus is compatible with the group in the plane. That is, structure induced by the additive group

Remark 7: We point out that for any integer it is possible , such that to find vectors and . This means that it is always possible to place of points which are on a flat torus a signal set of any number cyclically labeled (Proposition 4). A dual reading of Proposition 3 can be done after transforming and into orthogonal vectors and . This is obtained via the linear mapping determined by and . The previous tessellaassociated with tion by unit squares generated by the lattice is then taken into a tessellation by the canonical basis parallelograms associated with where . All results concerning the number of vertices, cyclic labeling, and expression for (graph) distance profile are preserved. This dual view allows the use of the embedding of any flat graph vertices will torus in a sphere of . In this case, the be taken into a cyclically labeled spherical code. The labeling described through a transversal translation will be realized in by a cyclic matrix of order . The performance analysis of is the object of a forthcodes based on those signal sets in coming paper. , . This 2) Special Subcase: subcase arises from a graph distance profile analysis of the which has very special properties. Through configuration of these properties we can establish connection with partition chains [12] and provide the basic setting for the construction of perfect codes (Section III-D). , , we can list the For following properties of the graph . i) The representatives of nearest to the origin are a full ball of radius in the graph metric. there are vertices at distance , ii) For each vertex of , that is, the ball of radius is complete in with this metric.

Fig. 5. Cyclic labeling of representatives of 0 for u = (4; 3) and v = 3 is complete in 0 with the graph metric.

(03; 4). Each ball of radius k, 1 k

Fig. 6.

=

A 100-point signal constellation based on partition chain

=2

, where

=

4 3

03 4

.

iii) On its cyclic labeling group , of are ,

. and

, the neighbors .

Example 8: We illustrate this subcase considering and . Fig. 5 shows by its representatives nearest . to the origin and its cyclic labeling by Remark 9: This subcase can be related to partition chains, which are used in modulation schemes associated with coset and trellis codes [12]. The above example, for instance, can be viewed as part of the construction of the partition chain where , and , where

In this case, we have a -point constellation partitioned in four -point constellations labeled by the cyclic group . Fig. 6 illustrates how a constellation of 100 points is partitioned and labeled through this scheme.


Fig. 7. The graph

hT ; T i 3

=4

.

S on the flat torus T

=

where G

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=

Fig. 9. The Voronoi polygon underlying the ball of radius k = 3 in the graph by the lattice group 3 , metric is the fundamental region of a tessellation in = w ; w , where w = (4; 3) and w = ( 3; 4). This tessellation is . used to determinate a perfect code of order 25 in the Lee space

f

0

Fig. 8. Labeling of 0 , u = (2; 2), and v = ( 2; 2), by a cyclic group of isometries of 0 that is not induced by a plane Euclidean group of isometries.

3) Other GU Codes Viewed on a Flat Torus: Other GU codes can also be viewed as signal sets in flat tori tessellated by squares with different labelings. a) Classes of partition-based GU codes: As an exin and ample let us consider the lattice , , and the groups , where is translation by , is rotation around the origin by , is the reflection on vertical , and is the reflection on horizontal axis axis through . These three groups of symmetry preserve and through in common. in have the subgroup the torus , , can be seen as the vertex set of a regular graph on . Fig. 7 illustrates the signal set and the flat torus . Each group induces a labeling of , all distinct in this case, , , and . As the for we have labelings suggest, we obtain signal sets that are realizations of and the Lee space as vertex sets of the Hamming space the same regular graph. and , b) The biorthogonal code: For consider the homogeneous distribution of eight points in the . Although the group corresponding flat torus , is not cyclic, has a cyclic labeling by a graph isometry which is not induced by any plane isometry. First, we define a one-to-one correspondence between the vertices of and the circular graph (Fig. 8). This graph has clearly an iso(clockwise rotation). Since and are metric action by graph-isometric, this induces a labeling which is an action of (as a group of graph isometries) in . If we identify the flat with the proper subset of , will be a biorthogonal torus code and this cyclic labeling will be realized by an orthogonal matrix.

