Geometrically uniform hyperbolic codes

Lazari, Henrique; Palazzo, Reginaldo

doi:10.1590/s0101-82052005000200002

Cited by 13 publications

(11 citation statements)

References 12 publications

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“…In this section, some basic and important concepts regarding geometrically uniform signal sets, quaternion algebras, quaternion orders, and arithmetic Fuchsian groups for this paper's development are presented. For a detailed description of these concepts we refer the reader to [1], [2] and [16]- [22].…”

Section: Preliminariesmentioning

confidence: 99%

“…), in the Euclidean sense. Theorem 2.1: [2] If S is a geometrically uniform signal set, then all the Dirichlet regions have the same shape.…”

Section: A Geometrically Uniform Signal Setsmentioning

confidence: 99%

“…Let α ∈ A be given by α = a 0 + a 1 i + a 2 j + a 3 ij, where a 0 , a 1 , a 2 , a 3 ∈ K. The conjugate of α, denoted byᾱ, is defined byᾱ = a 0 − a 1 i − a 2 j − a 3 ij. Thus, the reduced norm of α, is defined as N rd(α) = α.ᾱ = a 2 0 − aa 2 1 − ba 2 2 + aba 2 3 , and the reduced trace of α as T rd(α) = α+ᾱ = 2a 0 . Let A = (a, b) K be an algebra and M 0 , M 1 , M 2 and M 3 be linearly independent matrices in M 2 (K( √ a)), given by…”

Section: B Quaternion Algebras and Quaternion Ordersmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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Construction of Signal Sets From Quotient Rings of the Quaternion Orders Associated With Arithmetic Fuchsian Groups

Queiroz

Palazzo

2020

IEEE Access

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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 4g , 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 4g , which tessellate the hyperbolic plane D 2 . The corresponding tessellations are the self-dual tessellations {4g, 4g}. Knowing the generators of the quaternion orders which realize the edgepairings 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.

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Section: Preliminariesmentioning

confidence: 99%

“…), in the Euclidean sense. Theorem 2.1: [2] If S is a geometrically uniform signal set, then all the Dirichlet regions have the same shape.…”

Section: A Geometrically Uniform Signal Setsmentioning

confidence: 99%

Section: B Quaternion Algebras and Quaternion Ordersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Construction of Signal Sets From Quotient Rings of the Quaternion Orders Associated With Arithmetic Fuchsian Groups

Queiroz

Palazzo

2020

IEEE Access

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“…The main potential for coding in the hyperbolic plane is the infinitude of essentially distinct tessellations, in contrast to a Euclidean case. After [1], several papers connecting hyperbolic geometry with communication and coding theory have been published, [2], [3], [4], [5], among others. In [6] it is stated that it is possible to devise more efficient error correcting codes, in terms of error probability, if they are elaborated from two-dimensional varieties with genus g ≥ 2.…”

Section: Introductionmentioning

confidence: 99%