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.
Although codes related to Fuchsian groups have been considered before, our construction is original in that it describes the complete construction and decoding process, whereas earlier work has largely concentrated on the constellation design while giving little attention to the decoding and performance aspects. Another key difference to the aforementioned works is that we are studying codes on the arising from quaternion algebras and Fuchsian groups, and our aim is to apply the codes to the classical (euclidean) channel models such as the aforementioned AWGN channel, with possible future extension to fading channels @cite_2 @cite_23 . We do not use hyperbolic metric as our design metric, but use the Fuchsian group as a starting point to the code generation. Nevertheless, our decoder will rely on hyperbolic geometry as opposed to the classical decoders based on euclidean geometry.
@cite_23 @cite_2
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A novel construction of lattices is proposed. This construction can be thought of as Construction A with codes that can be represented as the Cartesian product of L linear codes over Fp1 , . . . ,FpL , respectively; hence, is referred to as the product construction. The existence of a sequence of such lattices that are good for quantization and Poltyrev-good under multistage decoding is shown. This family of lattices is then used to generate a sequence of nested lattice codes which allows one to achieve the same computation rate of Nazer and Gastpar for computeand-forward under multistage decoding, which is referred to as lattice-based multistage compute-and-forward. Motivated by the proposed lattice codes, two families of signal constellations are then proposed for the separation-based compute-and-forward framework proposed by together with a multilevel coding multistage decoding scheme tailored specifically for these constellations. This scheme is termed separation-based multistage compute-and-forward and is shown having a complexity of the channel coding dominated by the greatest common divisor of the constellation size (may not be a prime number) instead of the constellation size itself. In this paper, we propose a new class of lattices constructed from polar codes, namely polar lattices, to achieve the capacity @math of the additive white Gaussian-noise (AWGN) channel for any signal-to-noise ratio (SNR). Our construction follows the multilevel approach of , where on each level we construct a capacity-achieving polar code. The component polar codes are naturally nested, thereby fulfilling the requirement of the multilevel lattice construction. We prove that polar lattices are AWGN-good, in the sense that the error probability (for infinite lattice decoding) vanishes for any fixed volume-to-noise ratio (VNR) greater than @math . Furthermore, using the technique of source polarization, we propose discrete Gaussian shaping over the polar lattice to satisfy the power constraint. The proposed polar lattices permit low-complexity multistage successive cancellation decoding. Both the construction and shaping are explicit, and the overall complexity of encoding and decoding is @math for any fixed target error probability. Multiple antennas at both the transmitter and receiver ends of a wireless digital transmission channel may increase both data rate and reliability. Reliable high rate transmission over such channels can only be achieved through Space-Time coding. Rank and determinant code design criteria have been proposed to enhance diversity and coding gain. The special case of full-diversity criterion requires that the difference of any two distinct codewords has full rank. Extensive work has been done on Space–Time coding, aiming at finding fully diverse codes with high rate. Division algebras have been proposed as a new tool for constructing Space–Time codes, since they are non-commutative algebras that naturally yield linear fully diverse codes. Their algebraic properties can thus be further exploited to improve the design of good codes. The aim of this work is to provide a tutorial introduction to the algebraic tools involved in the design of codes based on cyclic division algebras. The different design criteria involved will be illustrated, including the constellation shaping, the information lossless property, the non-vanishing determinant property, and the diversity multiplexing trade-off. The final target is to give the complete mathematical background underlying the construction of the Golden code and the other Perfect Space–Time block codes. We construct a new family of space-time codes suited for multiple-antenna non-coherent communications over Rayleigh flat fading channels. These codes use all the complex degrees of freedom of the system, i.e. (M times(1-M T)) symbols per channel use, where T is the code length and M is the number of transmit antennas. Their codewords belong to the Grassmann manifold GTldrM (C), the set of the M-dimensional vector subspaces of CT. Our codes are built from space-time codes for coherent systems via a non linear map (parameterization) called the exponential map. The relationships between some properties of the non-coherent space-time codes and their corresponding coherent codes are investigated. We also propose a simplified decoding for these classes of unitary space-time codes. We analyse the performance of these codes in terms of complexity and error probability and we compare them with some current best propositions, in particular training-based codes. The performance of our codes is substantially equivalent or slightly better than the one of training-based codes, under GLRT (Generalized Likelihood Ratio Test) decoding. When a simplified decoder is applied at the receiver, the error probability curves of the two propositions seem to be basically equivalent in the SIMO case with small block size, high spectral efficiency and high number of receive antennas, and in the 2times2 MIMO case with low spectral efficiency. In the other cases the training codes with simplified decoding seem to outperform our proposition.