Multilevel block codes for Rayleigh-fading channels

New multilevel block codes for Rayleigh-fading channels are presented. At high signal-to-noise ratios (SNRs), the proposed block codes can achieve better bit error performance over TCM codes, optimum for fading channels, with comparable decoder complexity and bandwidth efficiency. The code construction is based on variant length binary component block codes. As component codes for the 8-PSK multilevel block construction, the authors propose two modified forms of Reed-Muller codes giving a good trade-off between the decoder complexity and the effective code rates. Code design criteria are derived from the error performance analysis. Multistage decoding shows very slight degradation of bit error performance relative to the maximum likelihood algorithm     

Multistage decoding of multilevel block M-PSK modulationcodes and its performance analysis

Multistage decoding of multilevel block multilevel phase-shift keying (M-PSK) modulation codes for the additive white Gaussian noise (AWGN) channel is investigated. Several types of multistage decoding, including a suboptimum soft-decision decoding scheme, are devised and analyzed. Upper bounds on the probability of an incorrect decoding of a code are derived for the proposed multistage decoding schemes. Error probabilities of some specific multilevel block 8-PSK modulation codes are evaluated and simulated. The computation and simulation results for these codes show that with multistage decoding, significant coding gains can be achieved with large reduction in decoding complexity. In one example, it is shown that the difference in performance between the proposed suboptimum multistage soft-decision decoding and the single-stage optimum decoding is small, only a fraction of a dB loss in SNR at the block error probability of 10^-6     

Product accumulate codes: a class of codes with near-capacity performance and low decoding complexity

We propose a novel class of provably good codes which are a serial concatenation of a single-parity-check (SPC)-based product code, an interleaver, and a rate-1 recursive convolutional code. The proposed codes, termed product accumulate (PA) codes, are linear time encodable and linear time decodable. We show that the product code by itself does not have a positive threshold, but a PA code can provide arbitrarily low bit-error rate (BER) under both maximum-likelihood (ML) decoding and iterative decoding. Two message-passing decoding algorithms are proposed and it is shown that a particular update schedule for these message-passing algorithms is equivalent to conventional turbo decoding of the serial concatenated code, but with significantly lower complexity. Tight upper bounds on the ML performance using Divsalar's (1999) simple bound and thresholds under density evolution (DE) show that these codes are capable of performance within a few tenths of a decibel away from the Shannon limit. Simulation results confirm these claims and show that these codes provide performance similar to turbo codes but with significantly less decoding complexity and with a lower error floor. Hence, we propose PA codes as a class of prospective codes with good performance, low decoding complexity, regular structure, and flexible rate adaptivity for all rates above 1/2.     

Multilevel turbo coding with short interleavers

The impact of the interleaver, embedded in the encoder for a parallel concatenated code, called the turbo code, is studied. The known turbo codes consist of long random interleavers, whose purpose is to reduce the value of the error coefficients. It is shown that an increased minimum Hamming distance can be obtained by using a structured interleaver. For low bit-error rates (BERs), we show that the performance of turbo codes with a structured interleaver is better than that obtained with a random interleaver. Another important advantage of the structured interleaver is the short length required, which yields a short decoding delay and reduced decoding complexity (in terms of memory). We also consider the use of turbo codes as component codes in multilevel codes. Powerful coding structures that consist of two component codes are suggested. Computer simulations are performed in order to evaluate the reduction in coding gain due to suboptimal iterative decoding. From the results of these simulations we deduce that the degradation in the performance (due to suboptimal decoding) is very small     

Box and match techniques applied to soft-decision decoding

In this paper, we improve the ordered statistics decoding algorithm by using matching techniques. This allows us: to reduce the worst case complexity of decoding (the error performance being fixed) or to improve the error performance (for a same complexity); to reduce the ratio between average complexity and worst case complexity; to achieve practically optimal decoding of rate-1/2 codes of lengths up to 128 (rate-1/2 codes are a traditional benchmark, for coding rates different from 1/2, the decoding is easier); to achieve near-optimal decoding of a rate-1/2 code of length 192, which could never be performed before.     

