Two-dimensional Fire codes

To improve the reliability of two-dimensional information, codes that can correct two-dimensional bursts (or spots) may be useful. In this paper a class of two-dimensional burst-correcting codes, called two-dimensional Fire codes, is proposed. The definition of these codes is a natural extension of that of the conventional Fire codes. The two-dimensional Fire code is a two-dimensional cyclic code designed for single two-dimensional burst correction. Several important properties such as the burst-correcting capability and the positions of the check symbols are presented. Also, encoding and decoding methods are presented. It is shown that the encoding and decoding are easily implemented by using two-dimensional feedback shift registers.     

Filterbank Decompositions for (Non)-Systematic Reed–Solomon Codes With Applications to Soft Decoding

This paper focuses on Reed-Solomon (RS) codes, which are the most widespread classical error correcting codes. Recently, we have shown that an finite-impulse response (FIR) critically subsampled filterbank representation can be derived for some RS codes. However, this work only addresses RS codes with a non-coprime codeword and dataword length, seriously limiting its practical usability. In this paper, an alternative purely algebraic method is presented to construct such a filterbank. Apart from providing additional insight into the algebraic structure of (non-systematic) RS codes, this method is suited to eliminate the non-coprimeness constraint mentioned before. Using this filterbank decomposition, a RS code is broken into smaller subcodes that can subsequently be used to build a soft-in soft-out (SISO) RS decoder. It is shown how any RS code, written as an FIR filterbank, can be SISO decoded using the filterbank based decoder. Owing to the importance of systematic RS codes, it is shown that any systematic RS code can be decoded using the FIR filterbank decomposition. This leads to better decoding performance in addition with a slightly lower complexity. A further extension towards systematic RS codes is also presented in this paper resulting in an infinite-impulse response critically subsampled filterbank representation.     

A new series of concatenated codes subject to matrix type-Bcodes

Some important aspects of a series of concatenated codes subjected to matrix type-Bcodes are investigated. Concatenated matrix codes and the concatenated quadratic residue codes especially are emphasized. An analysis of the error patterns, which can be corrected with the matrix coding, also is given. These codes are suitable for compound channels with memory (i.e., channels on which burst, cluster, and random errors occur). Explicit formulas are given for the number of bursts, cluster, and random errors that can be corrected with these codes. Decoding schemes and techniques for studying error propagation in the proposed codes are given. In particular a new decoding algorithm for a concatenated matrix code is given. The performance of coding and decoding schemes of the various types of concatenated codes can be tested in practice.     

非二进制Turbo级联码的性能分析

该文在分析了非二进制Turbo码的译码算法的基础上,提出了非二进制Turbo码和具有良好纠突发错误的RS码级联的非二进制Turbo级联码系统模型。该系统采用缩短的RS码(204, 188)作为外码,非二进制Turbo码作为内码的串行级联方式,内码和外码之间用深度为12的交织器隔开,译码时, 非二进制Turbo码分别采用Symbol-by-Symbol Log-MAP和SOVA算法。仿真结果显示,与二进制Turbo级联码系统相比,该系统具有误码率低,延时时间短,频带利用率高等优点。     

High-rate burst-error-correcting recurrent codes (Corresp.)

A bound and construction are presented for high-rate burst-error-correcting recurrent codes. The bound is an upper bound on the block length in terms of the total redundancy used in decoding, the redundancy per block, and the burst length. The construction uses a block-code parity-check matrix as the first block of the recurrent code parity-check matrix. For a block code it is typical to find that only a portion of the redundancy need be used to detect a burst. Any block code for which this is true can be used in the construction. The recurrent code is then related as follows to the block code from which it is constructed. 1) The recurrent code block length is the same as the block-code block length. 2) The total redundancy used in decoding the recurrent code is the same as the block-code redundancy per block. 3) The recurrent code redundancy per block is the same as the block-code redundancy required for burst detection only. 4) The recurrent code is of higher rate than the block code. 5) The recurrent code requires a guard space between bursts but otherwise corrects the same bursts as the block code. It is shown that, when certain well-known cyclic codes are used in the construction, the resulting recurrent codes are close to the upper bound.     

Spectrum of incorrectly decoded bursts for cyclic burst error codes

The enumeration of the incorrectly decoded bursts for cyclic burst error-correcting codes is reported here. The enumeration yields closed formulas for long bursts, whereas an efficient algorithm for the enumeration is given for the short bursts. The enumeration is carried out for two different decoding rules. Under the first rule, the cyclic code is a full length code, and split decodable patterns are decoded by the decoder in spite of the fact that a split decodable pattern can be a burst that exceeds the decoding capability of the decoder. The analysis for the second rule assumes split patterns are not decoded. This analysis is valid for the class of shortened cyclic codes.     

