Random codes: minimum distances and error exponents

Minimum distances, distance distributions, and error exponents on a binary-symmetric channel (BSC) are given for typical codes from Shannon's random code ensemble and for typical codes from a random linear code ensemble. A typical random code of length N and rate R is shown to have minimum distance N/spl delta//sub GV/(2R), where /spl delta//sub GV/(R) is the Gilbert-Varshamov (GV) relative distance at rate R, whereas a typical linear code (TLC) has minimum distance N/spl delta//sub GV/(R). Consequently, a TLC has a better error exponent on a BSC at low rates, namely, the expurgated error exponent.     

Approximately Lower Triangular Ensembles of LDPC Codes With Linear Encoding Complexity

The complexity of brute-force encoding of low-density parity-check (LDPC) codes is proportional to the square value of the block length. Richardson and Urbanke have proposed efficient encoding algorithms for LDPC codes. These algorithms permute the parity-check matrix of the code iteratively, such that it becomes approximately lower triangular. We propose a new approach for efficient encoding of LDPC codes in which we modify the code ensemble to force an approximate lower triangular structure, thus eliminating the need to apply the algorithms of Richardson and Urbanke in this ensemble. We prove that the new ensemble has the same asymptotic threshold as the corresponding standard ensemble. The new ensemble can be used for linear time encoding of an arbitrary code profile. Computer simulations confirm that the performances of the standard and new ensembles are also very similar when using finite length codes     

基于等差数列的LDPC码编码器设计

本文提出一种基于等差数列构造LDPC码的新码类,称为D-LDPC码.文中给出了D-LDPC码的 D 矢量和 D 矩阵的定义,提供一个不含4线循环的确定结构的稀疏奇偶校验矩阵 H 的通用结构,提出一种递归形式的D-LDPC码编码器设计算法.D-LDPC码的编码计算复杂度为O(M),低于卷积码的O(N)复杂度;在中、低码长,任意码率时,性能可与卷积码比美,甚至超越卷积码;编码参数设计灵活,既能与现有标准兼容,又能满足未来发展的需求.     

Complexity versus performance of capacity-achieving irregular repeat-accumulate codes on the binary erasure channel

We derive upper and lower bounds on the encoding and decoding complexity of two capacity-achieving ensembles of irregular repeat-accumulate (IRA1 and IRA2) codes on the binary erasure channel (BEC). These bounds are expressed in terms of the gap between the channel capacity and the rate of a typical code from this ensemble for which reliable communications is achievable under message-passing iterative (MPI) decoding. The complexity of the ensemble of IRA1 codes grows like the negative logarithm of the gap to capacity. On the other hand, the complexity of the ensemble of IRA2 codes with any choice of the degree distribution grows at least like the inverse square root of the gap to capacity, and at most like the inverse of the gap to capacity.     

Performance of low-density parity-check codes with linear minimum distance

This correspondence studies the performance of the iterative decoding of low-density parity-check (LDPC) code ensembles that have linear typical minimum distance and stopping set size. We first obtain a lower bound on the achievable rates of these ensembles over memoryless binary-input output-symmetric channels. We improve this bound for the binary erasure channel. We also introduce a method to construct the codes meeting the lower bound for the binary erasure channel. Then, we give upper bounds on the rate of LDPC codes with linear minimum distance when their right degree distribution is fixed. We compare these bounds to the previously derived upper bounds on the rate when there is no restriction on the code ensemble.     

Structured IRA Codes: Performance Analysis and Construction

In this letter, we present design techniques for structured irregular repeat-accumulate (S-IRA) codes with low error-rate floors. These S-IRA codes need not be quasi-cyclic, permitting flexibility in code dimension, length, and rate. We present a simple ensemble estimate of the level of the error-rate floor of finite-length IRA codes on the additive white Gaussian noise channel. This performance estimate provides guidance on the choice of IRA code column weights which yield low floors. We also present two design algorithms for S-IRA codes accompanied by software- and hardware-based performance results which demonstrate their low floors. Lastly, we present two design algorithms for multirate S-IRA code families implementable by a single encoder/decoder     

Near-capacity coding in multicarrier modulation systems

We apply irregular low-density parity-check (LDPC) codes to the design of multilevel coded quadrature amplitude modulation (QAM) schemes for application in discrete multitone systems in frequency-selective channels. A combined Gray/Ungerboeck scheme is used to label each QAM constellation. The Gray-labeled bits are protected using an irregular LDPC code with iterative soft-decision decoding, while other bits are protected using a high-rate Reed-Solomon code with hard-decision decoding (or are left uncoded). The rate of the LDPC code is selected by analyzing the capacity of the channel seen by the Gray-labeled bits and is made adaptive by selective concatenation with an inner repetition code. Using a practical bit-loading algorithm, we apply this coding scheme to an ensemble of frequency-selective channels with Gaussian noise. Over a large number of channel realizations, this coding scheme provides an average effective coding gain of more than 7.5 dB at a bit-error rate of 10/sup -7/ and a block length of approximately 10/sup 5/ b. This represents a gap of approximately 2.3 dB from the Shannon limit of the additive white Gaussian noise channel, which could be closed to within 0.8-1.2 dB using constellation shaping.     