g

0

D. Perfect Codes We recall now some usual definitions for graph codes [6]. Let be a graph with vertex set and the usual distance defined on it as the minimum number of edges connecting two vertices. A code in is a nonempty subset of . The ball of radius centered at in is the set . is The number called the minimum distance of ; is called an -error correcting code if . The distance of to is ; the Voronoi region defined as associated with is the subset formed by the elements of for which is the closest point in . The number is called the covering radius of . Clearly, the covering radius is the smallest number such that balls of radius centered at points cover . Those quantities are related by the inequality of ; and equality holds precisely when the balls around the points of form a partition of . A of radius code with this property is called perfect. It is said to correct errors. If we consider a ball of radius in the graph metric associated with a regular tessellation in , the condition on a code to be perfect is equivalent to have another tessellation with the Voronoi polygon underlying one of these balls (that is, the union of vertices of ) as a fundamental of all Voronoi regions in region. Example 10: In the graph metric given by the plane lattice , a ball of radius has 25 vertices. The Voronoi polygon underlying this ball is the fundamental region of a tessellation , where and by the lattice group , . This tessellation induces another one on the flat generated by , , and . torus induces (under a cyclic action) a The translation given by

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TABLE I EXAMPLES OF PERFECT CODES IN 0

-error correcting perfect code of order in the Lee space

(Fig. 9).

This last example comes from a more general result which can be stated as a consequence of Proposition 2. Proposition 11: (A recipe for perfect codes) Let be a by a lattice graph associated with a regular tessellation in group of rank , and the Voronoi polygon underlying a ball of radius in this graph metric. If there is a tessellation of by through a lattice group and if is a lattice group of , then induces a perfect -error correcrank , tion code on the flat torus with labeling group . We now consider by unit squares.

and the associated tessellation of

Corollary 12: Let be the graph associated with the tesselby . Let , lation of (or , ), , and let , , be the lattice generated by , , , , , . Then, if (or ), is a -perof order fect code in

Proof: A ball of radius in with the graph metric is a rotated square with vertices. The lattice generated by , gives rise to a plane tessellation with fundamental region , the Voronoi polygon underlying , which is a jagged square. For

we consider the equation to

Thus, if . So

, which is equivalent

,

for some is the smallest positive integer such that for .

Remark 13: Under the conditions of the above corollary, ex, we will have a perfect code in cept that with a noncyclic labeling group: .

Example 14: (Perfect Codes in Lee spaces) If in the above , and , corollary we consider and . we get Then is a -perfect code in the Lee space . These codes are algebraically described in [4, Theorem 13.25]. We also have -perfect codes in labeled by . In [18] it is shown that for they are the unique perfect codes in the two-dimensional Lee space. Example 15: In Table I, we give a few examples of perfect obtained by applying Corollary 12. codes in IV. TRIANGULAR AND HEXAGONAL GRAPHS ON FLAT TORI Hexagonal and triangular tessellations, as well as perfect codes on them, have been a benchmark for some channel and source coding applications (see [20]). In this section, we consider flat tori tessellations by triangles and hexagons. A special case to be discussed here (Sections IV-A and -B) is when the flat torus is generated by vectors and such that derives from by a rotation of . Such a flat torus, , , can be viewed either as a lozenge (rhombus) or a regular hexagon with its opposite sides identified [3]. This last approach comes from considering the torus coset classes inside . the hexagonal Voronoi region of a vertex of the lattice We now start from a grid on the plane generated by vectors and and consider the triangular tessellation and also a hexagonal tessellation associated with this grid. The groups of translations (crystallographic groups) associated with these tessellations are and , respectively, and we deand the underlying triangular and note by hexagonal planar graphs. Specializing Proposition 2 for this case, we can state the following. , and Proposition 16: Let be the associated equilateral triangle of side one. The triangular and induces a triangular regular graph, grid generated by , , and a tessellation by ( is the quotient map) on the flat torus generated by and if and , where , , , are integers. The is . number of vertices of


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and

Those flat tori may be called hexagonal since they can also , be obtained by the Voronoi set of a vertex of the lattice , which is an hexagon of side . We consider both the triangular and the hexagonal tessellations. In what follows , , and are used for the number of vertices, edges, and faces, respectively; is the number of sides of a face; and the number of edges meeting at each vertex. Fig. 10. A 1-perfect code of order 7 over a Shikhande-type graph 0 on the flat torus T , = f7w ; 7w g. The thick lines around the signals represent the balls of radius 1, and the gray tinted polygon underlying the ball is a Voronoi polygon on the flat torus.

