Quantum Information and Computation, Vol. 14, No. 15&16 (2014) 1424–1440
c Rinton Press
FAMILIES OF CLASSES OF TOPOLOGICAL QUANTUM CODES
FROM TESSELLATIONS {4i + 2, 2i + 1}, {4i, 4i}, {8i − 4, 4} and {12i − 6, 3}
CLARICE DIAS DE ALBUQUERQUE∗
Department of Telematics, Faculty of Electrical and Computer Engineering
State University of Campinas, 13083-852, Campinas - SP, Brazil.
REGINALDO PALAZZO JR.†
Department of Telematics, Faculty of Electrical and Computer Engineering
State University of Campinas, 13083-852, Campinas - SP, Brazil.
EDUARDO BRANDANI DA SILVA‡
Department of Mathematics, State University of Maringá
87020-900, Maringá - PR, Brazil.
Received May 26, 2013
Revised May 28, 2014
In this paper we present some classes of topological quantum codes on surfaces
with genus g ≥ 2 derived from hyperbolic tessellations with a specific property.
We find classes of codes with distance d = 3 and encoding rates asymptotically
going to 1, 21 and 13 , depending on the considered tessellation. Furthermore,
these codes are associated with embedding of complete bipartite graphs. We also
analyze the parameters of these codes, mainly its distance, in addition to show
a class of codes with distance 4. We also present a class of codes achieving the
quantum Singleton bound, possibly the only one existing under this construction.
Keywords: Topological quantum codes, Quantum error-correcting codes, Surface
codes, Tessellations.
Communicated by: R Jozsa & C Williams
1. General
The use of properties of quantum mechanics theoretically allows faster quantum computation than
classic computation to obtain solutions of certain computational problems, including factoring of
prime numbers [1]. Various alternatives have been considered for obtaining a feasible quantum
computer, and among them we have the topological quantum computing, which involves exotic
states of matter, as fractional quantum Hall effect, high-temperature superconductivity and Majorana
fermions, see [2].
Nevertheless, the construction of a quantum computer is a great challenge. One of the reasons for
this difficulty is that decoherence exists in a quantum system. Decoherence is the decay phenomenon
∗
[email protected]
[email protected]
‡
[email protected]
†
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C. D. Albuquerque, R. Palazzo Jr., and E. B. Silva
1425
of states in superposition and occurs due to the interactions between the system and the surround
environment [3] and act in a very destructive way on the quantum bits or qubits. Qubits, differently
of bits, may assume, beyond the states 0 and 1, many others distinct states called superpositions and
these states are very fragile.
This problem may be solved through quantum error correcting codes (or simply QECC). Initially,
Shor considered a quantum code analogous to the classic repetition code [3]. These codes were shown
to belong to the class of CSS codes, as proposed by Calderbank and Shor [5], and Steane [6]. In [7],
Gottesman considered a more general class of codes called stabilizer codes which includes the CSS
codes. The stabilizer codes are based on group theory.
A very important subclass of the stabilizer codes is the so called topological quantum errorcorrecting codes (TQECC), introduced by Kitaev [8], where it is considered the construction of
quantum error correcting codes especially well adaptable for a fault tolerant implementation. His
proposal was to use certain properties of particles confined in a plane to carry out topological quantum
computation. The term originated because these properties are related to the topology of the physical
system. Thus, continuous deformations caused by the environment would not be capable to modify
such properties, consequently providing a fault tolerant quantum computation. This computation
is carried out by means of entwined “strings” that represent the particle’s movement in space-time.
The particles involved in this process are quasi-particles, known as anyons, and they only exist in a
bidimensional world.
Experimental research, as in [9], has shown that topological quantum computing and TQECC
are practical and a powerful promise, and in [10] topological quantum computation is thoroughly
explored. Also, as established in [11] and [12], TQECC is the main tool to obtain quantum faulttolerant computation.
In addition to these facts, many good topological quantum codes constructions were proposed
having the Kitaev toric codes as the paradigm, namely the homological codes [13] and the generalization of the toric codes to g-tori, with g ≥ 2, [14]. In this generalization all possible tessellations
of the corresponding surfaces is considered. As a consequence, six classes of codes with distance 3,
resulting from the embedding of complete graphs and complete bipartite graphs were obtained, [15].
In this paper, families of classes of topological quantum codes derived from self-dual tessellations
({4i, 4i}), orthogonal tessellations ({8i−4, 4}), densest tessellations ({12i−6, 3}) and the tessellations
{4i + 2, 2i + 1} achieving good minimum distances are constructed. In addition, the unequal error
protection these classes of codes may provide to the storage and to the transmission of quantum
information is considered. Such codes are associated with the embedding of complete bipartite
graphs on orientable compact surfaces whose plane model is a 4g-gon. Moreover, the asymptotic
coding rate nk → 1 as n → ∞ for the self-dual tessellations; nk → 21 as n → ∞ for the orthogonal
tessellation; and nk → 13 as n → ∞ for the densest tessellations. We call attention to the fact that
the codes derived in this paper come from different tessellations as the codes shown in [14], as well
as the construction of such codes is based on the number of faces, nf .
This paper is organized as follows. In Section 2, the relevant aspects of the TQC codes are reviewed. In Section 3, the new classes of codes derived from self-dual, orthogonal, densest tessellations
and the {4i + 2, 2i + 1} tessellations and the embedding of the associated complete bipartite graphs
are shown. Finally, in Section 4 some properties of the families of codes, the special properties of each
tessellation, the unequal error protection provided by these codes, and the code’s minimum distance
dT QC are drawn.
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2. Hyperbolic Geometry and Topological Quantum Codes
Surfaces are deeply connected with polygons in Euclidean and hyperbolic geometries, see [16]. In
this work we are considering the Poincaré disc model, given by D = {z ∈ C : |z| < 1}. With an
appropriate metric, D is a model of the hyperbolic plane. Geodesics in D are the paths with least
hyperbolic distance between two points. A hyperbolic polygon P ′ with p′ edges, or a p′ -gon, is a
convex closed set consisting of p′ hyperbolic geodesic segments whose intersection of two adjacent
geodesics is called vertex of the polygon. A p′ -gon whose edges have the same length and the internal
angles are equal, is called a regular p′ -gon. Furthermore, a regular tessellation of the Euclidean or
hyperbolic plane, is a covering of the whole plane by regular polygons, all with the same number
of edges, without superposition of such polygons, meeting completely only on edges or vertices. We
denote a regular tessellation by {p, q}, where q regular polygons with p edges meet in each vertex. In
particular, if p = q the tessellation is said to be self-dual. In Figure 1 one has the self-dual tessellation
{10, 10}.
