Performance of a general decoding technique over the class ofrandomly chosen parity check codes

The paper extends a general decoding technique developed by Metzner and Kapturowski (1990) for concatenated code outer codes and for file disagreement location. That work showed the ability to correct most cases of d-2 or fewer erroneous block symbols, where d is the outer code minimum distance. Any parity check code can be used as the basis for the outer codes, and yet decoding complexity increases at most as the third power of the code length. In this correspondence, it is shown that, with a slight modification and no significant increase in complexity, the general decoding technique can be applied to the correction of many other cases beyond the code minimum distance. By considering average performance over all binary randomly chosen codes, it is seen that most error patterns of t_M or fewer block errors can be corrected, where: 1) t_M in most cases is much greater than the code minimum distance, and 2) asymptotically, the ratio of t_M to the theoretical maximum (the number of parity symbol blocks) approaches 1. Moreover, most cases of noncorrectable error block patterns are detected     

Bidirectional decoding of convolutional codes for indoor cellularradio

Bidirectional suboptimal breadth-first decoding of convolutional codes is an attractive technique for slowly varying and quasi-static fading channels as it restricts the extent of decoding errors due to correct path loss to very heavy noise or interference regions. This paper compares the performance of such a decoding scheme to the Viterbi algorithm over wideband TDMA indoor radio links where equalization and space diversity are also used to combat dispersive fading and cochannel interference. On the basis of equal computational complexity and equal decoding delay, suboptimal, breadth-first, bidirectional decoding of a long constraint length convolutional code is shown to be superior to Viterbi decoding of a shorter constraint length code. Furthermore, this advantage increases as the outage criterion (in terms of bit error rate) becomes more stringent which makes bidirectional decoding particularly attractive for data applications and makes channel coding a more attractive alternative to increasing the space diversity order at the receiver     

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     

Design and decoding of optimal high-rate convolutional codes

This correspondence deals with the design and decoding of high-rate convolutional codes. After proving that every (n,n-1) convolutional code can be reduced to a structure that concatenates a block encoder associated to the parallel edges with a convolutional encoder defining the trellis section, the results of an exhaustive search for the optimal (n,n-1) convolutional codes is presented through various tables of best high-rate codes. The search is also extended to find the "best" recursive systematic convolutional encoders to be used as component encoders of parallel concatenated "turbo" codes. A decoding algorithm working on the dual code is introduced (in both multiplicative and additive form), by showing that changing in a proper way the representation of the soft information passed between constituent decoders in the iterative decoding process, the soft-input soft-output (SISO) modules of the decoder based on the dual code become equal to those used for the original code. A new technique to terminate the code trellis that significantly reduces the rate loss induced by the addition of terminating bits is described. Finally, an inverse puncturing technique applied to the highest rate "mother" code to yield a sequence of almost optimal codes with decreasing rates is proposed. Simulation results applied to the case of parallel concatenated codes show the significant advantages of the newly found codes in terms of performance and decoding complexity.     

Majority-logic-like decoding of vector symbols

The use of the structure of one-step decodable majority logic codes for enhanced and simplified vector symbol decoding, such as outer code decoding of concatenated codes, is proposed. For J equations checking a particular symbol, the technique to be described almost always corrects the symbol if there are J-1 or fewer symbol errors, and often corrects cases of far more than J symbol errors. Ordinarily, majority level decoding with J equations for a symbol corrects the symbol in all cases where there are up to [J/2] errors. The decoding power is comparable to Reed-Solomon codes, but decoding is simpler than for Reed-Solomon codes     

Burst-trapping techniques for a compound channel

An adaptive decoding technique called burst trapping is presented to correct both random and burst errors. Two decoding algorithms are used, one for random errors, and the other for bursts. The former is based on a conventional correction technique, the latter utilizes an encoding procedure for which each information digit appears twice in the data stream, first unchanged, and second combined with (addition modulo2) a check digit of a widely separated later block. Whenever the number of errors within a code block are detected to be too large to correct with the random-error-correcting algorithm, the burst-correcting algorithm corrects these errors by recovering the information from later blocks where it appears in combination with check digits. It is shown that the scheme requires very limited guard space and has limited error propagation. Furthermore, the storage requirement is even smaller than the guard space. This is the only known coding system that has this desirable feature. Results of simulation of such codes over telephone channels indicate that the performance of such codes, when compared with interleaved block codes, offers better results at significantly lower cost.     

