We propose the construction of signal space codes over the quaternion orders from a graph associated with the arithmetic Fuchsian group
In the study of two-dimensional lattice codes, it is known that the lattice
The concept of geometrically uniform codes (GU codes) was proposed in [
In this paper, we propose the construction of signal space codes over the quaternion orders from graphs associated with the arithmetic Fuchsian group
This paper is organized as follows. In Section
In this section, some basic and important concepts regarding quaternion algebras, quaternion orders, and arithmetic Fuchsian groups with respect to the development of this paper are presented. For a detailed description of these concepts, we refer the reader to [
Let
Let
Notice that the reduced norm is a quadratic form such that
Let
Let
Let
Now, from the identification of
Furthermore, as the reduced norm of an element is given by the determinant of the isomorphism
Let
We have that
Let
Given
Given
Let
The lattices
An invariant of an order
Let
One of the main objectives of this paper is to identify the arithmetic Fuchsian group in a quaternion order. Once this identification is realized, then the next step is to show the codewords of a code over graphs or the signals of a signal constellation (quotient of an order by a proper ideal). However, for the algebraic labeling to be complete, it is necessary that the corresponding order be
If
Let
Consider the upper-half plane
The set of linear fractional Möbius transformations of
A Fuchsian group
Another Euclidean model of the hyperbolic plane is given by the Poincaré disc
For each order
Now, note that the Fuchsian groups may be obtained by the isomorphism
As a consequence, consider the following.
These previous concepts and results lead to the concept of arithmetic Fuchsian groups. Since every Fuchsian group may be obtained in this way, we say that a Fuchsian group is derived from a quaternion algebra if there exists a quaternion algebra
Theorem
Let if if
In this section, we identify the arithmetic Fuchsian group
From [
To verify if a Fuchsian group associated with an order as specified in the previous paragraph is in fact arithmetic, it suffices to show that the quaternion algebra is not ramified at
Consider the Fuchsian group
On the other hand, the order
Consider the self-dual tessellation
The ring of integers of
Observe that
Given the genus
Given
From (
Hence,
Now, considering the order
Since the proof of the next result is similar to the proof of Proposition
Given
When there is no confusion in the notation being used, we will denote for simplicity the reduced norm of
Let
Let
Now, given two elements
Now, since
We denote the number of elements of
Finally, the quaternion conjugation is an antiautomorphism, which implies that
Let
Given
We are not interested in orders such as
If
Note from Corollary
Given
As can be seen in Example
Let
Let
Let
Let
Let
Let
In this section some concepts of graphs and codes over graphs are considered which will be useful in the next section.
Let
For
Given the distance
Let
Given
Note that the distance between two signal points
Given a graph
A code derived from a graph is defined as geometrically uniform if for any two-code sequences, there exists an isometry that takes a code sequence into the other, while it leaves the code invariant. Hence, geometrically uniform codes partition a set of vertices of a graph by the Voronoi regions.
Given an element
For
The procedures considered may be extended to surfaces with any genus once the associated quaternion order is known. This allows us to construct new geometrically uniform codes over different signal constellations.