Signalling enhancement on multilevel turbo codes

The design of multilevel turbo codes using M‐PSK is optimized to achieve a low bit error rate (BER). Unequal error protection is employed via group set partitioning in multi‐stage decoding to minimize the error propagation and BER. Simulation results are performed under Gaussian and Rayleigh fading channels to depict the superiority of the new scheme. Copyright © 2008 John Wiley & Sons, Ltd.     

之型码和级联之型码

本文介绍了一种新型的纠错码，称之为“之型码”。之型码可以形成非常简捷的软输入/软输出译码规则，我们在Max-Log-MAP(MLM)译码方法的基础上提出了一种译码规则，这一译码规则的计算复杂度为每次迭代计算单位信息比特大约需要20次加法运算操作。在仿真实验中，我们用最优译码器和更简捷的次最优译码器进行译码时，其性能在误比特率为10^-5处分别距香农理论极限仅0.9dB和1.4dBH。此外，上述码字与2维turbo码相比具有更低的误码基底值（error floor）。     

Reducing the complexity of distance measurement methods for circular turbo codes that use structured interleavers

The knowledge of turbo code's minimum Hamming distance (d_min) and its corresponding codeword multiplicity (A_min) is of a great importance because the error correction capability of a code is strongly tied to the values of d_min and A_min. Unfortunately, the computational complexity associated with the search for d_min and A_min can be very high, especially for a turbo code that has high d_min value. This paper introduces some useful properties of turbo codes that use structured interleavers together with circular encoding. These properties allow for a significant reduction of search space and thus reduce significantly the computational complexity associated with the determination of d_min and A_min values. © 2014 The Authors. International Journal of Communication Systems published by John Wiley & Sons, Ltd.     

Analysis and performance evaluation of new coding options for space telecommand links ‐ Part I: AWGN channels

We investigate and compare, over the additive white Gaussian noise channel, different options for updating the error correcting code currently used in space mission telecommand links. Besides some more consolidated proposals, using low‐density parity‐check codes, we characterize and assess the performance of alternative schemes, based on parallel turbo codes and soft‐decision decoded Bose‐Chaudhuri‐Hocquenghem (BCH) codes. The analysis considers relevant metrics like the codeword error rate, the frame error rate, and the undetected frame error rate. The considered codes include binary and non‐binary low‐density parity‐check codes, parallel turbo codes and extended BCH codes, with different decoding algorithms. The complexity of the various schemes and possible limits for their application are considered and discussed. Several numerical examples are provided. International Journal of Satellite Communications and Networking. Copyright © 2014 John Wiley & Sons, Ltd.     

Turbo码的优选扩散交织器设计

该文提出了Turbo码交织器的一种新的设计方法,适用于任意长度的交织设计,称为优选扩散交织,文中阐述了它的设计原则与步骤,以及对16状态Turbo码交织器的设计分析,并对其性能进行了仿真。     

Generation of codes with good autocorrelation properties

The letter describes a conceptually simple method of generating impulse equivalent burst and cyclic codes, using an all-pass autoregressive moving average (ARMA) filter. These infinite impulse response burst codes may be converted into cyclic codes of arbitrary length, while retaining their low autocorrelation time sidelobe performance. Their extension to orthogonal coding is explored.     

APPLICATION OF TURBO CODES IN HIGH-SPEED REAL-TIME CHANNEL

I. Introduction Turbo code has obtained comprehensive atten-tion and research due to its near-Shannon perform-ance since it was proposed in 1993[1], and has be-come a research hotspot in information and coding theory area. Application and realization methods of turbo codes in various communication systems have also attracted great interest of researchers. The good BER performance of turbo codes provides it a wide application prospect in deep space and mobile com- munication systems. The IT…     

Skew cyclic codes over ring F p +vF p

In this paper, we study skew cyclic codes over the ring F ＋ vFp, where p is a odd prime and v2 = 1. We give the generators of skew cyclic codes, with the consideration of the dual of skew cyclic codes.     

Computing the free distance of turbo codes and seriallyconcatenated codes with interleavers: algorithms and applications

We present a new algorithm for computing the free distance d_free of parallel and serially concatenated codes with interleavers, the parameter that dominates the code performance at very high signal-to-noise ratios (SNRs). The knowledge of d_free allows one to analytically estimate the error floor, which may prevent the use of concatenated codes in applications requiring very low error rates. The algorithm is based on the new notion of constrained subcodes, and permits the computation of large distances for large interleavers without a constraint on the input sequence weight (e.g., up to d_free=40 for a rate-1/3 turbo code with interleaver length N=3568). Applications to practical cases of relevant interest, i.e., (1) the new Consultative Committee for Space Data Systems (CCSDS) standard for deep-space telemetry and (2) the new UMTS/3GPP standard for third-generation personal communications, are presented for the first time. Other related aspects, like a study on the free distance distribution of turbo codes with small/medium interleaver length, and a comparison between parallel and serial concatenation behavior, are also discussed     