Suboptimum decoding of decomposable block codes

To decode a long block code with a large minimum distance by maximum likelihood decoding is practically impossible because the decoding complexity is simply enormous. However, if a code can be decomposed into constituent codes with smaller dimensions and simpler structure, it is possible to devise a practical and yet efficient scheme to decode the code. This paper investigates a class of decomposable codes, their distance and structural properties. It is shown that this class includes several classes of well-known and efficient codes as subclasses. Several methods for constructing decomposable codes or decomposing codes are presented. A two-stage (soft-decision or hard-decision) decoding scheme for decomposable codes, their translates or unions of translates is devised, and its error performance is analyzed for an AWGN channel. The two-stage soft-decision decoding is suboptimum. Error performances of some specific decomposable codes based on the proposed two-stage soft-decision decoding are evaluated. It is shown that the proposed two-stage suboptimum decoding scheme provides an excellent trade-off between the error performance and decoding complexity for codes of moderate and long block length     

一种基于奇偶校验码级联极化码的低复杂度译码算法

极化码作为一种纠错码,具有较好的编译码性能,已成为5G短码控制信道的标准编码方案.但在码长较短时,其性能不够优异.作为一种新型级联极化码,奇偶校验码与极化码的级联方案提高了有限码长的性能,但是其译码算法有着较高的复杂度.该文针对这一问题,提出一种基于奇偶校验码级联极化码的串行抵消局部列表译码(PC-PSCL)算法,该算...     

Linear-time encodable/decodable codes with near-optimal rate

We present an explicit construction of linear-time encodable and decodable codes of rate r which can correct a fraction (1-r-/spl epsiv/)/2 of errors over an alphabet of constant size depending only on /spl epsiv/, for every 00. The error-correction performance of these codes is optimal as seen by the Singleton bound (these are "near-MDS" codes). Such near-MDS linear-time codes were known for the decoding from erasures; our construction generalizes this to handle errors as well. Concatenating these codes with good, constant-sized binary codes gives a construction of linear-time binary codes which meet the Zyablov bound, and also the more general Blokh-Zyablov bound (by resorting to multilevel concatenation). Our work also yields linear-time encodable/decodable codes which match Forney's error exponent for concatenated codes for communication over the binary symmetric channel. The encoding/decoding complexity was quadratic in Forney's result, and Forney's bound has remained the best constructive error exponent for almost 40 years now. In summary, our results match the performance of the previously known explicit constructions of codes that had polynomial time encoding and decoding, but in addition have linear-time encoding and decoding algorithms.     

A multilevel approach to the design of DC-free line codes

A multilevel approach to the design of DC-free line codes is presented. The different levels can be used for different purposes, for example, to control the maximum accumulated charge or to guarantee a certain minimum distance. The advantages of codes designed by this method over similar codes are the improved run-length/accumulated-charge parameters, higher transmission rate, and the systematic nature of the code construction. The multilevel structure allows the redundancy in the signal selection procedure to be allocated efficiently among the different levels. It also allows the use of suboptimal staged decoding procedures that have performance/complexity advantages over maximum-likelihood decoding     

Noncoherent block-coded MPSK

For coherent detection, block-coded modulation is a bandwidth efficient scheme. In this paper, we propose theorems about the error performance of block-coded modulation using M-ary phase-shift keying (MPSK) for noncoherent detection. Based on these theorems, we propose a novel block-coded modulation scheme for noncoherent detection called noncoherent block-coded MPSK. The proposed scheme provides flexible designs of noncoherent block codes with different code rate, block length and error performance. Good noncoherent block codes can be easily obtained by properly choosing binary linear block codes as the component codes. Moreover, noncoherent block codes of this new scheme can be decoded by multistage decoding, which has the advantage of low complexity and satisfactory error performance. In this paper, two algorithms of multistage decoding for noncoherent detection are proposed as well. The error performance of some designed codes and decoding algorithms is verified by computer simulation.     

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     

Accumulate codes based on 1+D convolutional outer codes

A new construction of good, easily encodable, and soft-decodable codes is proposed in this paper. The construction is based on serially concatenating several simple 1+D convolutional codes as the outer code, and a rate-1 1/(1+D) accumulate code as the inner code. These codes have very low encoding complexity and require only one shift-forward register for each encoding branch. The input-output weight enumerators of these codes are also derived. Divsalar?s simple bound technique is applied to analyze the bit error rate performance, and to assess the minimal required signal-to-noise ratio (SNR) for these codes to achieve reliable communication under AWGN channel. Simulation results show that the proposed codes can provide good performance under iterative decoding.     