On the Asymptotic Weight and Stopping Set Distribution of Regular LDPC Ensembles

In this correspondence, we estimate the variance of weight and stopping set distribution of regular low-density parity-check (LDPC) ensembles. Using this estimate and the second moment method we obtain bounds on the probability that a randomly chosen code from regular LDPC ensemble has its weight distribution and stopping set distribution close to respective ensemble averages. We are able to show that a large fraction of total number of codes have their weight and stopping set distribution close to the average     

Subspace-based active user identification for a collision-free slotted ad hoc network

We propose a novel spreading code scheme, transmitter-receiver-based code, for wireless ad hoc networks. The design facilitates collision resolution using multiuser detection at each node, and is more bandwidth efficient than creating orthogonal channels in time or frequency. A subspace-based receiver structure is introduced, which identifies users of interest, or "active" users, with minimal prior information on the spreading code ensemble. A subspace-based blind multiuser detector can then be implemented to suppress multiaccess interference. The performance of the proposed active user identifier is studied by investigating its false alarm rate P/sub f/ and miss rate P/sub m/. Tradeoffs between P/sub f/ and P/sub m/ are discussed, and a graphical method to determine the threshold value d/sub th/ of the decision statistic used in discriminating between active and inactive channels is introduced.     

The serial concatenation of rate-1 codes through uniform random interleavers

Until the analysis of repeat accumulate codes by Divsalar et al. (1998), few people would have guessed that simple rate-1 codes could play a crucial role in the construction of "good" binary codes. We construct "good" binary linear block codes at any rate r<1 by serially concatenating an arbitrary outer code of rate r with a large number of rate-1 inner codes through uniform random interleavers. We derive the average output weight enumerator (WE) for this ensemble in the limit as the number of inner codes goes to infinity. Using a probabilistic upper bound on the minimum distance, we prove that long codes from this ensemble will achieve the Gilbert-Varshamov (1952) bound with high probability. Numerical evaluation of the minimum distance shows that the asymptotic bound can be achieved with a small number of inner codes. In essence, this construction produces codes with good distance properties which are also compatible with iterative "turbo" style decoding. For selected codes, we also present bounds on the probability of maximum-likelihood decoding (MLD) error and simulation results for the probability of iterative decoding error.     

The random coding bound is tight for the average code (Corresp.)

The random coding bound of information theory provides a well-known upper bound to the probability of decoding error for the best code of a given rate and block length. The bound is constructed by upper-bounding the average error probability over an ensemble of codes. The bound is known to give the correct exponential dependence of error probability on block length for transmission rates above the critical rate, but it gives an incorrect exponential dependence at rates below a second lower critical rate. Here we derive an asymptotic expression for the average error probability over the ensemble of codes used in the random coding bound. The result shows that the weakness of the random coding bound at rates below the second critical rate is due not to upperbounding the ensemble average, but rather to the fact that the best codes are much better than the average at low rates.     

Cutoff rate of 16-DAPSK over a Rayleigh fading channel

The cutoff rate of 16-DAPSK modulation and differential detection over a Rayleigh fading channel is calculated, which provides the ensemble performance of coded data transmission. Given the code rate, the minimum SNR required for reliable communication is obtained with parameters such as the ring ratio and the diversity number     

基于 Q -矩阵的LDPC码编码器设计

本文给出 Q 矩阵的定义,在此基础上提出由 Q 矩阵构造的LDPC码新码族;研究 Q 矩阵的性质,根据 Q 矩阵的性质和变化形式,提出一种构造稀疏奇偶校验矩阵 H 的算法,同时给出一种基于 Q 矩阵的LDPC码编码器设计算法.模拟仿真表明,采用和积迭代解码算法,在0.5码率,6144码长,10^-5以下误码率时, Q 矩阵LDPC码目前的最好性能达到离香农限1.5dB.本文还研究了快速搜索 Q 矩阵的算法.如果对 Q 矩阵采用离线搜索,在线存储 Q 矢量的方式,可使构造 H 矩阵的计算复杂度为零,编码器算法复杂度与编码长度N成线性关系. Q 矩阵LDPC码不同于现有其它结构LDPC码的独特之处在于,对码长和码率参数的设计具有高度灵活性,使其能与现有标准兼容.     

Exponential error bounds for random codes in the arbitrarily varying channel

Random codes for the arbitrarily varying channel are investigated. The code ensemble is restricted to increase only exponentially with codeword length; a quantity called key rate is used as a measure of the rate of increase. The reliability function for such codes is related to the reliability function for codes with unlimited key rate. Explicit results are obtained in some examples of practical interest.     

