Cyclic division algebras: a tool for space-time coding

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

Golden Space-Time Trellis Coded Modulation

In this paper, we present a concatenated coding scheme for a high rate $2\times 2$ multiple-input multiple-output (MIMO) system over slow fading channels. The inner code is the Golden code \cite{Golden05} and the outer code is a trellis code. Set partitioning of the Golden code is designed specifically to increase the minimum determinant. The branches of the outer trellis code are labeled with these partitions. Viterbi algorithm is applied for trellis decoding. In order to compute the branch metrics a lattice sphere decoder is used. The general framework for code optimization is given. The performance of the proposed concatenated scheme is evaluated by simulation. It is shown that the proposed scheme achieves significant performance gains over uncoded Golden code.Comment: 33 pages, 13 figure

On Pseudocodewords and Improved Union Bound of Linear Programming Decoding of HDPC Codes

In this paper, we present an improved union bound on the Linear Programming (LP) decoding performance of the binary linear codes transmitted over an additive white Gaussian noise channels. The bounding technique is based on the second-order of Bonferroni-type inequality in probability theory, and it is minimized by Prim's minimum spanning tree algorithm. The bound calculation needs the fundamental cone generators of a given parity-check matrix rather than only their weight spectrum, but involves relatively low computational complexity. It is targeted to high-density parity-check codes, where the number of their generators is extremely large and these generators are spread densely in the Euclidean space. We explore the generator density and make a comparison between different parity-check matrix representations. That density effects on the improvement of the proposed bound over the conventional LP union bound. The paper also presents a complete pseudo-weight distribution of the fundamental cone generators for the BCH[31,21,5] code

Design and Performance Analysis for LDPC Coded Modulation in Multiuser MIMO Systems

The channel capacity can be greatly increased by using multiple transmit and receive antennas, which is usually called multi-input multi-output (MIMO) systems. Iterative processing has achieved near-capacity on a single-antenna Gaussian or Rayleigh fading channel. How to use the iterative technique to exploit the capacity potential in single-user and/or multiuser MIMO systems is of great interest. We propose a low-density parity-check (LDPC) coded modulation scheme in multiuser MIMO systems. The receiver can be regarded as a serially concatenated iterative detection and decoding scheme, where the LDPC decoder performs the role of outer decoder and the multiuser demapper does that of the inner decoder. For the proposed scheme, appropriate selection of a bit-to-symbol mapping is crucial to achieve a good performance, so we investigate and find the best mapping under various cases.Analytical bound serves as a useful tool to assess system performance. The search for powerful codes has motivated the introduction of efficient bounding techniques tailored to some ensembles of codes. We then investigate combinatorial union bounding techniques for fast fading multiuser MIMO systems. The union upper bound on maximum likelihood (ML) decoding error probability provides a prediction for the system performance, with which the simulated system performance can be compared. Closed-form expression for the union bound is obtained, which can be evaluated efficiently by using a polynomial expansion. In addition, the constrained channel capacity and the threshold obtained from extrinsic information transfer (EXIT) chart can also serve as performance measures. Based on the analysis for fast fading case, we generalize the union upper bound to the block fading case

On the Performance of Sphere Decoding of Block Codes

The performance of sphere decoding of block codes over a variety of channels is investigated. We derive a tight bound on the performance of maximum likelihood decoding of linear codes on q-ary symmetric channels. We use this result to bound the performance of q-ary hard decision sphere decoders. We also derive a tight bound on the performance of soft decision sphere decoders on the AWGN channel for BPSK and M-PSK modulated block codes. The performance of soft decision sphere decoding of arbitrary finite lattices or block codes is also analyzed