Universally optimum block codes and convolutional codes with maximum likelihood decoding

A new proof is presented for the existence of block codes whose error probability under maximum likelihood decoding is bounded asymptotically by the random coding bound universally over all discrete memoryless channels. On the basis of this result, the existence of convolutional codes with universally optimum performance is shown. Furthermore the existence of block codes which attain the expurgated bound universally over all discrete memoryless channels is proved under the use of maximum likelihood decoding.     

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.     

Universal decoding for finite-state channels

Universal decoding procedures for finite-state channels are discussed. Although the channel statistics are not known, universal decoding can achieve an error probability with an error exponent that, for large enough block length (or constraint length in case of convolutional codes), is equal to the random-coding error exponent associated with the optimal maximum-likelihood decoding procedure for the given channel. The same approach is applied to sequential decoding, yielding a universal sequential decoding procedure with a cutoff rate and an error exponent that are equal to those achieved by the classical sequential decoding procedure.     

Improved upper bounds on the ML decoding error probability ofparallel and serial concatenated turbo codes via their ensemble distancespectrum

The ensemble performance of parallel and serial concatenated turbo codes is considered, where the ensemble is generated by a uniform choice of the interleaver and of the component codes taken from the set of time-varying recursive systematic convolutional codes. Following the derivation of the input-output weight enumeration functions of the ensembles of random parallel and serial concatenated turbo codes, the tangential sphere upper bound is employed to provide improved upper bounds on the block and bit error probabilities of these ensembles of codes for the binary-input additive white Gaussian noise (AWGN) channel, based on coherent detection of equi-energy antipodal signals and maximum-likelihood decoding. The influence of the interleaver length and the memory length of the component codes is investigated. The improved bounding technique proposed here is compared to the conventional union bound and to a alternative bounding technique by Duman and Salehi (1998) which incorporates modified Gallager bounds. The advantage of the derived bounds is demonstrated for a variety of parallel and serial concatenated coding schemes with either fixed or random recursive systematic convolutional component codes, and it is especially pronounced in the region exceeding the cutoff rate, where the performance of turbo codes is most appealing. These upper bounds are also compared to simulation results of the iterative decoding algorithm     

Majority-logic-like vector symbol decoding with alternative symbolvalue lists

Majority-logic-like decoding is an outer concatenated code decoding technique using the structure of a binary majority logic code. It is shown that it is easy to adapt such a technique to handle the case where the decoder is given an ordered list of two or more prospective candidates for each inner code symbol. Large reductions in failure probability can be achieved. Simulation results are shown for both block and convolutional codes. Punctured convolutional codes allow a convenient flexibility of rate while retaining high decoding power. For example, a (856, 500) terminated convolutional code with an average of 180 random first-choice symbol errors can correct all the errors in a simple manner about 97% of the time, with the aid of second-choice values. A (856, 500) maximum-distance block code could correct only up to 178 errors based on guaranteed correction capability and would be extremely complex     

Performance of truncated type-II hybrid ARQ schemes with noisyfeedback over block fading channels

This paper considers truncated type-II hybrid automatic repeat-request (ARQ) schemes with noisy feedback over block fading channels. With these ARQ techniques, the number of retransmissions is limited, and, similar to forward error correction (FEC), error-free delivery of data packets cannot be guaranteed. Bounds on the average number of transmissions, the average coding rate as well as the reliability of the schemes are derived using random coding techniques, and the performance is compared with FEC. The random coding bounds reveal the achievable performance with block codes and maximum-likelihood soft-decision decoding. Union upper bounds and simulation results show that over block fading channels, these bounds can be closely approached with simple terminated convolutional codes and soft-decision Viterbi decoding. Truncated type-II hybrid ARQ and the corresponding FEC schemes have the same probability of packet erasure; however, the truncated ARQ schemes offer a trade-off between the average coding rate and the probability of undetected error. Truncated ARQ schemes have significantly higher average coding rates than FEC at high and medium signal-to-noise ratio even with noisy feedback. Truncated ARQ can be viewed as adaptive FEC that adapts to the instantaneous channel conditions     

A Study on Universal Codes With Finite Block Lengths

Based on random codes and typical set decoding, an alternative proof of Root and Varaiya's compound channel coding theorem for linear Gaussian channels is presented. The performance limit of codes with finite block length under a compound channel is studied through error bounds and simulation. Although the theorem promises uniform convergence of the probability of error as the block length approaches infinity, with short block lengths the performance can differ considerably for individual channels. Simulation results show that universal performance can be a practical goal as the block lengths become large.     