Considering the graph distance, a ball of radius , , in the triangular tessellation on the plane has vertices. The Voronoi polygon of this ball is a jagged hexagon (see Fig. 10) which tessellates the plane via the and group of translations generated by . Hence, using the same arguments as those of Corollary 12, we can assert the following. Corollary 17: (A class of cyclically labeled perfect codes) , , , Let , and , . Then, (or if ), if (or ) is an -perfect code of order

in

,

Proposition 19: The triangular tessellation induces a graph , , on the torus generated by and which has the following characteristics: , the number of triangular i) the number of vertices is and the number of edges is ; faces is can be labeled by . ii) Proof: i) The area of the parallelogram that generates the torus must be equal to the area of the union of all its faces. This area is

times the area of a single face. From the regular graph relations , with and , we get for the number of vertices. Finally, the Euler relation for the flat torus takes care of the number of edges. Item can be labeled by ii) is stated by noting that the vertices of (periodicity ) the group generated by the translations (periodicity ). This labeling can be written as and

.

Remark 18: Under the conditions of the above corollary, except that , we will have a perfect code in with a noncyclic labeling group: . A. Shikhande-Type Graphs , , Consider the flat torus generated by . By Proposition 16, the triangular grid on the plane generated by and induces a graph and triangular tessellation on , . For this , is naturally labeled by is the Shikhande graph. This graph, . Cyclic -perfect codes in this graph, which we may call with this triangular graph metric, can be given by for This is a consequence of Corollary 17, since we have , , , . Fig. 10 illustrates the Shikhande-type graph for the associated -error correcting vertex code.

Remark 20: (Perfect codes on hexagonal tori) We note that if in the notation of Corollary 17 we take , , , , , we get and with , . Hence, is an -perfect code on the triangular graph given by the last . proposition. Fig. 11 illustrates a perfect code for Proposition 21: For , , , the hexagonal tessellation induces a graph , , in the flat torus which has the following characteristics: has faces, vertices, and edges; can be labeled by the semi-direct product ; there is no labeling by an Abelian group of isometries; iii) for even , there is a -perfect code of order

i) ii)

and

B. Triangular and Hexagonal Tessellations on Hexagonal Flat Tori The tessellations considered here are those induced on the flat tori generated by vectors

with labeling group . There is no perfect code correcting more than a single error on . The proof of this proposition is included in the Appendix. An example of a -perfect code (for ) of order in the hexagonal grid is illustrated in Fig. 12.

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is a prime number, we have a cyclic la, that is, the label is mapped to , (due to the fact that the order of any nonzero ). translation must be Something else can be said about the labeling group and, more generally, about . The order of an element of is determined in Proposition 23, which is developed in the Appendix, and therefore we can decide whether a translation generates or does not generate a labeling of . by Specializing the result of Proposition 23 to the tessellation by , unit hypercubes where is the standard basis of , we obtain the next proposition. and be a basis for , where . Let Let be the generating matrix of over , i.e., and (which is the order of ).

In particular, if by beling of

f

Fig. 11. A 1-perfect code over 0 , = 7(w + w ); 7(2w The thick lines around the signals delimitate the balls of radius 1.

0 w )g.

Proposition 22: Let be the generating matrix of over , and let be the absolute value of the cofactor of associated with the element . Then we have the following. and such that a) If, for some , there are , then the translation generated by . by provides a cyclic labeling of for some , , we also have that the b) If translation generated by provides a cyclic labeling of by . We remark that Proposition 4 is a particular case of Proposition 22 for , where is a generating matrix for , with respect to the basis : in this case, the order of is , where

f

0

g

Fig. 12. A 1-perfect code in 0 , = 6(w + w ); 6( w + 2w ) . The gray tinted polygon over the signal is a Voronoi polygon in the flat torus.

V. TESSELLATIONS AND GRAPHS ON FLAT TORI IN HIGHER DIMENSIONS At this point, we return to the concepts developed at the beginning of this paper. Although we treated only the planar case, similar results are valid in higher dimensions. To begin with, it should be mentioned that, as was stated in Proposition 2, given and , if is a sublattice of , then the two lattices -tessellation of (with the polytope spanned by as fun-tessellation in . The same damental region) induces a is valid for the associated graphs. and proceeding as in Proposition 3, we Considering obtain the following results: the standard -dimensional tesselinduces a graph lation by hypercubes given by the lattice and a tessellation on for any basis of vectors with integer is given by , coordinates. The number of vertices of where is the square matrix which has the elements of as is a labeling group for . columns vectors, and