A compact topological surface M may be obtained from a polygon P ′ by pairwise edge identifications. Such edge-pairing transformations are isometries γ 6= Id of an orientation preserving isometry
group Γ, taking an edge s of P ′ to another edge γ(s) = s′ of P ′ . Furthermore, γ −1 ∈ Γ \ {Id} takes
γ(s) = s′ to s. If s is identified with s′ , and s′ is identified with s′′ , then s is identified with s′′ . Such
a chain of identifications may also occur with vertices, and so we call a maximal set {v1 , v2 , . . . vk } of
identified vertices a vertex cycle. For more information on hyperbolic geometry we refer the reader
to [16, 17, 18, 19].
Fig. 1. The tessellation {10, 10}
Topological quantum codes (TQC) form a subclass of the stabilizer codes, where each qubit is
associated, in a one-to-one correspondence, to each edge of a tessellation of an orientable compact
surface, whereas the stabilizer operators, defining the code, are associated with each vertex and
each face of the tessellation. The latter are local operators, constituting a Hamiltonian with local
interactions, whose ground state coincides with the protected space of the code. The operations
described by the Hamiltonian control an intrinsic mechanism of protection of the encoded quantum
states. One advantage of these codes is related to the locality property of its operators which may
facilitate the physical implementation of these systems. For more details, the reader may see [8], [13]
and [14].
Formally, let M be an orientable compact surface and {p, q} a tessellation of M with E edges,
V vertices, F faces and genus g. Given a vertex v ∈ V and a face f ∈ F , we define the operators
Av as the tensor product of σx corresponding to each edge having v as the common vertex and the
C. D. Albuquerque, R. Palazzo Jr., and E. B. Silva
1427
operators Bf as the tensor product of σz corresponding to each edge forming the border of the face f .
A topological quantum code C with length n = |E|, with stabilizer S = {Av | v ∈ V } ∪ {Bf | f ∈ F },
encodes k = 2g qubits and its distance is d = min{d, d∗ }, where d denotes the minimum distance in
the tessellation {p, q}, whereas d∗ denotes the minimum distance in the dual tessellation {q, p}.
As an example, let us consider the toric codes, [8]. In this case, M is a flat torus with g = 1,
obtained from square lattice with l × l squares. The qubits are associated with edges of the square
lattice l × l.
f
v
Fig. 2. The l × l flat torus
σ j and Bf =
σ j . and the code is C = {|ψi :
The parity-check operators are Av =
j∈Ef z
j∈Ev x
Av |ψi = |ψi, Bf |ψi = |ψi ∀ v, f }. Figure 2 illustrates the flat torus for l=5 as also shown in [14].
⌋ errors. The distance is the number of
The toric codes detect l − 1 errors and correct ⌊ l−1
2
edges contained in the shortest homologically nontrivial cycle on the tessellation or dual tessellation.
Hence d = l, and the code has parameters [[2l2 , 2, l]]. The identifications of opposite sides of the
region limited by Zl × Zl result in the identification with the flat torus. Considering effectively the
qubits associates with the length of code word, we have n = 2l2 qubits.
The generalization of Kitaev’s toric code presented in [14] considers orientable compact surfaces
with genus g ≥ 2 and the corresponding geometry, the hyperbolic geometry.
This construction consists in selecting a regular hyperbolic polygon (plane model P ′ of the surface
derived from a hyperbolic tessellation {4g, 4g}, where g is the genus of the surface) and its possible
tilling {p, q}. Next, we briefly review such a construction.
An edge-pairing of P ′ defines an identification space SP ′ making it a hyperbolic surface if the
angles of each vertex cycle adds up to 2π. SP ′ in turn can be identified with a complete and connected
hyperbolic surface H2 /Γ, where Γ is a Fuchsian group, since P ′ is compact, [19].
Now, P SL(2, R) is the multiplicative group of Möbius transformations T : C → C defined by
, where a, b, c, d ∈ R such that ad − bc = 1, and a Fuchsian group Γ is a discrete
T (z) = az+b
cz+d
subgroup of P SL(2, R). In this case, Γ is an orientation preserving isometry group whose elements
are edge-pairing transformations γ.
On the other hand, a compact hyperbolic surface M ≡ H2 /Γ is the identification space of a
polygon P ′ if P ′ is the fundamental region [
for Γ, that is, a closed subset of a metric space X which
γ(P ′ ) = X and
Γ acts, with non-empty interior, such that
N
N
γ∈Γ
int(P ′ )
\
γ(int(P ′ )) = ∅,
This holds if the following conditions are satisfied:
∀ γ ∈ Γ − {Id}.
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Edge and Angle conditions [19]: If a compact polygon P ′ is the fundamental region for an orientation preserving isometry group Γ of S2 (sphere surface), R2 (Euclidean plane), or H2 (hyperbolic
plane), then
(i) For each edge s of P ′ there exists a unique edge s′ of P ′ such that s′ = γ(s), for γ ∈ Γ;
(ii) Given edge-pairings of P ′ , for each set of the identified vertices, the sum of the angles has to
be equal to 2π. This set is a vertex cycle.
Theorem 1 (Poincaré), [19] A compact polygon P ′ satisfying the edge and angle conditions is
a fundamental region for the group Γ generated by the edge-pairing transformations of P ′ , and Γ is a
Fuchsian group.
The procedure proposed in [14] takes into consideration polygons P ′ of the type 4g-gon (fundamental region of the self-dual tessellation {4g, 4g}) as the plane models of the corresponding surfaces
which we denote in this paper by P 4g . In these polygons the edge-pairing transformations are defined
by, γ : S → S; γ(si ) = si+2g , where S = {s1 , . . . , s4g } is the set of edges of P 4g , i = 1, 2, . . . , 4g,
and the sum of the subscripts of s is realized modulo 4g. Such isometry γ realizes the pairings of
opposite edges of P ′ . The selection of these edge-pairing transformations leads to a code distance
having the greatest hyperbolic distance between the identified edges of P 4g . Since p′ = q ′ = 4g, the
unique cycle of vertices obtained from these edge-pairing transformations has the sum of the internal
angles equal to (p′ /q ′ )(2π) = 2π, and so satisfying the necessary and sufficient conditions for P 4g to
be a fundamental region of the group of these edge-pairing transformations Γ.
2.1. Conditions under which a compact surface is generated
We call attention to the fact that fundamental regions (polygons) associated with different tessellations other than the tessellation {4g, 4g} may be used as planar models for surfaces with genus g.