Boosting the error performance of suboptimal tailbiting decoders

Tailbiting is an attractive method to terminate convolutional codes without reducing the code rate. Maximum-likelihood and exact a posteriori probability decoding of tailbiting codes implies, however, a large computational complexity. Therefore, suboptimal decoding methods are often used in practical coding schemes. It is shown that suboptimal decoding methods work better when the slope of the active distances of the generating convolutional encoder is large. Moreover, it is shown that considering quasi-cyclic shifts of the received channel output can increase the performance of suboptimal tailbiting decoders. The findings are most relevant to tailbiting codes where the number of states is not small relative to the block length.     

高性能时不变LDPC卷积码构造算法研究

该文基于由QC-LDPC码获得时不变LDPC卷积码的环同构方法,设计了用有限域上元素直接获得时不变LDPC卷积码多项式矩阵的新算法。以MDS卷积码为例,给出了一个具体的构造过程。所提构造算法可确保所获得的时不变LDPC卷积码具有快速编码特性、最大可达编码记忆以及设计码率。基于滑动窗口的BP译码算法在AWGN信道上的仿真结果表明,该码具有较低的误码平台和较好的纠错性能。     

Erasure-free sequential decoding of trellis codes

An erasure-free sequential decoding algorithm for trellis codes, called the buffer looking algorithm (BLA), is introduced. Several versions of the algorithm can be obtained by choosing certain parameters and selecting a resynchronization scheme. These can be categorized as block decoding or continuous decoding, depending on the resynchronization scheme. Block decoding is guaranteed to resynchronize at the beginning of each block, but suffers some rate loss when the block length is relatively short. The performance of a typical block decoding scheme is analyzed, and we show that significant coding gains over Viterbi decoding can be achieved with much less computational effort. A resynchronization scheme is proposed for continuous sequential decoding. It is shown by analysis and simulation that continuous sequential decoding using this scheme has a high probability of resynchronizing successfully. This new resynchronization scheme solves the rate loss problem resulting from block decoding. The channel cutoff rate, demodulator quantization, and the tail's influence on performance are also discussed. Although this paper considers only the decoding of trellis codes, the algorithm can also be applied to the decoding of convolutional codes     

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.     

Bounds on the error probability ofml decoding for block and turbo-block codes

The performance of either structured or random turbo-block codes and binary, systematic block codes operating over the additive white Gaussian noise (Awgn) channel, is assessed by upper bounds on the error probalities of maximum likelihood (Ml) decoding. These bounds on the block and bit error probability which depend respectively on the distance spectrum and the input-output weight enumeration function (Iowef) of these codes, are compared, for a variety of cases, to simulated performance of iterative decoding and also to some reported simulated lower bounds on the performance ofMl decoders. The comparisons facilitate to assess the efficiency of iterative decoding (as compared to the optimalMl decoding rule) on one hand and the tightness of the examined upper bounds on the other. We focus here on uniformly interleaved and parallel concatenated turbo-Hamming codes, and to that end theIowefs of Hamming and turbo-Hamming codes are calculated by an efficient algorithm. The usefulness of the bounds is demonstrated for uniformly interleaved turbo-Hamming codes at rates exceeding the cut-off rate, where the results are compared to the simulated performance of iteratively decoded turbo-Hamming codes with structured and statistical interleavers. We consider also the ensemble performance of ‘repeat and accumulate’ (Ka) codes, a family of serially concatenated turbo-block codes, introduced by Divsalar, Jin and McEliece. Although, the outer and inner codes possess a very simple structure: a repetitive and a differential encoder respectively, our upper bounds indicate impressive performance at rates considerably beyond the cut-off rate. This is also evidenced in literature by computer simulations of the performance of iteratively decodedRa codes with a particular structured interleaver.     

低复杂度的TBCC自适应循环VA译码算法

咬尾是一种将卷积码转换为块码的技术,它消除了归零状态所造成的码率损失,同时避免了截尾带来的性能降低,在短块编码中具有明显优势.针对咬尾卷积码(TBCC)现有译码算法复杂度过大和收敛性问题,提出一种低复杂度的TBCC自适应循环维特比(VA)译码算法.该算法根据信道变化自适应调整译码迭代次数,使咬尾路径收敛到最佳.通过仿真...     

Finite-length analysis of low-density parity-check codes on thebinary erasure channel

In this paper, we are concerned with the finite-length analysis of low-density parity-check (LDPC) codes when used over the binary erasure channel (BEC). The main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of LDPC codes when decoded iteratively. We also give expressions for upper bounds on the average bit and block erasure probability for regular LDPC ensembles and the standard random ensemble under maximum-likelihood (ML) decoding. Finally, we present what we consider to be the most important open problems in this area     

An analysis of the block error probability performance of iterative decoding

Asymptotic iterative decoding performance is analyzed for several classes of iteratively decodable codes when the block length of the codes N and the number of iterations I go to infinity. Three classes of codes are considered. These are Gallager's regular low-density parity-check (LDPC) codes, Tanner's generalized LDPC (GLDPC) codes, and the turbo codes due to Berrou et al. It is proved that there exist codes in these classes and iterative decoding algorithms for these codes for which not only the bit error probability P/sub b/, but also the block (frame) error probability P/sub B/, goes to zero as N and I go to infinity.     