Some results concerning the construction of spherical codes are also valid in higher dimensions. In Section II-A, it was shown that if is an orthogonal basis, there is an isometric embedding of the flat torus in a sphere in . A special case is a rotated “ -hypercubic” torus graph, which is obtained . Such whenever is an orthogonal frame where vertices . This genergraphs have always alizes the bidimensional case , where , and the graphs have vertices. If is odd, these frames are in general hard to find. As an these torus graphs can be given example, in dimension by where , , and , . For , we have a torus graph of 27 vertices and Proposition 23 can be used to show that in is labeled by , , and . For even dimensions, very special cases are and , where we also make use of the algebraic structure of quaternions and Cayley numbers. In both cases, we can find such a frame


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starting from any given vector of integer coordinates. In conwe may start from any vector . sidering (quaternion) and taking Identifying , , and , we obtain an orthogonal . In order to prove this last statement, basis we briefly digress to discuss two basic properties of the quaternionic norm. If is a unitary vector of , right (and left) multiplication by is an isometry . In fact, the conjugate of a quaternion is defined as . The , which is equal to the quaternionic norm of is Euclidean norm of . . Moreover, the quaternionic norm satisfies is an isometry, beHence, if is a unitary quaternion, then cause

construction of -error correcting block codes, keeping the geometrical interpretation of the errors, over fields not previously with The main difference is considered, such as that, in our labeling , the multiplicative structure of is not related to a multiplicative structure on the plane lattice. Therefore, one has to find the neighbors of and then find the equation that they satisfy in , with as small as possible. For instance, let . We consider the lattice with the graph given by the squared tessellation, and the sublatgenerated by , , and . tice is isomorphic to , which will be regarded The group as the field . We now define a block code over this field. In the general case , suppose that the neighbors of are roots of , and let be a primitive element of . The code defined by the parity-check matrix

Now let and consider the unitary . Since is an orthonormal vector is basis, an orthonormal basis (angles are also preserved by isometries). is an orthogonal Finally, basis. , the modulus function (1) that defines For is (1). Note that this function can also be expressed, like in the complex case, in terms of “Gaussian quaternionic integers,” by

corrects one error, where every neighbor of in satisfies the in . In fact, an error vector of weight equation one is a vector of the form , where is a neighbor of . A straightforward calculation shows that if and . and and Back to our example, the neighbors are their equation is . If we take a primitive element of , then the parity-check matrix

Besides providing a geometric construction of torus signal sets, the use of quaternion and Cayley algebraic structure is very likely to provide more tools in the construction of block codes, similarly to the complex case. A remarkable result from elementary number theory tells us that any integer can be written as a sum of four squares. This implies that, given any positive integer , we can construct graphs (and spherical codes) which are quotients of and have vertices. In particular, for prime, the natural labeling or . groups will be either VI. BLOCK CODES Given that we have dealt with metrics that are compatible with , we may also consider block codes over the labeling groups fields (or even modular rings). When a graph is labeled by a cyclic group , prime, we can consider as the field of elements and define a distance over by

Block codes for the above metric were studied in [15] and or . An important [16], where is a quotient field of step in the construction of these codes is that the labelings used there are also ring homomorphisms, and then the error vectors , where belongs to a specof weight one are of the type ified set of roots of unity. This is further extended in [11] to quotient fields of some algebraic integer rings , and the codes are made to correct one error coming from a specified group of roots of unity. The labelings defined here can also be used for the

defines a code over that corrects one error. We remark that these codes depend on the particular sublatcan lead to codes with different tice . A different choice of is genparameters, or even to no code at all. For instance, if erated by and , one of the neighbors is a primitive , and no code is obtained. element of VII. CONCLUSION Geometrically uniform codes have been considered here in a wider context than originally, to encompass signal sets in regular graphs arising from classes of Euclidean regular tessellations by and codes translations. These codes include the Lee space over Gauss integers with Mannheim distance (see [15]) and are related to spherical and coset codes. The contributions presented here include the following. • The placement of coset codes as homogeneous subsets of a surface (flat torus) thus allowing a strongly geometric view and extensions of previous results in this matter and connections with spherical codes. QAM-like signal sets can be viewed as perfectly homogeneous codes in this context. • Construction of homogeneous signal sets of any cardinality labeled by a cyclic group, and the establishment of the corresponding metric on this group. • Construction of classes of cyclically labeled perfect codes on triangular, square, and hexagonal graphs on flat tori. • Extension to higher dimensions and construction of block codes over finite fields in special cases. It was shown in [9] that codes on graphs on flat tori obtained by identifying the opposite sides of a parallelogram, when viewed as Slepian-type codes in , achieve better performance