However, they have to satisfy the Poincaré theorem. Equivalently, let P 4g be a fundamental region
of the Fuchsian group Γ with generators the edge-pairing transformations of P 4g . Hence, P 4g seen
as the fundamental region of the associated Fuchsian group leads to the Euler characteristic of the
surface χ(M ) = 2 − 2g. Now, P 4g seen as the polygon associated with the tessellation {p′ , q ′ } leads to
the Euler characteristic χ(M ) = V − E + F . Now, it is not difficult to see that from the edge-pairing
′
′
identifications V = pq′ , E = p2 and F = 1. Therefore, p′ must be even and divisible by q ′ .
Recall that the Euclidean tessellations have to satisfy (p′ − 2)(q ′ − 2) = 4 whereas the hyperbolic
tessellations have to satisfy (p′ − 2)(q ′ − 2) > 4. From these facts we conclude that if q ′ is even,
then q ′ = 4, 8, 12, . . ., that is, q ′ = 4r, for r = 1, 2, 3, . . .. Hence, p′ has to be of the form (2r′ − 1)q ′ ,
. On the other hand, if q ′
r′ = 1, 2, 3, . . ., and consequently the genus is given by g = 1 − r + q(2r−1)
4
′
′
′
′
′
is odd, then q = 2r + 1, therefore p = 2(2r − 1)q , where r, r = 1, 2, 3, . . .. Note that for each value
of q ′ , by varying the value of r′ leads to a different p′ . For instance, for q ′ = 4 we have g = r and
p′ = 8g − 4, whereas for q ′ = 3 we have g = r and p′ = 12g − 6. Table 1 illustrates some examples of
possible tessellations whose fundamental region (face) generates a compact surface.
Theoretically, any polygon which generates a compact surface can be employed in the construction
of such codes. Nevertheless, one of the reasons in selecting the model {4g, 4g}, is that all the pairings
are from opposite edges, and so the code achieves its greatest minimum distance.
We call special attention to the notation: P 4g is a regular hyperbolic polygon used as the plane
model of the surface, equivalently, the polygon associated with the fundamental region of the tessellation {4g, 4g}, whereas the tessellation {p, q} is the one which will tile the fundamental region
C. D. Albuquerque, R. Palazzo Jr., and E. B. Silva
1429
Table 1. Polygons generating compact surfaces.
{p, q}
{4, 4}
{6, 3}
{8, 8}
{10, 5}
{12, 4}
{12, 12}
{14, 7}
{16, 16}
{18, 3}
{18, 9}
{20, 4}
{20, 20}
{22, 11}
{24, 8}
{24, 24}
{26, 13}
{28, 4}
{28, 28}
{30, 3}
{30, 5}
{30, 15}
{32, 32}
{34, 17}
{36, 4}
{36, 12}
{36, 36}
{38, 19}
{40, 8}
{40, 40}
{42, 7}
{42, 3}
{60, 12}
g
1
1
2
2
2
3
3
4
2
4
3
5
5
5
6
6
4
7
3
5
7
8
8
5
8
9
9
8
10
8
4
13
Tessellation
{4g, 4g}
{12g − 6, 3}
{4g, 4g}
{4g + 2, 2g + 1}
{8g − 4, 4}
{4g, 4g}
{4g + 2, 2g + 1}
{4g, 4g}
{12g − 6, 3}
{4g + 2, 2g + 1}
{8g − 4, 4}
{4g, 4g}
{4g + 2, 2g + 1}
, 8}
{ 16g−8
3
{4g, 4g}
{4g + 2, 2g + 1}
{8g − 4, 4}
{4g, 4g}
{12g − 6, 3}
{ 20g−10
, 5}
3
{4g + 2, 2g + 1}
{4g, 4g}
{4g + 2, 2g + 1}
{8g − 4, 4}
{ 24g−12
, 12}
5
{4g, 4g}
{4g + 2, 2g + 1}
, 8}
{ 16g−8
3
{4g, 4g}
{ 28g−14
, 7}
5
{12g − 6, 3}
{ 24g−12
, 12}
5
P 4g .
2.2. Tessellations, operators and parameters
Every possible tiling {p, q} of the polygon P 4g satisfies the following equation:
µ(P 4g ) = nf µ(P ),
(1)
in addition to the following constraint (p − 2)(q − 2) > 4. In (1) µ(P 4g ) denotes the area of the
polygon P 4g , µ(P ) denotes the area of the polygon with p edges associated with the tiling {p, q}, and
nf is a positive integer which denotes the number of faces of the tessellation {p, q}. Note that, given
a tessellation {p, q}, the dual tessellation {q, p} has to satisfy the same previous conditions.
The area of a hyperbolic polygon is given by, [18, 19],
µ(P 4g ) = 4π(g − 1),
(2)
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where g is the genus of the surface. Moreover, by using the Gauss-Bonnet Theorem, [18, 19], we have
2π
π
2pπ
− 2 ] = (p − 2)π −
.
p
q
q
Thus equation (1) may be rewritten as:
µ(P ) = p[π −
4π(g − 1)
=
nf
2pπ
.
(p − 2)π −
q
(3)
Hence, the number of faces, nf , associated with the tiling {p, q} of P 4g is given by
nf =
4q(g − 1)
.
pq − 2p − 2q
(4)
Note that the tessellation {p, q} tiles the polygon P 4g for those values of p and q such that (4) is a
positive integer.
Given a vertex v of the tessellation, the vertex operator acts nontrivially on the q qubits having
v as the common vertex and the identity operator acts on the remaining qubits. Given a face f of
the tessellation, the face operator acts nontrivially on the p qubits forming the border of this face,
and the identity operator acts on the remaining qubits of the tessellation.
The parameters of these codes are: the code length is n = nf p/2 qubits, the number of encoded
qubits is k = 2g, and the dimension of the code C is 22g = 4g . The minimum distance is the smallest
homologically nontrivial cycle of the tessellation {p, q} or its dual {q, p}. This cycle is given by the
geodesic of least length connecting paired edges of P 4g , we denote this length by dh . Since the
distance of the code is a function of the edges of the tessellation covering the regular P 4g , it must be
considered the path over the edges closest to the chosen geodesic. Then, we consider the least integer
dh
, that is, the minimum distance of the code is
greater than or equal to the ratio l(p,q)
dT QC = ⌈
dh
⌉,
l(p, q)
(5)
where
dh = 2 arccosh
cos(π/4g)
,
sin(π/4g)
(6)
and
l(p, q)
=
arccosh
cos2 (π/q) + cos(2π/p)
.
sin2 (π/q)
(7)
These equations follow from hyperbolic trigonometry, [18].