Nonsystematic turbo codes

In this paper, we introduce the concept of nonsystematic turbo codes and compare them with classical systematic turbo codes. Nonsystematic turbo codes can achieve lower error floors than systematic turbo codes because of their superior effective free distance properties. Moreover, they can achieve comparable performance in the waterfall region if the nonsystematic constituent encoder has a low-weight feedforward inverse. A uniform interleaver analysis is used to show that rate R=1/3 turbo codes using nonsystematic constituent encoders have larger effective free distances than when systematic constituent encoders are used. Also, mutual information-based transfer characteristics and extrinsic information transfer charts are used to show that rate R=1/3 turbo codes with nonsystematic constituent encoders having low-weight feedforward inverses achieve convergence thresholds comparable to those achieved with systematic constituent encoders. Catastrophic encoders, which do not possess a feedforward inverse, are shown to be capable of achieving low convergence thresholds by doping the code with a small fraction of systematic bits. Finally, we give tables of good nonsystematic turbo codes and present simulation results comparing the performance of systematic and nonsystematic turbo codes.     

Fast turbo codes with space‐time block codes in fast fading channels

In this paper, space‐time block coding has been used in conjunction with Turbo codes to provide good diversity and coding gains. A new method of dividing turbo encoder and decoder into several parallel encoding and decoding blocks is considered. These blocks work simultaneously and yield a faster coding scheme in comparison to classical Turbo codes. The system concatenates fast Turbo coding as an outer code with Alamouti's G2 space‐time block coding scheme as an inner code, achieving benefits associated with both techniques including acceptable diversity and coding gain as well as short coding delay. In this paper, fast fading Rayleigh and Rician channels are considered for discussion. For Rayleigh fading channels, a fixed frame size and channel memory length of 5000 and 10, respectively, the coding gain is 7.5 dB and bit error rate (BER) of 10^?4 is achieved at 7 dB. For the same frame size and channel memory length, Rician fading channel yields the same BER at about 4.5 dB. Copyright © 2013 John Wiley & Sons, Ltd.     

Combinatorial analysis of the minimum distance of turbo codes

In this paper, new upper bounds on the maximum attainable minimum Hamming distance of turbo codes with arbitrary-including the best-interleavers are established using a combinatorial approach. These upper bounds depend on the interleaver length, the code rate, and the scramblers employed in the encoder. Examples of the new bounds for particular turbo codes are given and discussed. The new bounds are tighter than all existing ones and prove that the minimum Hamming distance of turbo codes cannot asymptotically grow at a rate more than the third root of the codeword length     

Asymptotic performance comparison of concatenated (turbo) codes over GF(4)

In this paper, we investigate and compare the asymptotic performance of concatenated convolutional coding schemes over GF(4) over additive white Gaussian noise (AWGN) channels. Both parallel concatenated codes (PCC) and serial concatenated codes (SCC) are considered. We construct such codes using optimal non‐binary convolutional codes where optimality is in the sense of achieving the largest minimum distance for a fixed number of encoder states. Code rates of the form k₀/(k₀ + 1) for k₀=1, 8, and 64 are considered, which suite a wide spectrum of communications applications. For all of these code rates, we find the minimum distance and the corresponding multiplicity for both concatenated code systems. This is accomplished by feeding the encoder with all possible weight‐two and weight‐three input information patterns and monitoring, at the output of the encoder, the weight of the corresponding codewords and their multiplicity. Our analytical results indicate that the SCC codes considerably outperform their counterpart PCC codes at a much lower complexity. Inspired by the superiority of SCC codes, we also discuss a mathematical approach for analysing such codes, leading to a more comprehensive analysis and allowing for further improvement in performance by giving insights on designing a proper interleaver that is capable of eliminating the dominant error patterns. Copyright © 2004 John Wiley & Sons, Ltd.     

Easily decodable binary cyclic codes

This paper presents a class of binary cyclic codes with block length n = 2^m? 1, having n?k = 2^m? parity checks and a minimum distance d=m +1, where m is an integer. These codes are shown to be majority logic decodable in one step by making use of the concept of a quasi-perfect finite difference set.     

Algebraic properties of BCH codes useful for decoding

In this paper theorems are presented which allow the simplified decoding of (n, k, δ) BCH codes in certain cases of practical interest. Such results are in a way implicit in the theory of BCH codes, but so far have not appeared explicitly in the literature. It is shown that any t₀ errors, 1 ? t₀ ? δ-1, can be detected by using any set of only t₀ consecutive coefficients of the syndrome polynomial. The correction of any t₀ errors, 1 ? t₀ ? [(δ-1)/2], can be performed by using any set of 2t₀ consecutive coefficients of the syndrome polynomial, where [x] means the integer part of x. Similar results are derived for punctured BCH codes. In this case sets of t₀ or 2t₀ consecutive coefficients, respectively, for detecting or correcting t₀ errors, are selected from the δ-1-p higher-order coefficients of the modified syndrome polynomial, where p is the number of digits punctured from a code word. These results hold true even when the punctured digits are not consecutive.     

