A reduction point algorithm for cocompact Fuchsian groups and applications

Bayer, Pilar; Remón, Dionís

doi:10.3934/amc.2014.8.223

Cited by 4 publications

(13 citation statements)

References 6 publications

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“…One of the main features of the codes in [4,2] is that they have logarithmic decoding complexity. We have shown that by adapting the point reduction algorithm introduced in [3] to the present, more general case, the decoding complexity of the corresponding Fuchsian codes remains logarithmic in the codebook size, provided that we have a fundamental domain and a representation of the Fuchsian group.…”

Section: Discussionmentioning

confidence: 99%

“…As the decoding of Fuchsian codes is based on the point reduction algorithm, let us summarize here its main features. See [3] for validity and complexity proofs.…”

Section: The Point Reduction Algorithm (Pra)mentioning

confidence: 99%

“…Recently, in [2] and [4], a family of nonlinear and nonuniform Fuchsian codes were constructed based on Fuchsian groups of the first kind defined from quaternion algebras over Q. The decoding procedure of these codes is based on a modified point reduction algorithm having logarithmic complexity in the codebook size, which was shown to imply logarithmic decoding complexity for the Fuchsian codes [2], [3].…”

Section: Introductionmentioning

confidence: 99%

“…In Section 4, we explore such Fuchsian codes for a number of Fuchsian groups derived from those of signature (1; e), classified by Takeuchi [19]. This task has required to explicitly construct suitable fundamental domains for these groups and to adapt the point reduction algorithm [3] to our case. We also provide some numerical evidence to justify the study of higher rate Fuchsian codes by showing that increasing the rate may indeed increase the codebook size and hence the data rate.…”

Section: Introductionmentioning

confidence: 99%

“…As already mentioned, a general algorithm for the point reduction problem was recently proposed in [3]. Some specific examples can be found in [2].…”

mentioning

confidence: 99%

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Fuchsian codes with arbitrarily high code rates

Blanco-Chacón

Hollanti

Alsina

et al. 2016

Journal of Pure and Applied Algebra

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Abstract. Recently, so-called Fuchsian codes have been proposed in [I. BlancoChacón et al., "Nonuniform Fuchsian codes for noisy channels", J. of the Franklin Institute 2014] for communication over channels subject to additive white Gaussian noise (AWGN). The two main advantages of Fuchsian codes are their ability to compress information, i.e., high code rate, and their logarithmic decoding complexity. In this paper, we improve the first property further by constructing Fuchsian codes with arbitrarily high code rates while maintaining logarithmic decoding complexity. Namely, in the case of Fuchsian groups derived from quaternion algebras over totally real fields we obtain a code rate that is proportional to the degree of the base field. In particular, we consider arithmetic Fuchsian groups of signature (1; e) to construct explicit codes having code rate six, meaning that we can transmit six independent integers during one channel use.

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Section: Discussionmentioning

confidence: 99%

“…As the decoding of Fuchsian codes is based on the point reduction algorithm, let us summarize here its main features. See [3] for validity and complexity proofs.…”

Section: The Point Reduction Algorithm (Pra)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…As already mentioned, a general algorithm for the point reduction problem was recently proposed in [3]. Some specific examples can be found in [2].…”

mentioning

confidence: 99%

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Fuchsian codes with arbitrarily high code rates

Blanco-Chacón

Hollanti

Alsina

et al. 2016

Journal of Pure and Applied Algebra

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Nonuniform Fuchsian codes for noisy channels

Blanco-Chacón

Remón

Hollanti

et al. 2014

Journal of the Franklin Institute

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We develop a new transmission scheme for additive white Gaussian noisy (AWGN) channels based on Fuchsian groups from rational quaternion algebras. The structure of the proposed Fuchsian codes is nonlinear and nonuniform, hence conventional decoding methods based on linearity and symmetry do not apply. Previously, only brute force decoding methods with complexity that is linear in the code size exist for general nonuniform codes. However, the properly discontinuous character of the action of the Fuchsian groups on the complex upper half-plane translates into decoding complexity that is logarithmic in the code size via a recently introduced point reduction algorithm.Peer ReviewedPostprint (author’s final draft

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