In [14], all the best codes on compact surfaces with genus 2, 3, 4 and 5 are tabulated. In [15] it is
reproduced one class from [13] and five original classes of topological quantum codes with distance
3 are shown. These codes correct one error and detect two errors. The coding rate asymptotically
goes to 1. This is the best one can achieve for such a rate. There is also a class with asymptotically
coding rate going to 31 . These classes are determined from the self-dual tessellations ({4i − 3, 4i − 3},
{4i − 2, 4i − 2}, and {4i − 1, 4i − 1} for i = 2, 3, · · ·) quasi-self-dual tessellations ({p, p + 1}) and
from the densest tessellations({p, 3}). In addition, it has been observed a connection of these classes
with the embedding of either the self-dual complete graphs or the complete bipartite graphs on the
corresponding compact surfaces. The resulting classes are shown below.
C. D. Albuquerque, R. Palazzo Jr., and E. B. Silva
1431
• [[ s(s − 1)/2, (s(s − 1)/2) − 2(s − 1), 3 ]],
• [[ s(s − 3)/2, (s(s − 3)/2) − 2(s − 1), 3 ]]
• [[s(s − 5), s(s − 5) − 2(s − 1), 3 ]],
• [[ s(s − 2)/4, (s(s − 2)/4) − 2(s − 2), 3 ]],
• [[ 3s/2, (s − 20)/2, 3 ]],
where s = nf .
3. Families of Classes of TQC Derived from Special Tessellations
To obtain these new families of classes of topological quantum codes from the construction shown in
[14], we consider the following tessellations {4i + 2, 2i + 1}, {4i, 4i}, {8i − 4, 4} and {12i − 6, 3}, where
i = 2, 3, . . .. In particular, these tessellations have to satisfy the conditions described in Section 2.1,
that is, they are capable of generating compact surfaces of genus i. However, the purpose is to use
these tessellations to tile the fundamental region of the 4g regular polygon. Since each one of these
tessellations has a fundamental region, P , its area is given by µ(P ) = 4π(i − 1).
In general for tessellations {4i + 2, 2i + 1}, {4i, 4i}, {8i − 4, 4} and {12i − 6, 3}, where i = 2, 3, . . .,
,
the area of its fundamental region, that is, the hyperbolic polygon with p faces and inner angles 2π
q
is 4π(i − 1). Therefore, the number of faces, nf , is determined as:
nf =
µ(P 4g )
4π(g − 1)
g−1
=
=
µ(P )
4π(i − 1)
i−1
(8)
Since nf is a positive integer, from (8) it follows that (i − 1) divides (g − 1).
Next, a brief review of the embedding of graphs, [20], is presented. After that, three classes
corresponding to nf = 20, nf = 30 and nf = 40 in each family of codes, are considered.
3.1. Embedding of graphs: a review
A bipartite graph is a graph where its set of vertices can be partitioned into two subsets U and
W such that the vertices in each one of the subsets are mutually nonadjacent, that is, every edge
connects a vertex in U to one in W . If every vertex of U is adjacent to every vertex of W , then the
graph is called a complete bipartite graph on the sets U and W . The complete bipartite graph on the
sets U and W with m′ and n′ vertices, respectively, is denoted by Km′ ,n′ .
A graph is embedded on a surface M if it can be drawn on M without crossing edges. For
m′ , n′ ≥ 2, the Euler characteristic of the complete bipartite graph Km′ ,n′ is given by:
χ(Km′ ,n′ ) = 2[(m′ + n′ −
m′ n ′
)/2],
2
(9)
where [a] denotes the greatest integer less than or equal to the real number a.
If there is no restriction of the embedding being a region homeomorphic to an open disk, then
the embedding may be realized on every orientable compact surface with characteristic greater than
or equal to the Euler characteristic of the given surface.
The dual of an embedding of a graph on a surface is obtained by considering the interior of each
face of the original embedding as a vertex of a new embedded graph. If two faces are adjacent along
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Families of codes of topological quantum codes from ...
an edge in the original graph, we connect these faces by a new edge crossing the old edge along which
the two faces are adjacent. The resulting embedding of the new graph on the same surface is called
the dual graph of the original embedding, and if the dual graph is isomorphic to the original one, so
the original and the new embedding are said to be self-dual.
3.2. Families of classes of codes derived from the tessellation {4i + 2, 2i + 1}
In this section, some examples of classes of topological quantum codes by tiling the fundamental
polygon P 4g by the tessellation {4i + 2, 2i + 1} are presented.
First, considering nf = 20, and substituting the values of p, q and nf in (4), yields g = 20(i−1)+1.
˙ = 20(2i + 1), and the number of encoded qubits
The length of the code is given by n = nf ṗ2 = 20 4i+2
2
is k = 2g = 40(i − 1) + 2. The minimum distance dT QC is 3. Therefore, the class of codes associated
with this tessellation has parameters:
[[n, k dT QC ]] = [[ 20(2i + 1), 40(i − 1) + 2, 3 ]].
Table 2 shows the codes of this class.
Since the number of edges of a complete bipartite graph Km′ ,n′ is m′ n′ , it follows that n = m′ n′ .
Thus, we may consider m′ = 20 and n′ = 2i + 1. Therefore, this class of codes is associated with the
embedding of the graph K20,2i+1 . This embedding is possible since the genus g = 20(i − 1) + 1 is
greater than the minimum value of the Euler characteristic of the graph K20,2i+1 , (9).
Table 2. Codes from the tessellation {4i + 2, 2i + 1} and nf = 20
i
2
3
4
5
6
{p, q}
{10,5}
{14,7}
{18,9}
{22,11}
{26,13}
nf
20
20
20
20
20
{q, p}
{5,10}
{7,14}
{9,18}
{11,22}
{13,26}
n∗f
40
40
40
40
40
g
21
41
61
81
101
d
4
4
3
3
3
d∗
3
3
3
3
2
[[n, k, dT QC ]]
[[100,42,3]]
[[140,82,3]]
[[180,122,3]]
[[220,162,3]]
[[260,202,2]]
k/n
0.42
0.5857
0.6777
0.7363
0.7769
d/n
0.04
0.0285
0.0166
0.01363
0.0115
d∗ /n
0.03
0.0214
0.0166
0.0136
0.0076
˙ = 30(2i + 1), and
Similarly, considering nf = 30 yields g = 30(i − 1) + 1, n = nf ṗ2 = 30 4i+2
2
k = 2g = 60(i − 1) + 2. The codes minimum distance dT QC is 3. Finally, the class of codes associated
with this tessellation has parameters:
[[n, k dT QC ]] = [[ 30(2i + 1), 60(i − 1) + 2, 3 ]].
Table 3 shows some examples of this class.
Since the number of edges of a complete bipartite graph Km′ ,n′ is m′ n′ , it follows that n = m′ n′ .