Efficient decoding algorithms for generalized Reed-Muller codes

Previously, a class of generalized Reed-Muller (RM) codes has been suggested for use in orthogonal frequency-division multiplexing. These codes offer error correcting capability combined with substantially reduced peak-to mean power ratios. A number of approaches to decoding these codes have already been developed. Here, we present low complexity, suboptimal alternatives which are inspired by the classical Reed decoding algorithm for binary RM codes. We simulate these new algorithms along with the existing decoding algorithms using additive white Gaussian noise and two-path fading models for a particular choice of code. The simulations show that one of our new algorithms outperforms all existing suboptimal algorithms and offers performance that is within 0.5 dB of maximum-likelihood decoding, yet has complexity comparable to or lower than existing decoding approaches     

On multilevel block modulation codes

The multilevel technique for combining block coding and modulation is investigated. A general formulation is presented for multilevel modulation codes in terms of component codes with appropriate distance measures. A specific method for constructing multilevel block modulation codes with interdependency among component codes is proposed. Given a multilevel block modulation code C with no interdependency among the binary component codes, the proposed method gives a multilevel block modulation code C' that has the same rate as C, a minimum squared Euclidean distance not less than that of C, a trellis diagram with the same number of states as that of C, and a smaller number of nearest neighbor codewords than that of C . Finally, a technique is presented for analyzing the error performance of block modulation codes for an additive white Gaussian noise (AWGN) channel based on soft-decision maximum likelihood decoding. Error probabilities of some specific codes are evaluated by simulation and upper bounds based on their Euclidean weight distributions     

Iterative reliability-based decoding of low-density parity checkcodes

In this paper, reliability based decoding is combined with belief propagation (BP) decoding for low-density parity check (LDPC) codes. At each iteration, the soft output values delivered by the BP algorithm are used as reliability values to perform reduced complexity soft decision decoding of the code considered. This approach allows to bridge the error performance gap between belief propagation decoding which remains suboptimum, and maximum likelihood decoding which is too complex to be implemented for the codes considered. Trade-offs between decoding complexity and error performance are also investigated. In particular, a stopping criterion which reduces the average number of iterations at the expense of very little performance degradation is proposed for this combined decoding approach. Simulation results for several Gallager (1963, 1968) LDPC codes and different set cyclic codes of hundreds of information bits are given and elaborated     

LT码和q-LDPC码级联方案在深空通信中的应用

该文针对深空通信对长纠删码的需求,提出了LT (Luby Transform)码和q-LDPC码的级联方案。在综合考虑性能和复杂度的情况下,选取8-LDPC码和8PSK的级联作为等效的删除信道,长度选择灵活、编译码简单的LT码实现纠删功能。文中设计了两种短8-LDPC码,并对整个级联系统的纠错性能进行了仿真。仿真结果表明8-LDPC码的性能优于信源信息速率和码率相同的二进制LDPC码,级联系统在等效包删除概率不超过0.1时,系统误比特率以概率1趋于0。     

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     

On the complexity of minimum distance decoding of long linear codes

We suggest a decoding algorithm of q-ary linear codes, which we call supercode decoding. It ensures the error probability that approaches the error probability of minimum-distance decoding as the length of the code grows. For n→∞ the algorithm has the maximum-likelihood performance. The asymptotic complexity of supercode decoding is exponentially smaller than the complexity of all other methods known. The algorithm develops the ideas of covering-set decoding and split syndrome decoding     

A new multilevel coding method using error-correcting codes

A new multilevel coding method that uses several error-correcting codes is proposed. The transmission symbols are constructed by combining symbols of codewords of these codes. Usually, these codes are binary error-correcting codes and have different error-correcting capabilities. For various channels, efficient systems can be obtained by choosing these codes appropriately. Encoding and decoding procedures for this method are relatively simple compared with those of other multilevel coding methods. In addition, this method makes effective use of soft-decisions to improve the performance of decoding. The decoding error probability is analyzed for multiphase modulation, and numerical comparisons to other multilevel coding systems are made. When equally complex systems are compared, the new system is superior to other multilevel coding systems.     

一类性能好、复杂度低的纠错编码--乘加码

对一类性能好且复杂度低的纠错编码技术——乘加码进行了介绍。他是在单校验位的Turbo乘积码(Single Parity Check Turbo Product Code)的基础上改进而来的,即由单校验位的Turbo乘积码作为外码,码率为1的递归卷积码作为内码串行级联而成。介绍了乘加码的编码方式和译码方法,并给出了其性能分析。对于一定的分组长度,这类码表现出与Turbo码相近的性能,但其译码复杂度要远远低于Turbo码。