Turbo-SPC codes

This letter is concerned with a family of modified turbo-type codes, referred to as turbo-SPC (single parity check) codes. A technique based on the SPC code is introduced to replace puncturing for rate adjustment. A noticeable feature of the proposed scheme is the significantly reduced decoding complexity compared with the standard punctured turbo code. The cost reduction factor becomes more noticeable as rate increases (over ten times for rate →1). The error-rate floor problem known to turbo codes can also be improved using the proposed scheme. For moderate to high rates, performances of about 0.5 dB from the theoretical limits at BER=10^-5 have been observed for turbo-SPC codes     

Space-time codes with full antenna diversity using weighted nonbinary repeat-accumulate codes

Like turbo codes, repeat-accumulate codes have remarkably good performance when r/spl ges/3, where r is the number of repetition times. We present space-time codes with full antenna diversity using "weighted" nonbinary repeat-accumulate codes. Compared with the space-time turbo codes of Y. Liu et al. (see IEEE J. Select. Areas Commun., vol.19, p.969-80, 2001) and of H. Su and E. Geraniotis (see ibid., vol.49, p.47-57, 2001), the main advantage of this new scheme is to construct space-time codes with full diversity for any m/spl les/r and any length of frame without searching for interleavers, where m is the number of transmit antennas. These space-time codes have rate m/r and, so, have full rate when m=r. Furthermore, they have an efficient decoding based on the message passing algorithm.     

A Bandwidth Efficient Turbo Coding Scheme for VDSL Systems

The current forward error correction (FEC) scheme for very high bit-rate digital subscriber line (VDSL) systems in the ANSI standard employs a 16-state four-dimensional (4D) Wei code as the inner code and the Reed-Solomon (RS) code as the outer code. The major drawback of this scheme is that further improvement cannot be achieved without a substantial increase in the complexity and power penalty. Also, a VDSL system employing the 4D Wei-RS scheme operates far below the channel capacity. In 1993, powerful turbo codes were introduced whose performance closely approaches the Shannon limit. In this paper, we propose a bandwidth and power efficient turbo coding scheme for VDSL modems in order to obtain high data rates, extended loop reach and increased transmission robustness. We also propose a pipelined decoding scheme to reduce the latency at the receiver end. The objective of the proposed scheme is to provide a higher coding gain than that given by the 4D Wei-RS scheme, resulting in an improved performance of the VDSL modems in terms of bit rate, loop length and transmitting power. The scheme is investigated for various values of transmitting power, signaling frequencies and numbers of crosstalkers for a targeted bit error rate of 10⁻⁵ and is implemented in a system with a quadrature amplitude modulation in which a mixed set partitioning mapping is employed to reduce the decoding complexity. The effects of code complexity, interleaver length, the number of decoding iterations and the level of modulation on the performance of VDSL modems are explored. Simulation results are presented and compared to those of the 4D Wei-RS scheme. The results show that the choice of turbo codes not only provides a significant coding gain over the standard FEC scheme but also efficiently maximizes the loop length and bit rate at a very low transmitting power in the presence of dominant far-end crosstalk and intersymbol interference. In order to compare the hardware complexity, we synthesize the proposed and 4D Wei-RS schemes using SYNOPSYS with the target technology of Xilinx 4020e-3. The Xilinx field programmable gate array statistics of the proposed scheme is compared with that of the 4D Wei-RS scheme.     

A search for good convolutional codes to be used in theconstruction of turbo codes

Recursive systematic convolutional encoders have been shown to play a crucial role in the design of turbo codes. We recall some properties of binary convolutional encoders and apply them to a search for good constituent convolutional codes of turbo codes. Tables of the “best” recursive systematic convolutional encoders found are presented for various rates, together with the average bit-error probability performances of some turbo codes using them     

Novel construction of quasi-cyclic low-density parity-check codes with variable code rates for cloud data storage systems

This paper proposed a novel method for constructing quasi-cyclic low-density parity-check (QC-LDPC) codes of medium to high code rates that can be applied in cloud data storage systems, requiring better error correction capabilities. The novelty of this method lies in the construction of sparse base matrices, using a girth greater than 4 that can then be expanded with a lift factor to produce high code rate QC-LDPC codes. Investigations revealed that the proposed large-sized QC-LDPC codes with high code rates displayed low encoding complexities and provided a low bit error rate (BER) of 10⁻¹⁰ at 3.5 dB E_b/N₀ than conventional LDPC codes, which showed a BER of 10⁻⁷ at 3 dB E_b/N₀. Subsequently, implementation of the proposed QC-LDPC code in a software-defined radio, using the NI USRP 2920 hardware platform, was conducted. As a result, a BER of 10⁻⁶ at 4.2 dB E_b/N₀ was achieved. Then, the performance of the proposed codes based on their encoding–decoding speeds and storage overhead was investigated when applied to a cloud data storage (GCP). Our results revealed that the proposed codes required much less time for encoding and decoding (of data files having a 10 MB size) and produced less storage overhead than the conventional LDPC and Reed–Solomon codes.