Abstract
This paper aims to construct signal sets from quotient rings of the quaternion over a real number field associated with the arithmetic Fuchsian group Γ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4g</sub> , where g is the genus of the associated surface. These Fuchsian groups consist of the edge-pairing isometries of the regular hyperbolic polygons (fundamental region) P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4g</sub> , which tessellate the hyperbolic plane D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . The corresponding tessellations are the self-dual tessellations {4g, 4g}. Knowing the generators of the quaternion orders which realize the edge-pairings of the polygons, the signal points of the signal sets derived from the quotient rings of the quaternion orders are determined. It is shown by examples the relevance of adequately selecting the ideal in the maximal order to construct the signal sets satisfying the property of geometrical uniformity. The labeling of such signals is realized by using the mapping by set partitioning concept to solve the corresponding Diophantine equations (extreme quadratic forms). Trellis coded modulation and multilevel codes whose signal sets are derived from quotient rings of quaternion orders are considered possible applications.
Highlights
A generalization of the concept of geometrically uniform (GU) signal sets, [1], was done in [2] by considering the extension of the previous uniform tilings and the corresponding groups related to the Euclidean geometry to those corresponding to the hyperbolic geometry
This paper aims to construct signal sets from quotient rings of the quaternion over a real number field associated with the arithmetic Fuchsian group 4g
We provide the elements of a quaternion algebra A in order to determine a maximal order O in it to obtain the corresponding arithmetic Fuchsian group (A, O)
Summary
A generalization of the concept of geometrically uniform (GU) signal sets, [1], was done in [2] by considering the extension of the previous uniform tilings and the corresponding groups related to the Euclidean geometry to those corresponding to the hyperbolic geometry. Quest for the signal set leading to the best performance, some creative constructions of such signal sets associated with surfaces with genus g ≥ 2 for use in lattice codes were proposed, [6], [7] and [8] among others Such surfaces may be obtained by the quotient of Fuchsian groups of the first kind [9].
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