Theoretical analysis of the error correction performance ofmajority-logic-like vector symbol codes

The average codeword success probability of the majority-logic-like vector symbol (MLLVS) code is derived for the following two cases: (1) single-pass decoding and (2) upper bound of multipass decoding, when the received word has more than (J-1) symbol errors, where J is the number of check sum equations. The MLLVS code has been simulated by Metzner (1996), and it was concluded that the average error correcting capability of MLLVS codes exceed the decoding capability of Reed-Solomon codes, but is achieved with less complexity. Additionally, for codes that have larger structures, the error correcting capability is sustained even further with a high probability of decoding success through multipass decoding procedures. The mathematical derivations of the error correction performance beyond (J-1) symbol errors serve as theoretical proof of the MLLVS code error correcting capability that was shown only through simulation results until now by Metzner. One characteristic feature of this derivation is that it does not assume any specific inner code usage, enabling the derived decoding probability equations to be easily applied to any inner code selected, of a concatenated coding structure     

List Viterbi algorithms for continuous transmission

The conventional list Viterbi algorithm (LVA) produces a list of the L best output sequences over a certain block length in decoding a terminated convolutional code. We show in this paper that the LVA with a sufficiently long list is an optimum maximum-likelihood decoder for the concatenated pair of a convolutional code and a cyclic redundancy check (CRC) block code with error detection. The CRC is used to select the output. New LVAs for continuous transmission are proposed and evaluated, where no termination bits are required for the convolutional code for every CRC block. We also present optimum and suboptimum LVAs for tailbiting convolutional codes. Convolutional codes with Viterbi decoding were proposed for so-called hybrid in band on channel (hybrid IBOC) systems for digital audio broadcasting compatible with the frequency modulation band. For high-quality audio signals, it is beneficial to use error concealment/error mitigation techniques to avoid the worst type of channel errors. This requires a reliable error flag mechanism (error detection feature) in the channel decoder. A CRC on a block of audio information bits provides this mechanism. We demonstrate how the LVA can significantly reduce the flag rate compared to the regular Viterbi algorithm (VA) for the same transmission parameters. At the expense of complexity, a receiver optional LVA can reduce the flag rate by more than an order of magnitude. The difference in audio quality is dramatic. The LVA is backward compatible with a VA     

On turbo decoding of nonbinary codes

Symbol-by-symbol maximum a posteriori (MAP) decoding algorithms for nonbinary block and convolutional codes over an extension field GF(p ^a) are presented. Equivalent MAP decoding rules employing the dual code are given which are computationally more efficient for high-rate codes. It is shown that these algorithms meet all requirements needed for iterative decoding as the output of the decoder can be split into three independent estimates: soft channel value, a priori term and extrinsic value. The discussed algorithms are then applied to a parallel concatenated coding scheme with nonbinary component codes in conjunction with orthogonal signaling     

Reliable communication over channels with insertions, deletions,and substitutions

A new block code is introduced which is capable of correcting multiple insertion, deletion, and substitution errors. The code consists of nonlinear inner codes, which we call “watermark"” codes, concatenated with low-density parity-check codes over nonbinary fields. The inner code allows probabilistic resynchronization and provides soft outputs for the outer decoder, which then completes decoding. We present codes of rate 0.7 and transmitted length 5000 bits that can correct 30 insertion/deletion errors per block. We also present codes of rate 3/14 and length 4600 bits that can correct 450 insertion/deletion errors per block     

Hybrid block- self-orthogonal convolutional codes

Two convolutional-code construction schemes that utilize block codes are given. In the first method the generators of a self-orthogonal convolutional code (SOCC) are expanded. The generators of a block code whose block length is longer than that of the SOCC code replace the nonzero blocks of the convolutional code. The zero blocks are extended to the longer block length. There results a convolutional code whose blocks are self-orthogonal and which has a lower transmission rate. In the second scheme the parity constraints of an SOCC are expanded. The parity constraints of a block code replace some of the individual nonzero elements of the SOCC parity-check matrix, so that the convolutional code rate is greater than the block code rate. The resulting codes retain the SOCC advantages of simple implementation and limited error propagation. Both the encoding and the decoding can be based on the underlying block code. If a block code is majority decodable, then the resulting "hybrid" codes are majority decodable. Optimum majority-decodable block codes with up to five information bits per block are given, and from these codes several majority-decodable convolutional codes that are "optimum" with respect to the proposed construction are obtained.     