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than standard -PSK -PSK signal sets. Good performance may also be expected for other signal sets endowed with the graph metric as proposed here. Directions for further research can be drawn from the fact that codes on graphs can always be placed as signal sets on a two-dimensional surface. When restricted to two dimensions, GU codes can then be obtained from tessellations on surfaces with constant curvature. Slepian group codes are built on a sphere (the simplest surface with genus and positive curvature). The “next” surface ) is the flat torus (zero curvature), considered in (genus this paper, which reveals more possibilities for regular graph codes. Moreover, we point out that even natural spaces like with the Hamming metric require, as graphs, surfaces of higher for ), and on homogeneous surfaces genus (e.g., , the distance underlying the graph of higher genus codes to be considered is the hyperbolic distance, rather than the Euclidean distance. Some preliminary results on the performance of signal sets on the hyperbolic plane, through which the higher genus surface are constructed, have been found quite encouraging ([5] and [2]). APPENDIX A. Proof of Proposition 2 a) The proof follows essentially from the fact that, under the restricted to a region involving any conditions required, is a local isometry (and, hence, is continuous tile of and open) and a bijection on the interior of . Therefore, has the same area as and also all conditions for as showed next. tessellations are fulfilled for acts as a discrete group of isometries To see that , we can show that each isometry induces on an isometry in the flat torus defined by . In fact, since is a commutative group of translations, we can write

(and ) . therefore is injective when restricted to iii) Since

b) The vertices and edges of will be defined as images by of vertices and edges of . The homogeneity of (the same number of edges and -dimensional polytopes starting from each vertex) follows since it is constructed by a quotient group of isometries. The proof that this graph underlines the tessellation on the flat torus follows from a) restricted to the interior of and, again, from the fact that a tile is an injective mapping. B. Order of an Element of the Labeling Group In what follows are bases for and of over is .

and , respectively, the generating matrix , where , and

Proposition 23: Let and be two bases of , and the lattices generated by and , respectively, and suppose that . Let be the generating matrix of over , a vector of , and the matrix obtained from by substituting for the th column of . Then the order of in is given by . , if and only if , Proof: For that is, if and only if there is an such that . Therefore, the order of an element is the least positive integer such that the system has a solution with integer coordinates (Cramer formulas). is invertible, by the Cramer formulas, the system Since has a unique solution given by This means that

To see that the group of the induced isometries is isomorphic to , we must show that if and only if

,

has the solution

Since , if then divides . . We have , and Now, let be the order of the unique solution of is given by . . Now, for another integer such Hence, divides each , let be given by that divides . Then , which implies that . This shows that , and that

Considering ( at the th position) in the preceding proposition, we get the conclusions of Proposition 22.

such that The conditions of Definition 1 are deduced as follows. i) ii)

. If

then

and for some

where ,

C. Proof of Proposition 21 with Item i) follows from the relations and , and from the Euler relation on the torus. In order to prove ii), we start proving that there is no labeling by an Abelian group of isometries. This proof deals with Cayley graphs arguments on groups of isometries (see [17] for a more detailed exposition). A Cayley graph on a group is a graph


that has

as vertex set and adjacency relation given by , where is a fixed set of generators of . If a group of isometries labels , this labeling induces a Cayley graph on which is isomorphic to this same graph . If were Abelian, we would have cycles of order in , which . cannot happen for , we start by remarking To establish a labeling for that half of the vertices can be labeled by the lattice , and the other half can be labeled by reflecting the previous vertices on the central axis perpendicular to . Our initial point is . The first half of the vertices can be . The second written as (where is the complex conhalf is given by ). The group of isometries associated with this lajugate of (the action beling is isomorphic to a semidirect product in that defines this product is given by of , where and ). iii) The ball of radius one on the plane hexagonal grid has four vertices. Its Voronoi polygon (a triangle) tessellates the plane with , , where and group , and represents the reflection group taking the Voronoi triangle into the other part of the lozenge and . If and , then generated by , , as a normal subgroup. Proposition 2 assures then the first part of assertion three. Finally, we observe , the Voronoi polygon of the ball of radius that for does not tessellate the plane, and so there is no -perfect code in a hexagonal grid. for ACKNOWLEDGMENT The authors wish to thank the Associate Editor at Large, G. Battail and the referees for pertinent comments and fruitful suggestions. REFERENCES [1] E. Agustini, S. I. R. Costa, and R. Palazzo Jr, “Codes on graphs on flat tori,” in Proc. VII Int. Workshop on Algebraic Coding Theory–ACCT 2000, Bansko, Bulgaria, June 18–24, 2000, pp. 11–14.