Thus, we may consider m′ = 30 and n′ = 2i + 1. Therefore, this class of codes is associated with the
embedding of the graph K30,2i+1 . This embedding is possible since the genus g = 30(i − 1) + 1 is
greater than the minimum value of the Euler characteristic of the graph K30,2i+1 , (9).
Likewise for nf = 40, we obtain the class
[[ 40(2i + 1), 2(40(i − 1) + 1), 3 ]].
This class is associate with the embedding of the graph K40,2i+1 and it is illustrated in Table 4.
For each nf we can obtain new classes of codes, however some of these classes are too small. In
general, family of classes of codes with parameters [[ s(2i + 1), 2(s(i − 1) + 1), 3 ]], may be obtained,
C. D. Albuquerque, R. Palazzo Jr., and E. B. Silva
1433
Table 3. Codes from the tessellation {4i + 2, 2i + 1} and nf = 30
i
2
3
4
5
6
7
8
9
10
{p, q}
{10,5}
{14,7}
{18,9}
{22,11}
{26,13}
{30,15}
{34,17}
{38,19}
{42,21}
i
2
3
4
5
6
7
8
9
10
11
12
13
{p, q}
{10,5}
{14,7}
{18,9}
{22,11}
{26,13}
{30,15}
{34,17}
{38,19}
{42,21}
{46,23}
{50,25}
{54,27}
nf
30
30
30
30
30
30
30
30
30
{q, p}
{5,10}
{7,14}
{9,18}
{11,22}
{13,26}
{15,30}
{17,34}
{19,38}
{21,42}
n∗f
60
60
60
60
60
60
60
60
60
g
31
61
91
121
151
181
211
241
271
d
5
4
4
3
3
3
3
3
2
d∗
3
3
3
3
3
3
3
3
2
[[n, k, dT QC ]]
[[150,62,3]]
[[210,122,3]]
[[270,182,3]]
[[330,242,3]]
[[390,302,3]]
[[450,362,3]]
[[510,422,3]]
[[570,482,3]]
[[630,542,2]]
k/n
0.4133
0.5809
0.6740
0.7333
0.7743
0.8044
0.8274
0.8456
0.8603
d/n
0.0333
0.019
0.0148
0.00909
0.0076
0.0066
0.0058
0.0052
0.0047
d∗ /n
0.02
0.0142
0.0111
0.0090
0.0076
0.0066
0.0058
0.0052
0.0031
Table 4. Codes from the tessellation {4i + 2, 2i + 1} and nf = 40
nf
40
40
40
40
40
40
40
40
40
40
40
40
where s = nf , with
the graph Ks,2i+1 .
k
n
{q, p}
{5,10}
{7,14}
{9,18}
{11,22}
{13,26}
{15,30}
{17,34}
{19,38}
{21,42}
{23,46}
{25,50}
{27,54}
=
n∗f
80
80
80
80
80
80
80
80
80
80
80
80
2(s(i−1)+1)
s(2i+1))
g
41
81
121
161
201
241
281
321
361
401
441
481
d
5
4
4
4
3
3
3
3
3
3
3
3
d∗
3
3
3
3
3
3
3
3
3
3
3
3
[[n, k, dT QC ]]
[[200,82,3]]
[[280,162,3]]
[[360,242,3]]
[[440,322,3]]
[[520,402,3]]
[[600,482,3]]
[[680,562,3]]
[[760,642,3]]
[[840,722,3]]
[[920,802,3]]
[[1000,882,3]]
[[1080,962,3]]
k/n
0.41
0.5785
0.6722
0.7318
0.7730
0.8033
0.8264
0.8447
0.8595
0.8717
0.882
0.8907
d/n
0.025
0.0142
0.0111
0.0090
0.0057
0.005
0.0044
0.0039
0.0035
0.0032
0.003
0.0027
d∗ /n
0.015
0.0107
0.0083
0.0068
0.0057
0.005
0.0044
0.0039
0.0035
0.0032
0.003
0.0027
→ 1 as i → ∞. This family is associated with the embedding of
3.3. Families of classes of codes derived from the tessellation {4i, 4i}
Some examples of families of topological quantum codes by tiling the fundamental polygon P 4g by
the tessellation {4i, 4i} are presented.
If nf = 20, by substituting the values of p, q and nf in (4), one has g = 20(i − 1) + 1. The value of
˙ = 20(2i), and the number of encoded qubits is k = 2g = 40(i − 1) + 2, coinciding
n is n = nf ṗ2 = 20 4i
2
with the later case. Again the codes minimum distance dT QC is 3. Therefore, the class of codes
associated with this tessellation has parameters: [[n, k dT QC ]] = [[20(2i), 2(20(i − 1) + 1), 3]] .
In Table 5, it is shown some examples of codes of this class.
Since the number of edges of a complete bipartite graph Km′ ,n′ is m′ n′ , it follows that n = m′ n′ .
Thus, we may consider m′ = 20 and n′ = 2i. Therefore, this class of codes is associated with the
embedding of the graph K20,2i . This embedding is possible, once the genus g = 20(i − 1) + 1 is greater
than the minimum value of the Euler characteristic of K20,2i , (9).
Similarly, in the Tables 6 e 7, codes are explained for nf = 30 e nf = 40, respectively. These
values of nf give rise to the classes with parameters [[n, k dT QC ]] = [[30(2i), 2(30(i − 1) + 1), 3]]
and [[n, k dT QC ]] = [[40(2i), 2(40(i − 1) + 1), 3]], respectively.
Note that in this case the tessellation {4i, 4i} is self-dual, consequently having less computational
1434
Families of codes of topological quantum codes from ...
complexity in its implementation.
