Efficient Error Control Decoder Architectures for Noncoherent Random Linear Network Coding

Random linear network coding is an efficient technique for disseminating information in networks, but it is highly susceptible to errors. Kötter-Kschischang (KK) codes and Mahdavifar-Vardy (MV) codes are two important families of subspace codes that provide error control in noncoherent random linear network coding. List decoding has been used to decode MV codes beyond half distance. Existing hardware implementations of the rank metric decoder for KK codes suffer from limited throughput, long latency and high area complexity. The interpolation-based list decoding algorithm for MV codes still has high computational complexity, and its feasibility for hardware implementations has not been investigated. In this paper we propose efficient decoder architectures for both KK and MV codes and present their hardware implementations. Two serial architectures are proposed for KK and MV codes, respectively. An unfolded decoder architecture, which offers high throughput, is also proposed for KK codes. The synthesis results show that the proposed architectures for KK codes are much more efficient than rank metric decoder architectures, and demonstrate that the proposed decoder architecture for MV codes is affordable.     

Scarce-state-transition error-trellis decoding of block codes withBPSK signals

A novel decoding technique for linear block codes with coherent BPSK signals is proposed. The new system has the same error performance as and similar complexity to the conventional trellis decoding of block codes. Like the scarce-state-transition Viterbi decoding of convolutional codes, the proposed system is also well suited for CMOS VLSI implementation and has a lower power consumption     

Nonlinear rotationally invariant trellis codes for multidimensionalmodulation

The concept of a general nonlinear parity-check equation was first proposed by Pietrobon et al. who used them to construct rotationally invariant trellis codes for two dimensional signal sets. This paper applies this technique to trellis-coded modulation (TCM) codes with multidimensional MPSK and MQAM constellations to construct nonlinear phase rotationally invariant multi-D codes. Based on this construction technique, a series of the best fully phase rotationally invariant multi-D trellis codes for a variety of signal constellations is obtained through a systematic code search program. The code search results show that most codes achieve the same asymptotic coding gain as the best linear, but nonphase rotationally invariant codes found by Pietrobon et al     

Iterative maximum-likelihood trellis decoding for block codes

An iterative trellis search technique is described for the maximum-likelihood (ML) soft decision decoding of block codes. The proposed technique derives its motivation from the fact that a given block code may be a subcode for a parent code whose associated trellis has substantially fewer edges. Through the use of list-Viterbi (1967) decoding and an iterative algorithm, the proposed technique allows for the use of a trellis for the parent code in the ML decoding of the desired subcode. Complexity and performance analyses, as well as details of potential implementations, indicate a substantial reduction in decoding complexity for linear block codes of practical length while achieving ML or near-ML soft decision performance     

On channel orthogonalization using space-time block coding with partial feedback

Orthogonal space-time block codes (OSTBCs) yield full diversity gain even while requiring only a linear receiver. Such full-rate (rate-one) orthogonal designs are available for complex symbol constellations only for N=2 transmit antennas. In this paper, we propose a new family of full-rate space-time block codes (STBCs) using a single parameter feedback for communication over Rayleigh fading channels for N=3,4 transmit antennas and M receive antennas. The proposed rate-one codes achieve full diversity, and the performance is similar to maximum receiver ratio combining. The decoding complexity of these codes are only linear even while performing maximum-likelihood decoding. The partial channel information is a real phase parameter that is a function of all the channel gains, and has a simple closed-form expression for N=3,4. This feedback information enables us to derive (channel) orthogonal designs starting from quasi-orthogonal STBCs. The feedback complexity is significantly lower than conventional closed-loop transmit beamforming. We compare the proposed codes with the open-loop OSTBCs and also with the closed-loop equal gain transmission (EGT) scheme which uses equal power loading on all antennas. Simulated error-rate performances indicate that the proposed channel orthogonalized STBCs significantly outperform the open-loop orthogonal designs, for the same spectral efficiency. Moreover, even with significantly lower feedback and computational complexity, the proposed scheme outperforms the EGT technique for M>N.     

A class of linear codes with the same performance as those based upon the logical Hadamard transform (Corresp.)

Banta has recently proposed a class of nonlinear codes that use the logical Hadamard transform for encoding and decoding. The same performance can be achieved by class of linear codes exhibited in this correspondence. The decoder for these linear codes is simpler than the decoder for the Banta codes, while the encoder is almost trivial.     

Generalized rational interpolation over commutative rings and remainder decoding

We propose a new decoding procedure for Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Solomon (RS) codes over Z/sub m/ where m is a product of prime powers. Our method generalizes the remainder decoding technique for RS codes originally introduced by Welch and Berlekamp and retains its key feature of not requiring the prior evaluation of syndromes. It thus represents a significant departure from other algorithms that have been proposed for decoding linear block codes over integer residue rings. Our decoding procedure involves a Welch-Berlekamp (WB)-type algorithm for solving a generalized rational interpolation problem over a commutative ring R with identity. The solution to this problem includes as a special case, the solution to the WB key equation over R which is central to our decoding procedure. A remainder decoding approach for decoding cyclic codes over Z/sub m/ up to the Hartmann-Tzeng bound is also presented.     

