Arithmetic fuchsian groups and space time block codes

Carvalho, Edmir Daniel; Andrade, Antonio Aparecido de; Palazzo, Reginaldo; Filho, Jozué Vieira

doi:10.1590/s1807-03022011000300001

Cited by 7 publications

(8 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The division algebra structure associated with these orders of quaternions from Fuchsian arithmetic groups provided an efficient algebraic technique in the process of combating the diversity that appears in antenna-to-antenna transmission problems. This allowed the construction of new families of space-time block codes that satisfy the property of full diversity [18,26,27].…”

Section: Introdutionmentioning

confidence: 99%

Hyperbolic Geometrically Uniform Codes and Ungerboeck Partitioning on the Double Torus

et al. 2022

View full text Add to dashboard Cite

Current research builds labelings for geometrically uniform codes on the double torus through tiling groups. At least one labeling group was provided for all of the 11 regular tessellations on the double torus, derived from triangular Fuchsian groups, as well as extensions of these labeling groups to generate new codes. An important consequence is that such techniques can be used to label geometrically uniform codes on surfaces with greater genera. Furthermore, partitioning chains are constructed into geometrically uniform codes using soluble groups as labeling, which in some cases results in an Ungerboeck partitioning for the surface. As a result of these constructions, it is demonstrated that, as in Euclidean spaces, modulation and encoding can be combined in a single step in hyperbolic space.

show abstract

Section: Introdutionmentioning

confidence: 99%

Hyperbolic Geometrically Uniform Codes and Ungerboeck Partitioning on the Double Torus

et al. 2022

View full text Add to dashboard Cite

show abstract

“…Our interest in Fuchsian groups as a basis for code construction stems from a series of recent papers by Palazzo et al In [20,6,16,17], among others, various interesting connections between Fuchsian groups and signal constellation design are presented. In [20], the authors construct Fuchsian groups suitable for signal constellation design.…”

Section: Introductionmentioning

confidence: 99%

Fuchsian codes with arbitrarily high code rates

Blanco-Chacón

Hollanti

Alsina

et al. 2016

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

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.

show abstract

“…Shimura curves are also present in the theory of error-correcting codes [11]. More recently, Fuchsian groups have made an appearance [19,23,6,21] in the context of signal constellation design with potential applications in communications.…”

Section: Introductionmentioning

confidence: 99%

“…• Finally, we present an alternative method for constructing Fuchsian codes by certain parametrization of the integer tuples defining the Möbius transformations used for the code construction. Our interest in Fuchsian groups as a basis for code construction stems from a series of recent papers by Palazzo et al In [23,6,19,21], among others, various interesting connections between Fuchsian groups and signal constellation design are presented. In [23], the authors construct Fuchsian groups suitable for signal constellation construction.…”

Section: Introductionmentioning

confidence: 99%

“…Building on this work, in [21] the authors construct dense tessellations and counting Dirichlet domains in tessellations of certain type. In [6] the authors use units of quaternion orders to construct space-time matrices with the potential use case being wireless multi-antenna (MIMO) communications. We refer the reader to [17,14] as the early references to the use of division algebras and maximal orders in MIMO, and to [3] for a more general introduction to the topic.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Nonuniform Fuchsian codes for noisy channels

Blanco-Chacón

Remón

Hollanti

et al. 2014

Journal of the Franklin Institute

View full text Add to dashboard Cite

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

show abstract

Arithmetic fuchsian groups and space time block codes

Cited by 7 publications

References 12 publications

Hyperbolic Geometrically Uniform Codes and Ungerboeck Partitioning on the Double Torus

Hyperbolic Geometrically Uniform Codes and Ungerboeck Partitioning on the Double Torus

Fuchsian codes with arbitrarily high code rates

Nonuniform Fuchsian codes for noisy channels

Contact Info

Product

Resources

About