Symbol-by-symbol MAP decoding algorithm for high-rate convolutionalcodes that use reciprocal dual codes

A symbol-by-symbol maximum a posteriori (MAP) decoding algorithm for high-rate convolutional codes applying reciprocal dual convolutional codes is presented. The advantage of this approach is a reduction of the computational complexity since the number of codewords to consider is decreased. All requirements for iterative decoding schemes are fulfilled. Since tail-biting convolutional codes are equivalent to quasi-cyclic block codes, the decoding algorithm for truncated or terminated convolutional codes is modified to obtain a soft-in/soft-out decoder for high-rate quasi-cyclic block codes which also uses the dual code because of complexity reasons. Additionally, quasi-cyclic block codes are investigated as component codes for parallel concatenation. Simulation results obtained by iterative decoding are compared with union bounds for maximum likelihood decoding. The results of a search for high-rate quasi-cyclic block codes are given in the appendix     

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.     

Bit error probability calculations for convolutional codes withshort constraint lengths on very noisy channels

A technique for estimating convolutional code performance on very noisy channels is considered. Specifically, the performance of short constraint length codes operating near the channel cutoff rate is estimated. Decoding convolutional codes with a sliding window decoder (SWD) are considered. This decoder is an optimal (maximum likelihood) symbol decoder as the window size grows toward infinity, while the Viterbi decoder is the maximum-likelihood sequence estimator. The difference in the decoded BERs (bit error rates) between the two decoders is very small and approaches zero asymptotically as the channel BER decreases. Therefore, an estimate on the decoded BER for the SWD can also be used as an estimate of the decoded BER for Viterbi decoding     

Implementation of burst-correcting convolutional codes

A general procedure is formulated for decoding any convolutional code with decoding delayNblocks that corrects all bursts confined toror fewer consecutive blocks followed by a guard space of at leastN-1consecutive error-free blocks. It is shown that all such codes can be converted to a form called "doubly systematic" which simplifies the decoding circuitry. The decoding procedure can then be implemented with a circuit of the same order of complexity as a parity-checking circuit for a block-linear code. A block diagram of a complete decoder is given for an optimal burst-correcting code. It is further shown that error propagation after a decoding mistake is always terminated by the occurrence of a double guard space of error-free blocks.     

迭代译码的级联Reed-Solomon乘积码与卷积码

该文提出用Reed Solomon(RS)乘积码作为外码,卷积码作为内码的级联码方案并且内外码间用Congruential向量生成的交织图案对RS码符号进行重排列。对此级联码采用的迭代译码基于成员码的软译码算法。当迭代次数达到最大后,通过计算RS码的校正子,提出一种纠正残余错误的方法,进一步提高了系统的误比特性能。仿真结果表明,在AWGN信道中与迭代译码的级联RS/卷积码相比,当误比特率为1e-5时,新系统的编码增益大约有0.4 dB。     

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...     

Distributed decoding of cyclic block codes using a generalizationof majority-logic decoding

One-step majority-logic decoding is one of the simplest algorithms for decoding cyclic block codes. However, it is an effective decoding scheme for very few codes. This paper presents a generalization based on the “common-symbol decoding problem.” Suppose one is given M (possibly corrupted) codewords from M (possibly different) codes over the same field; suppose further that the codewords share a single symbol in common. The common-symbol decoding problem is that of estimating the symbol in the common position. This is equivalent to one-step majority logic decoding when each of the “constituent” codes is a simple parity check. This paper formulates conditions under which this decoding is possible and presents a simple algorithm that accomplishes the same. When applied to decoding cyclic block codes, this technique yields a decoder structure ideal for parallel implementation. Furthermore, this approach frequently results in a decoder capable of correcting more errors than one-step majority-logic decoding. To demonstrate the simplicity of the resulting decoders, an example is presented     

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     

Optimal decoding of linear codes for minimizing symbol error rate (Corresp.)

The general problem of estimating the a posteriori probabilities of the states and transitions of a Markov source observed through a discrete memoryless channel is considered. The decoding of linear block and convolutional codes to minimize symbol error probability is shown to be a special case of this problem. An optimal decoding algorithm is derived.