基于Logistic混沌序列的错误图样对纠错码的影响

为了在“比特级”层面对纠错码进行更深入的研究,提出了一种服从Logistic混沌序列的错误图样分布模型。通过将混沌序列映射成二进制序列,并采用一定的方法使序列中“1”的数目可控,以此来生成错误图样。将卷积码、RS码以及RS+卷积级联码作为研究对象,以随机错误为参照,分别研究了该错误图样下三者的译码性能。仿真结果表明,基于Logistic混沌序列的错误图样对卷积码和级联码的影响要大于随机错误图样,而随机错误图样对RS码的影响更大。     

一种自适应数据逐层分解的Reed-Solomon码迭代纠错方法及应用

该文针对Reed-Solomon码纠错算法计算复杂度较高、运算时间较长等问题,提出一种自适应数据逐层分解的Reed-Solomon码的迭代译码纠错方法。首先,接收码通过逐层分解将随机错误或突发错误分散于不同的子序列中,缩小突发或随机错误的查找范围;其次,制定约束规则确定错误数目,同时根据不同的伴随矩阵维数自适应选择迭代求解关键方程的方法,定位子序列中误码的位置;最后,计算正确码字,结束纠错。实验测试表明,该算法在保证不漏检误码的前提下,能够有效简化计算多项式的维数,减少计算量和复杂度,纠错时效优于DFT(Discrete Fourier Transform)算法和BM(Berlekamp-Massey)算法。特别是对2维码数据的纠错测试中,与传统算法相比,该算法纠错时效可提升一个数量级。     

Frequency-hopped multiple access communications with coding andside information

The authors consider frequency-hopped spread-spectrum multiple-access communications using M-ary modulation and error-correction coding. The major concerns are multiple-access interference and the network capacity in terms of the number of users that can transmit simultaneously for a given level of codeword error probability. Block coding is studied in detail. The authors first consider the use of Q-ary Reed-Solomon (RS) codes in combination with M-ary modulation with mismatched alphabets so that Q>M. It is shown that the network capacity is drastically reduced in comparison with the system with matched alphabets. As a remedy, the use of matched M-ary BCH codes is proposed as an alternative to mismatched RS codes. It is shown that when the number of users in the system is large, a BCH code outperforms an RS code with a comparable code rate and decoding complexity. The authors consider the use of a robust technique for generation of reliable side information based on a radio-threshold test. They analyze its performance in conjunction with MFSK and error-erasure correction decoding. It is shown that this nonideal ratio-threshold method can increase the network capacity in comparison with the system with perfect side information     

Turbo codes-based image transmission for channels with multiple types of distortion

Product codes are generally used for progressive image transmission when random errors and packet loss (or burst errors) co-exist. However, the optimal rate allocation considering both component codes gives rise to high-optimization complexity. In addition, the decoding performance may be degraded quickly when the channel varies beyond the design point. In this paper, we propose a new unequal error protection (UEP) scheme for progressive image transmission by using rate-compatible punctured Turbo codes (RCPT) and cyclic redundancy check (CRC) codes only. By sophisticatedly interleaving each coded frame, the packet loss can be converted into randomly punctured bits in a Turbo code. Therefore, error control in noisy channels with different types of errors is equivalent to dealing with random bit errors only, with reduced turbo code rates. A genetic algorithm-based method is presented to further reduce the optimization complexity. This proposed method not only gives a better performance than product codes in given channel conditions but is also more robust to the channel variation. Finally, to break down the error floor of turbo decoding, we further extend the above RCPT/CRC protection to a product code scheme by adding a Reed-Solomon (RS) code across the frames. The associated rate allocation is discussed and further improvement is demonstrated.     

Improved FEC Code Based on Concatenated Code for Optical Transmission Systems

The improved three novel schemes of the super forward error correction(super-FEC) concatenated codes are proposed after the development trend of long-haul optical transmission systems and the defects of the existing FEC codes have been analyzed. The performance simulation of the Reed-Solomon(RS)+Bose-Chaudhuri-Hocguenghem(BCH) inner-outer serial concatenated code is implemented and the conceptions of encoding/decoding the parallel-concatenated code are presented. Furthermore, the simulation results for the RS(255,239)+RS(255,239) code and the RS(255,239)+RS(255,223) code show that the two consecutive concatenated codes are a superior coding scheme with such advantages as the better error correction, moderate redundancy and easy realization compared to the classic RS(255,239) code and other codes, and their signal to noise ratio gains are respectively 2～3dB more than that of the RS(255,239)code at the bit error rate of 1×10 -13 . Finally, the frame structure of the novel consecutive concatenated code is arranged to lay a firm foundation in designing its hardware.     