Table 5. Codes from the tessellation {4i, 4i} and nf = 20
i
2
3
4
5
6
7
{p, q}
{8, 8}
{12, 12}
{16, 16}
{20, 20}
{24, 24}
{28, 28}
nf
20
20
20
20
20
20
g
21
41
61
81
101
121
[[n, k, dT QC ]]
[[80,42,3]]
[[120,82,3]]
[[160,122,3]]
[[200,162,3]]
[[240,202,3]]
[[280,242,2]]
k/n
0.525
0.683333
0.7625
0.81
0.841667
0.864286
d/n
0.0375
0.025
0.01875
0.015
0.0125
0.00714286
Table 6. Codes from the tessellation {4i, 4i} and nf = 30
i
2
3
4
5
6
7
8
9
10
11
{p, q}
{8, 8}
{12, 12}
{16, 16}
{20, 20}
{24, 24}
{28, 28}
{32, 32}
{36, 36}
{40, 40}
{44, 44}
nf
30
30
30
30
30
30
30
30
30
30
g
31
61
91
121
151
181
211
241
271
301
[[n, k, dT QC ]]
[[20,62,3]]
[[180,122,3]]
[[240,182,3]]
[[300,242,3]]
[[360,302,3]]
[[420,362,3]]
[[480,422,3]]
[[540,482,3]]
[[600,542,3]]
[[660,602,2]]
k/n
0.516667
0.677778
0.758333
0.806667
0.838889
0.861905
0.879167
0.892593
0.903333
0.912121
d/n
0.025
0.0166667
0.0125
0.01
0.008333
0.0071428
0.00625
0.0055555
0.005
0.00303
Table 7. Codes from the tessellation {4i, 4i} and nf = 40
i
2
3
4
5
6
7
8
9
10
11
12
13
14
15
{p, q}
{8, 8}
{12, 12}
{16, 16}
{20, 20}
{24, 24}
{28, 28}
{32, 32}
{36, 36}
{40, 40}
{44, 44}
{48, 48}
{52, 52}
{56, 56}
{60, 60}
nf
40
40
40
40
40
40
40
40
40
40
40
40
40
40
g
41
81
121
161
201
241
281
321
361
401
441
481
521
561
[[n, k, dT QC ]]
[[160,82,4]]
[[240,162,3]]
[[320,242,3]]
[[400,322,3]]
[[480,402,3]]
[[560,482,3]]
[[640,562,3]]
[[720,642,3]]
[[800,722,3]]
[[880,802,3]]
[[960,882,3]]
[[1040,962,3]]
[[1120,1042,3]]
[[1200,1122,2]]
k/n
0.5125
0.675
0.75625
0.805
0.8375
0.860714
0.878125
0.891667
0.9025
0.911364
0.91875
0.925
0.930357
0.935
d/n
0.025
0.0125
0.009375
0.0075
0.00625
0.00535714
0.0046875
0.00416667
0.00375
0.003409
0.003125
0.00288462
0.00267857
0.00166667
In general, a family of classes of codes with parameters [[s(2i), 2(s(i − 1) + 1), 3 ]] is obtained,
where s = nf , with nk = 2(s(i−1)+1)
→ 1 as i → ∞. This family is associated with the embedding of
s(2i))
the graph Ks,2i .
3.4. Families of classes of codes derived from the tessellation {8i − 4, 4}
C. D. Albuquerque, R. Palazzo Jr., and E. B. Silva
1435
Considering the tessellation {8i−4, 4} of P 4g , we obtain, for the same reason that g = 20(i−1)+1. The
˙ = 20(4i−2), and the number of encoded qubits is k = 2g = 40(i−1)+2,
length of the code n = 20 8i−4
2
coinciding with the previous cases. The codes minimum distance dT QC is 3. Therefore, this class of
codes has parameters: [[n, k dT QC ]] = [[20(4i − 2), 2(20(i − 1) + 1), 3]].
Some examples of these codes are shown in the Table 8.
In this case, the class of codes is associated with the embedding of the graph K20,4i−2 . This
embedding is possible since the genus g = 20(i − 1) + 1 is greater than the minimum value of the
Euler characteristic of K20,4i−2 , (9).
For nf = 30 and nf = 40, the corresponding classes of codes are:
[[n, k dT QC ]] = [[30(4i − 2), 2(30(i − 1) + 1), 3]] ,
[[n, k dT QC ]] = [[40(4i − 2), 2(40(i − 1) + 1), 3]] ,
as shown in Tables 9 and 10.
We observe that the tessellation {8i − 4, 4} is orthogonal, since the inner angles of the defining
polygon is π2 . This property may be useful in the decoding process.
Table 8. Codes from the tessellation {8i − 4, 4} and nf = 20
i
2
3
4
{p, q}
{12,4}
{20,4}
{28,4}
nf
20
20
20
{q, p}
{4,12}
{4,20}
{4,28}
n∗f
60
100
140
g
21
41
61
d
5
6
6
d∗
3
3
2
[[n, k, dT QC ]]
[[120,42,3]]
[[200,82,3]]
[[280,122,2]]
k/n
0.35
0.41
0.4357
d/n
0.0416
0.03
0.02142
d∗ /n
0.025
0.015
0.0071
Table 9. Codes from the tessellation {8i − 4, 4} and nf = 30
i
2
3
4
5
6
{p, q}
{12,4}
{20,4}
{28,4}
{36,4}
{44,4}
nf
30
30
30
30
30
i
2
3
4
5
6
7
8
{p, q}
{12,4}
{20,4}
{28,4}
{36,4}
{44,4}
{52,4}
{60,4}
nf
40
40
40
40
40
40
40
{q, p}
{4,12}
{4,20}
{4,28}
{4,36}
{4,44}
n∗f
90
150
210
270
330
g
31
61
91
121
151
d
6
6
7
7
7
d∗
3
3
3
3
2
[[n, k, dT QC ]]
[[180,62,3]]
[[300,122,3]]
[[420,182,3]]
[[540,242,3]]
[[660,302,2]]
k/n
0.3444
0.4066
0.4333
0.4481
0.4575
d/n
0.0333
0.02
0.0166
0.0129
0.0106
d∗ /n
0.0166
0.01
0.0071
0.0055
0.0030
Table 10. Codes from the tessellation {8i − 4, 4} and nf = 40
{q, p}
{4,12}
{4,20}
{4,28}
{4,36}
{4,44}
{4,52}
{4,60}
n∗f
120
200
280
360
440
520
600
g
41
81
121
161
201
241
281
d
6
7
7
7
8
8
8
d∗
3
3
3
3
3
3
2
[[n, k, dT QC ]]
[[240,82,3]]
[[400,162,3]]
[[560,242,3]]
[[720,322,3]]
[[880,402,3]]
[[1040,482,3]]
[[1200,562,2]]
k/n
0.3416
0.405
0.4321
0.4472
0.4568
0.4634
0.4683
d/n
0.025
0.0175
0.0125
0.0097
0.0090
0.0076
0.0066
d∗ /n
0.0125
0.0075
0.0053
0.0041
0.0034
0.0028
0.0016
In general, the family of classes of codes with parameters [[s(4i − 2), 2(s(i − 1) + 1), 3 ]] is
obtained, where s = nf , with nk = 2(s(i−1)+1)
→ 12 as i → ∞. This family is is associated with the
s(4i−2)
embedding of the graph Ks,4i−2 .
3.5. Families of classes of codes derived from the tessellation {12i − 6, 3}
1436
Families of codes of topological quantum codes from ...