On the computation of the performance probabilities for block codeswith a bounded-distance decoding rule

When a block code is used on a discrete memoryless channel with an incomplete decoding rule that is based on a generalized distance, the probability of decoding failure, the probability of erroneous decoding, and the expected number of symbol decoding errors can be expressed in terms of the generalized weight enumerator polynomials of the code. For the symmetric erasure channel, numerically stable methods to compute these probabilities or expectations are proposed for binary codes whose distance distributions are known, and for linear maximum distance separable (MDS) codes. The method for linear MDS codes saves the computation of the weight distribution and yields upper bounds for the probability of erroneous decoding and for the symbol error rate by the cumulative binomial distribution. Numerical examples include a triple-error-correcting Bose-Chaudhuri-Hocquenghem (BCH) code of length 63 and a Reed-Solomon code of length 1023 and minimum distance 31     

On type-B1 burst-error-correcting convolutional codes

Two new classes of type-B1 burst-error-correcting convolutional codes are introduced. One of them requires a shorter length of guard space and a smaller number of shift register stages than optimum type-B2 codes used for type-B1 burst correction. Another class of codes improves the required number of shift register stages considerably when the correctable burst length is very large. In addition, these codes require a very short length of additional guard space to restore the decoder to correct operation after a decoding failure. Both classes of codes are derived in a straightforward manner and their implementations are also very simple. Thus, we can avoid type-B2 code procedures to correct type-B1 bursts. The codes derived here result in the more efficient and simply implemented type-B1 burst-correcting convolutional codes.     

A Comparison of Block and Convolutional Codes in ARQ Error Control Schemes

ARQ methods of error control can considerably improve the reliablity of data transmission in such areas as satellite communications, computer networks, etc. A number of ARQ schemes using both block and convolutional codes have appeared in the literature. In this paper, the following problem is addressed. Given two different implementations of an ARQ scheme, one using a block code and the other using a convolutional code, such that the bit error probability of both implementations does not exceed some specific value, which implementation has the higher throughput and under what conditions will it be attained? The comparison is made for three basic retransmission schemes using both hybrid and pure ARQ: stop-and-wait, go-back-N, and selective repeat. Numerical estimates of the throughput were obtained using approximate theoretical expressions for BCH codes and simulation results for sequential decoding of rate 1/2 convolutional codes. Parameters optimizing the performance of both block and convolutional codes for different channel conditions and round trip delays were found and were used to obtain these numerical estimates. Comparison of the quantitative results indicates a trend toward preferring convolutional codes as delay and/or block length increases. A binary symmetric channel with noiseless feedback was assumed. Possible implications for the Gaussian channel are also discussed.     

A new approach to the construction of high-rate convolutional codes

This letter presents a new technique to construct high-rate convolutional codes using a structure formed by a high-rate block code and a simpler convolutional code. The goal is to obtain good convolutional codes in terms of free distance and number of nearest neighbors, with better performance than punctured codes. The obtained codes improve over the best known high-rate punctured codes with the same rate and memory in terms of both bit error probability and computational decoding complexity     

Coded modulation in the block-fading channel: coding theorems and code construction

We consider coded modulation schemes for the block-fading channel. In the setting where a codeword spans a finite number N of fading degrees of freedom, we show that coded modulations of rate R bit per complex dimension, over a finite signal set /spl chi//spl sube//spl Copf/ of size 2/sup M/, achieve the optimal rate-diversity tradeoff given by the Singleton bound /spl delta/(N,M,R)=1+/spl lfloor/N(1-R/M)/spl rfloor/, for R/spl isin/(0,M/spl rfloor/. Furthermore, we show also that the popular bit-interleaved coded modulation achieves the same optimal rate-diversity tradeoff. We present a novel coded modulation construction based on blockwise concatenation that systematically yields Singleton-bound achieving turbo-like codes defined over an arbitrary signal set /spl chi//spl sub//spl Copf/. The proposed blockwise concatenation significantly outperforms conventional serial and parallel turbo codes in the block-fading channel. We analyze the ensemble average performance under maximum-likelihood (ML) decoding of the proposed codes by means of upper bounds and tight approximations. We show that, differently from the additive white Gaussian noise (AWGN) and fully interleaved fading cases, belief-propagation iterative decoding performs very close to ML on the block-fading channel for any signal-to-noise ratio (SNR) and even for relatively short block lengths. We also show that, at constant decoding complexity per information bit, the proposed codes perform close to the information outage probability for any block length, while standard block codes (e.g., obtained by trellis termination of convolutional codes) have a gap from outage that increases with the block length: this is a different and more subtle manifestation of the so-called "interleaving gain" of turbo codes.