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     

RS级联编码在超短波通信与卫星通信信道的仿真分析

超短波波段数据通信与卫星通信已被广泛应用。在这些信道中都存在较强的多径衰落,并且为克服两个波段中的多种干扰,纠正随机错误和突发错误,RS编码被普遍选为信道编码方案。RS码是线性分组码中纠突发错误能力最强,效率最高的编码,Turbo码是可以接近香农限的编码方案。文章提出RS编码与Turbo码的级联码编码方案和解码算法,适当选择参数,并通过仿真验证了方案的正确性与可行性。     

A versatile time-domain Reed-Solomon decoder

A versatile Reed-Solomon (RS) decoder structure based on the time-domain decoding algorithm (transform decoding without transforms) is developed. The algorithm is restructured, and a method is given to decode any RS code generated by any generator polynomial. The main advantage of the decoder structure is its versatility, that is, it can be programmed to decode any Reed-Solomon code defined in Galois field (GF) 2^m with a fixed symbol size m. This decoder can correct errors and erasures for any RS code, including shortened and singly extended codes. It is shown that the decoder has a very simple structure and can be used to design high-speed single-chip VLSI decoders. As an example, a gate-array-based programmable RS decoder is implemented on a single chip. This decoder chip can decode any RS code defined in GF (2⁵) with any code word length and any number of information symbols. The decoder chip is fabricated using low-power 1.5-μ, two-layer-metal, HCMOS technology     

纠正随机错误与长突发删除的多进制乘积码研究

考虑多进制LDPC码的符号特性，以及对其残留错误和删除的分析，本文采用多进制LDPC码作为内码，相同Galois域下的高码率RS码作为外码来构造多进制乘积码；并提出了一种低复杂度的迭代译码方案，减少信息传输的各类错误。在译码时，只对前一次迭代中译码失败的码字执行译码，并对译码正确码字所对应的比特初始概率信息进行修正，增强下一次迭代多进制LDPC译码符号先验信息的准确性，减少内码译码后的判决错误，从而充分利用外码的纠错能力。仿真结果显示，多进制乘积码相较于二进制LDPC乘积码有较大的编码增益，并通过迭代进一步改善了性能，高效纠正了信道中的随机错误和突发删除。对于包含2%突发删除的高斯信道，在误比特率为10-6时，迭代一次有0.4 dB左右的增益。      

A new concatenated matrix code with cluster,burst and random error correcting abilities

A class of linear matrix codes for compound channels with memory (i.e. channels on which burst, cluster and random errors occur) ia given. Explicit formulas are given for the number of burst, cluster and random errors that can be corrected with tlicso codes. Decoding nchcmes and analysis techniques to study error propagation in the proposed codoa are given. In particular, a new decoding algorithm for a concatenated matrix codo is given. The algorithm utilizes the decoding algorithm for the corresponding concatenated matrix code. A coding–decoding procedure is first, followed. The codes can bo used to transmit information from fixed–rate sources through fixed–rate compound channels. Tho channel probabilities are related to channel error propagation.     

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.     

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.     

Performance analysis of coded M-ary orthogonal signaling usingerrors-and-erasures decoding over frequency-selective fading channels

The performance of M-ary orthogonal signaling schemes employing Reed-Solomon (RS) codes and redundant residue number system (RRNS) codes is investigated over frequency-selective Rayleigh fading channels. “Errors-and-erasures” decoding is considered, where erasures are judged based on two low-complexity, low-delay erasure insertion schemes-Viterbi's ratio threshold test (RTT) and the proposed output threshold test (OTT). The probability density functions (PDF) of the ratio associated with the RTT and that of the demodulation output in the OTT conditioned on both the correct detection and erroneous detection of M-ary signals are derived, and the characteristics of the RTT and OTT are investigated. Furthermore, expressions are derived for computing the codeword decoding error probability of RS codes or RRNS codes based on the above PDFs, The OTT technique is compared to Viterbi's RTT, and both of these are compared to receivers using “error-correction only” decoding over frequency-selective Rayleigh-fading channels. The numerical results show that by using “errors-and-erasures” decoding, RS or RRNS codes of a given code rate can achieve higher coding gain than that without erasure information, and that the OTT technique outperforms the RTT, provided that both schemes are operated at the optimum decision thresholds     

基于改进2-DGRS码的QC-LDPC码高效构造

利用GRS(generalized reed-solomon)码的生成多项式提出了基于改进的2-D GRS(two-dimensional GRS)码设计和构造QC-LDPC(quasi-cyclic low density parity-check)码的方法,使所构造的码具有较好的译码性能。同时在码的构造过程中,考虑到了准双对角线结构和合适的度分布。不同码率的LDPC码用于和新设计的QC-LDPC码进行测试和比较。实验结果表明,所提出的码构造方法可加快LDPC码校验矩阵的构造,同时基于所提出方法构造的QC-LDPC码可提高译码性能,并降低编码复杂度。