For the tessellation {12i − 6, 3} of P 4g , analogous to the previous cases, the genus is given by g =
˙ = 20(6i − 3), the number of encoded qubits is
20(i − 1) + 1. The length of the code is n = 20 12i−6
2
equal to k = 2g = 40(i − 1) + 2, and the codes minimum distance dT QC is 3. The parameters of this
class of codes are:
[[n, k dT QC ]] = [[20(6i − 3), 2(20(i − 1) + 1), 3]] .
Some examples are shown in the Table 11.
The class of codes is associated with the embedding of the graph K20,6i−3 , and such embedding
is possible if the genus g = 20(i − 1) + 1 is greater than the minimum value of Euler characteristic of
K20,6i−3 is satisfied, (9).
By considering nf = 30 and nf = 40, we have the classes:
[[n, k dT QC ]] = [[30(6i − 3), 2(30(i − 1) + 1), 3]] ,
[[n, k dT QC ]] = [[40(6i − 3), 2(40(i − 1) + 1), 3]] ,
respectively (see Tables 12 e 13).
The tessellation {12i − 6, 3} is the densest tessellation.
Table 11. Codes from the tessellation {12i − 6, 3} and nf = 20
{p, q}
{18,3}
{30,3}
{42,3}
i
2
3
4
{q, p}
{3,18}
{3,30}
{3,42}
nf
20
20
20
n∗f
120
200
280
g
21
41
61
d∗
3
3
2
d
8
9
10
[[n, k, dT QC ]]
[[180,42,3]]
[[300,82,3]]
[[420,122,2]]
k/n
0.2333
0.2733
0.2904
d/n
0.0444
0.03
0.0238
d∗ /n
0.0166667
0.01
0.0047
Table 12. Codes from the tessellation {12i − 6, 3} and nf = 30
i
2
3
4
5
6
{p, q}
{18,3}
{30,3}
{42,3}
{54,3}
{66,3}
nf
30
30
30
30
30
{q, p}
{3,18}
{3,30}
{3,42}
{3,54}
{3,66}
n∗f
180
300
420
540
660
g
31
61
91
121
151
d
9
10
11
11
11
d∗
3
3
3
3
2
[[n, k, dT QC ]]
[[270,62,3]]
[[450,122,3]]
[[630,182,3]]
[[810,242,3]]
[[990,302,2]]
k/n
0.2296
0.2711
0.2888
0.2987
0.3050
d/n
0.0333
0.0222
0.0174
0.0135
0.0111
d∗ /n
0.0111
0.0066
0.0047
0.0037
0.0020
Table 13. Codes from the tessellation {12i − 6, 3} and nf = 40
i
2
3
4
5
6
7
{p, q}
{18,3}
{30,3}
{42,3}
{54,3}
{66,3}
{78,3}
nf
40
40
40
40
40
40
{q, p}
{3,18}
{3,30}
{3,42}
{3,54}
{3,66}
{3,78}
n∗f
240
400
560
720
880
1040
g
41
81
121
161
201
241
d
9
10
11
12
12
12
d∗
3
3
3
3
3
2
[[n, k, dT QC ]]
[[360,82,3]]
[[600,162,3]]
[[840,242,3]]
[[1080,322,3]]
[[1320,402,3]]
[[1560,482,2]]
k/n
0.2277
0.27
0.2880
0.2981
0.3045
0.3089
d/n
0.025
0.0166
0.0130
0.0111
0.0090
0.0076
d∗ /n
0.0083
0.005
0.0035
0.0027
0.0022
0.0012
In general, the family of classes of codes with parameters [[s(6i − 3), 2(s(i − 1) + 1), 3 ]] is
obtained, where s = nf , with nk = 2(s(i−1)+1)
→ 13 as i → ∞. This family is associated with the
s(6i−3)
embedding of the graph Ks,4i−2 .
4. Discussion about distance dT QC
C. D. Albuquerque, R. Palazzo Jr., and E. B. Silva
1437
In this section we show the importance of the distance in the search for classes of codes; in the
determination of the properties associated with these codes; and finally to mention the existence of
only one class of TQC codes achieving the quantum Singleton bound, the class of MDS codes, by
exhaustive search. As previously mentioned the distance of the topological quantum codes is given
by the equation (5). However, when we seek for families of codes it is necessary to calculate the
distance for many values of g, as shown in the Table 14. From these calculations it is possible to find
patterns in nf identifying such a class of codes. Next, we show how to obtain such classes of TQC
codes by considering the classes of codes within the tessellation {8i − 4, 4}.
Table 14. Topological quantum codes obtained from the tessellation {8i − 4, 4} when i = 2
g
3
..
.
10
11
..
.
21
..
.
31
..
.
41
..
.
57
58
..
.
303
304
..
.
nf
2
..
.
9
10
..
.
20
..
.
30
..
.
40
..
.
56
57
..
.
302
303
..
.
n∗f
6
..
.
27
30
..
.
60
..
.
90
..
.
120
..
.
168
171
..
.
906
909
..
.
n
12
..
.
54
60
..
.
120
..
.
180
..
.
240
..
.
336
342
..
.
1812
1818
..
.
k
6
..
.
20
22
..
.
42
..
.
62
..
.
82
..
.
114
116
..
.
606
608
..
.
d
2.3954
..
.
3.8892
4.0046
..
.
4.7849
..
.
5.2539
..
.
00000
..
.
5.9867
6.0077
..
.
7.9962
8.0002
..
.
d∗
1.1977
..
.
1.9446
2.0023
..
.
2.3924
..
.
2.6270
..
.
00000
..
.
2.9934
3.0038
..
.
3.9981
4.0001
..
.
[[n, k, dT QC ]]
[[12,6,2]]
..
.
[[54,20,2]]
[[60,22,3]]
..
.
[[120,42,3]]
..
.
[[180,62,3]]
..
.
[[240,82,3]]
..
.
[[336,114,3]]
[[342,116,4]]
..
.
[[1812,606,4]]
[[1818,608,5]]
..
.
We start by determining the parameters nf , n∗f , n, k, d and d∗ for each tessellation belonging to
a family of tessellations by varying the genus g of the surface. This may be seen in Table 14 for
the case of the family of tessellation {8i − 4, 4} when i = 2 leading to the tessellation {12, 4}. Note
from Table 14 that the distance of the dual code d∗ increase logarithmically with the genus g, and
the parameters nf , n∗f , n, k increase in arithmetic progression. Recall that the distance dT QC of a
TQC code is the minimum distance obtained from the original tessellation and from its dual, this is
shown in the last column of Table 14. For each family of tessellation, {4i + 2, 2i + 1}, {4i, 4i} and
{12i − 6, 3}, we have used the same procedure as that used with the tessellation {8i − 4, 4}.