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[2] E. Agustini, “Signal sets on hyperbolic spaces,” Ph.D. dissertation (in Portuguese), State Univ. Campinas, Brazil, 2002. [3] M. Berger, Differential Geometry: Manifolds, Curves, and Surfaces. Berlin, Germany: Springer-Verlag, 1988. [4] E. R. Berlekamp, Algebraic Coding Theory. New York: McGraw-Hill, 1968. [5] E. Brandani, R. Palazzo Jr, and S. I. R. Costa, “Improving the performance of M-PAM signals constellations in the Euclidean space by embedding them in the hyperbolic space,” in Proc. 1998 Information Theory Workshop, Killarney, Ireland, 1998, pp. 98–100. [6] A. E. Brower, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs. New York: Springer-Verlag, 1989. [7] A. R. Calderbank and N. J. A. Sloane, “New trellis codes based on lattices and cosets,” IEEE Trans. Inform. Theory, vol. IT-33, pp. 177–195, Mar. 1987. [8] S. I. R. Costa, E. Agustini, and R. Palazzo Jr, “On knotted M-PSK correct reception performance,” in Proc. VII Int. Workshop–ACCT 2000, Bansko, Bulgaria, June 18–24, 2000, pp. 103–106. [9] S. I. R. Costa et al., “Slepian-type codes on flat tori,” in Proc. IEEE Int. Symp. Information Theory (ISIT-2000), Sorrento, Italy, June 25–30, 2000, pp. 58–58. [10] S. I. R. Costa, J. R. Gerônimo, R. Palazzo Jr, and M. Muniz, “The symin the Lee space and the —Linearity,” in Lecmetry group of ture Notes in Computer Science. New York: Springer-Verlag, 1997, vol. 1255, pp. 66–77. [11] X. Dong, C. B. Soh, and E. Gunawan, “Codes over finite fields for multidimensional signals,” J. Algebra, vol. 233, no. 1, pp. 105–121, Nov. 2000. [12] G. D. Forney, “Coset codes—Part I: Introduction and geometrical classification,” IEEE Trans. Inform. Theory, vol. 34, pp. 1123–1151, Sept. 1988. [13] , “Geometrically uniform codes,” IEEE Trans. Inform. Theory, vol. 37, pp. 1241–1260, Sept. 1991. [14] S. Gallot, D. Hulin, and J. Lafontaine, Riemannian Geometry. New York: Springer-Verlag, 1990. [15] K. Huber, “Codes over Gaussian integers,” IEEE Trans. Inform. Theory, vol. 40, pp. 207–216, Jan. 1994. [16] , “Codes over Eisenstein-Jacobi integers,” in Contemporary Mathematics. Providence, RI: Amer. Math Soc., 1994, vol. 168, pp. 165–179. [17] M. Muniz and S. I. R. Costa, “Labelings of Lee and hamming spaces,” Discr. Math., vol. 260, pp. 119–136, Jan. 2003. [18] A. Racsmány, “Correction to my paper: ‘Perfect single Lee-error-correcting code’,” Studia Sci. Math. Hungar., vol. 23, no. 1–2, pp. 295–296, 1988. [19] J. Stillwell, Geometry and Surfaces. New York: Springer-Verlag, 1992. [20] V. A. Vaishampayan, N. J. A. Sloane, and S. D. Servetto, “Multiple-description vector quantization with lattice codebooks: Design and analysis,” IEEE Trans. Inform. Theory, vol. 47, pp. 1718–1734, July 2001. [21] V. A. Vaishampayan and S. I. R. Costa, “Curves on a sphere, shift-map dynamics, and error control for continuous alphabet sources,” IEEE Trans. Inform. Theory, vol. 49, pp. 1658–1672, July 2003.