LDPC-coded cooperative relay systems: performance analysis and code design

We treat the problem of designing low-density parity-check (LDPC) codes to approach the capacity of relay channels. We consider an efficient analysis framework that decouples the factor graph (FG) of a B-block transmission into successive partial FGs, each of which denotes a two-block transmission. We develop design methods to find the optimum code ensemble for the partial FG. In particular, we formulate the relay operations and the destination operations as equivalent virtual MISO and MIMO systems, and employ a binary symmetric channel (BSC) model for the relay node output. For AWGN channels, we further develop a Gaussian approximation for the detector output at the destination node. Jointly treating the relay and the destination, we analyze the performance of the LDPC-coded relay system using the extrinsic mutual information transfer(EXIT) chart technique. Furthermore, differential evolution is employed to search for the optimum code ensemble. Our results show that the optimized codes always outperform the regular LDPC codes with a significant gain; in the AWGN case, when Protocol-II is employed and the relay is close to the source, the optimized code performs within 0.1dB to the capacity bound.     

Design methods for irregular repeat-accumulate codes

We optimize the random-like ensemble of irregular repeat-accumulate (IRA) codes for binary-input symmetric channels in the large block-length limit. Our optimization technique is based on approximating the evolution of the densities (DE) of the messages exchanged by the belief-propagation (BP) message-passing decoder by a one-dimensional dynamical system. In this way, the code ensemble optimization can be solved by linear programming. We propose four such DE approximation methods, and compare the performance of the obtained code ensembles over the binary-symmetric channel (BSC) and the binary-antipodal input additive white Gaussian noise channel (BIAWGNC). Our results clearly identify the best among the proposed methods and show that the IRA codes obtained by these methods are competitive with respect to the best known irregular low-density parity-check (LDPC) codes. In view of this and the very simple encoding structure of IRA codes, they emerge as attractive design choices.     

Generalized Low-Density Parity-Check Codes Based on Hadamard Constraints

In this paper, we consider the design and analysis of generalized low-density parity-check (GLDPC) codes in AWGN channels. The GLDPC codes are specified by a bipartite Tanner graph, as with standard LDPC codes, but with the single parity-check constraints replaced by general coding constraints. In particular, we consider imposing Hadamard code constraints at the check nodes for a low-rate approach, termed LDPC-Hadamard codes. We introduce a low-complexity message-passing based iterative soft-input soft-output (SISO) decoding algorithm, which employs the a posteriori probability (APP) fast Hadamard transform (FHT) for decoding the Hadamard check codes at each decoding iteration. The achievable capacity with the GLDPC codes is then discussed. A modified LDPC-Hadamard code graph is also proposed. We then optimize the LDPC-Hadamard code ensemble using a low-complexity optimization method based on approximating the density evolution by a one-dimensional dynamic system represented by an extrinsic mutual information transfer (EXIT) chart. Simulation results show that the optimized LDPC-Hadamard codes offer better performance in the low-rate region than low-rate turbo-Hadamard codes, but also enjoy a fast convergence rate. A rate-0.003 LDPC-Hadamard code with large block length can achieve a bit-error-rate (BER) performance of 10^-5 at -1.44 dB, which is only 0.15 dB away from the ultimate Shannon limit (-1.592 dB) and 0.24 dB better than the best performing low-rate turbo-Hadamard codes     

Turbo codes extended with outer BCH code

The `error floor' observed in several simulations with the turbo codes is verified by calculation of an upper bound to the bit error rate for the ensemble of all interleavers. Also an easy way to calculate the weight enumerator used in this bound is presented. An extended coding scheme is proposed including an outer BCH code correcting a few bit errors     

Lattices which are good for (almost) everything

We define an ensemble of lattices, and show that for asymptotically high dimension most of its members are simultaneously good as sphere packings, sphere coverings, additive white Gaussian noise (AWGN) channel codes and mean-squared error (MSE) quantization codes. These lattices are generated by applying Construction A to a random linear code over a prime field of growing size, i.e., by "lifting" the code to /spl Ropf//sup n/.     

On the Probability of Undetected Error for Linear Block Codes

The problem of computing the probability of undetected error is considered for linear block codes used for error detection. The recent literature is first reviewed and several results are extended. It is pointed out that an exact calculation can be based on either the weight distribution of a code or its dual. Using the dual code formulation, the probability of undetected error for the ensemble of all nonbinary linear block codes is derived as well as a theorem that shows why the probability of undetected error may not be a monotonic function of channel error rate for some poor codes. Several bounds on the undetected error probability are then presented. We conclude with detailed examples of binary and nonbinary codes for which exact results can be obtained. An efficient technique for measuring an unknown weight distribution is suggested and exact results are compared with experimental results.