With respect to the code distance one may notice from equations (5) and (6) the dependence of
dT QC with the genus of the surface, the greater the genus the greater will be the minimum distance
of the code. This fact may be observed in Table 14. However, within each class the code distance
depends not only on the genus g as also on the values taken on by p and q (which establish the
tessellation (see (5), (6) and (7))), which in turn depend on the parameter i. Consequently, the code
distance diminishes within each class when the genus g increases, see tables in Section 3. Although
1438
Families of codes of topological quantum codes from ...
it is possible to obtain codes with minimum distances 4, 5 or greater (see Table 15 for classes of
codes derived from the tessellation {4i, 4i}), in this paper, we have aimed at codes with distance 3,
for these cases the genus of the associated surface is not that high.
Table 15. Codes from the tessellation {4i, 4i} and nf = 600
i
2
3
4
5
6
7
8
9
10
11
..
.
234
{p, q}
{8, 8}
{12, 12}
{16, 16}
{20, 20}
{24, 24}
{28, 28}
{32, 32}
{36, 36}
{40, 40}
{44, 44}
..
.
{936, 936}
nf
600
600
600
600
600
600
600
600
600
600
..
.
600
g
601
1201
1801
2401
3001
3601
4201
4801
5401
6001
..
.
139801
[[n, k, dT QC ]]
[[2400,1202,5]]
[[3600,2402,5]]
[[4800,3602,4]]
[[6000,4802,4]]
[[7200,6002,4]]
[[8400,7202,4]]
[[9600,8402,4]]
[[10800,9602,4]]
[[12000,10802,3]]
[[13200,12002,3]]
..
.
[[280800,279602,3]]
k/n
0.5008
0.6672
0.7504
0.8003
0.8336
0.8574
0.8752
0.8891
0.9002
0.9092
..
.
0.9957
It was shown in [14] a class of MDS codes with distance d = 2, namely the class of codes
with parameters [[2(g + 1), 2g, 2]]. From an exhaustive search procedure, we infer that this is the
only possible class of TQC codes satisfying with equality the quantum Singleton bound, that is,
n − k ≥ 2(d − 1). In [21] quantum codes with minimum distance d = 2 is considered in detail
with the objective to explore the inherent properties and to generalize such techniques. One of the
posed questions is as follows: How large can be the code dimension when it is given d = 2 and the
code length n? A conjecture is as follows: any optimum code with d = 2 and even code length the
associated parameters are [[2m, 2(m − 1), 2]] for these codes have a large automorphism group. In
fact, the class of TQC codes we have found has parameters [[2(g + 1), 2g, 2]] which for m = g + 1
coincides with the optimum class of codes. To the best of our knowledge, this is the only class of
topological quantum codes achieving the quantum Singleton bound with equality.
4.1. Parameters n, k and Rates
A characteristic of the classes of codes presented in this paper, is that the number of faces of
the tessellation {p, q} of P 4g , nf , is a constant. Thus, the number of vertices (or equivalently, the
number of faces in the dual tessellation) and also the length of the codes, are both multiple of nf .
This number may be seen as the “design number of faces” to be considered in the determination
of the remaining parameters of the code in a similar way as it is done in the construction of the
BCH codes by knowing the “design distance”. It is possible to obtain all the necessary classes by
modifying the value taken on by nf . By specifying the genus and the tessellation nf and dT QC are
readily determined, and so a new class of topological quantum codes.
With respect to the tessellations, the codes derived from the self-dual tessellations have the least
implementation complexity due to the fact that the code and its dual are equal. On the other hand,
the densest tessellations are important due to the fact that they have the best packing parameter,
the least error rate, however the decoding complexity is a little greater than that of the self-dual
tessellations.The orthogonal tessellations are important since its decoding process is quite reliable due
C. D. Albuquerque, R. Palazzo Jr., and E. B. Silva
1439
to the fact that the distance among the qubits is the greatest possible. We call the attention to the fact
that the self-dual and the densest tessellations were not considered in [14]. Other tessellations may
, 8},{ 20g−10
, 5}, { 24g−12
, 12}
be considered in this construction, for instance the tessellations { 16g−8
3
3
5
28g−14
and { 5 , 7} shown in Table 1.
By realizing an analysis with respect to the parameters and rates we may observe that the
number of codewords with length k in different classes of codes will always have the same value
for all tessellations once we fix the same value nf for them and that n increases as an arithmetic
progression within each class of codes. The encoding rate nk of the code families derived from the
tessellations {4i, 4i} are the best found, starting with rate 0.52 and increasing up to 0.93, whereas
for other tessellations of the type {12i − 6, 3} are the lowest ones achieving the value 0.3. The rates
k
for the families derived from the tessellations {4i + 2, 2i + 1} vary from 0.4 up to 0.9, and the
n
rates for the tessellations {8i − 4, 4} are in the interval 0.34 and 0.46. We observe that the self-dual
tessellations achieve the highest encoding rates whereas the densest tessellations the lowest ones.
Another interesting fact is that one may find a great number of codes with distance 3 in self-dual
tessellations when compared to the densest tessellations. As expected, the codes derived from the
self-dual tessellation {4i, 4i} does not provide unequal error protection.
Regarding the unequal error protection that these codes may provide, note from Tables 2, 3, and
4 that the codes derived from the tessellation {4i + 2, 2i + 1} provide unequal error protection for
i = 2, 3, i = 2, 3, 4, and i = 2, 3, 4, 5, respectively. When considering the tessellation {8i − 4, 4} shown
in Tables 8, 9, and 10, and the tessellation {12i − 6, 3} in Tables 11, 12, and 13, any value taken
on by i leads to unequal error protection whose protection increases as the genus increases within
the given tessellation and the protection also increases from the tessellation going toward the densest
tessellation.
It is also noticed that different tessellations generate codes with the same parameters. For instance, the tessellations {6, 6} and {12, 4} with its dual {4, 12} generate the code with parameters
[[60, 22, 3]] for g = 11 and the code with parameters [[66, 24, 3]] for g = 12, whereas the tessellations
{8, 8} and {12, 6} with its dual {6, 12} generate the code with parameters [[72, 38, 3]] for g = 19, the
code with parameters [[84, 44, 3]] for g = 22, [[96, 50,3]] for g = 25 and the code [[108, 56, 3]] for
g = 28 and so on. This property opens up the opportunity to select a tessellation fitting properly to
a given problem.
Acknowledgements
This work has been supported by FAPESP under grants 2007/56052-8 and 2009/50837-9, and CNPq
under grants 303059/2010-9 and 160086/2012-4.
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