Histogram-Based Calibration of Capacitor Mismatch and Comparator Offset for 1-Bit/Stage Pipelined ADCs

An efficient two-phase calibration technique for 1-bit/stage pipelined Analog–to–Digital Converters (ADCs) is presented in this paper. The proposed technique employs linear histogram testing to collect the required information to calibrate the non-ideal ADC output behavior induced by capacitor mismatch and comparator offset. In the first phase, it calibrates the missing-decision-level errors by amplification gain reduction. Unlike previous works, which require large capacitor arrays, only few switches are added to the circuit. The second phase eliminates missing-transition levels (missing codes). It achieves better calibrated linearity and provides better mismatch tolerance than the traditional digital calibration technique. Simulation results show that the proposed technique effectively improves both static and dynamic performances.     

基于极小线性码上的秘密共享方案

从理论上说,每个线性码都可用于构造秘密共享方案,但是在一般情况下,所构造的秘密共享方案的存取结构是难以确定的.本文提出了极小线性码的概念,指出基于这种码的对偶码所构造的秘密共享方案的存取结构是容易确定的.本文首先证明了极小线性码的缩短码一定是极小线性码.然后对几类不可约循环码给出它们为极小线性码的判定条件,并在理论上研究了基于几类不可约循环码的对偶码上的秘密共享方案的存取结构.最后用编程具体求出了一些实例中方案的存取结构.     

Performance of polar codes with the construction using density evolution

Polar coding, proposed by Ar?kan, makes it possible to construct capacity-achieving codes for symmetric binaryinput discrete memoryless channels, with low encoding and decoding complexity. Complexity of the originally proposed code construction method, however, grows exponentially in the blocklength unless a channel is the binary erasure channel. Recently, the authors have proposed a new capacity-achieving code construction method with linear complexity in the blocklength for arbitrary symmetric binary-input memoryless channels. In this letter, we evaluate performance of polar codes designed with the new construction method, and compare it with that of the codes constructed with another heuristic method with linear complexity proposed by Ar?kan.     

A Class of High Rate Codes for Byte-Oriented Information Systems

In this paper we introduce a class of linear codes especially designed to provide additional error protection for data consisting of bytes all having even (or odd) parity (e.g., ASCII characters). The technique consists in adding an overall parity byte computed as a linear function of the information bytes. The linear function is designed such that the resulting codes can correct all single errors and all double errors occurring in distinct information bytes. It is shown that any code which can correct these latter mentioned error patterns has an overall length of at most 37 bytes, and a specific code of length 29 bytes is described. A practical decoding algorithm for the new class of codes is described. Finally, the performance of the codes, when used on the binary symmetric channel, is compared with that of the row-column codes for which the additional parity byte is simply the modulo-2 sum of the information bytes.     

Nonlinear Programming Approaches to Decoding Low-Density Parity-Check Codes

We consider the decoding problem for low-density parity-check codes, and apply nonlinear programming methods. This extends previous work using linear programming (LP) to decode linear block codes. First, a multistage LP decoder based on the branch-and-bound method is proposed. This decoder makes use of the maximum-likelihood-certificate property of the LP decoder to refine the results when an error is reported. Second, we transform the original LP decoding formulation into a box-constrained quadratic programming form. Efficient linear-time parallel and serial decoding algorithms are proposed and their convergence properties are investigated. Extensive simulation studies are performed to assess the performance of the proposed decoders. It is seen that the proposed multistage LP decoder outperforms the conventional sum-product (SP) decoder considerably for low-density parity-check (LDPC) codes with short to medium block length. The proposed box-constrained quadratic programming decoder has less complexity than the SP decoder and yields much better performance for LDPC codes with regular structure.     

Design methods for irregular repeat-accumulate codes

We optimize the random-like ensemble of irregular repeat-accumulate (IRA) codes for binary-input symmetric channels in the large block-length limit. Our optimization technique is based on approximating the evolution of the densities (DE) of the messages exchanged by the belief-propagation (BP) message-passing decoder by a one-dimensional dynamical system. In this way, the code ensemble optimization can be solved by linear programming. We propose four such DE approximation methods, and compare the performance of the obtained code ensembles over the binary-symmetric channel (BSC) and the binary-antipodal input additive white Gaussian noise channel (BIAWGNC). Our results clearly identify the best among the proposed methods and show that the IRA codes obtained by these methods are competitive with respect to the best known irregular low-density parity-check (LDPC) codes. In view of this and the very simple encoding structure of IRA codes, they emerge as attractive design choices.     

A New Error Control Scheme for Hybrid ARQ Systems

This paper introduces the concept of a generalized hybrid ARQ (GH-ARQ) scheme for adaptive error control in digital communication systems. This technique utilizes the redundant information available upon successive retransmissions in an efficient manner so as to provide high throughput during poor channel conditions. A new class of linear codes is proposed for the GH-ARQ system application. The main feature of this class of codes is that the encoder/decoder configuration does not change as the length of the code is varied. As a result, the receiver uses the same decoder for decoding the received information after every retransmission while the error correcting capability of the code increases, thereby leading to an improved performance and minimum complexity for the overall system implementation.     

Multidimensional trellis coded phase modulation using a multilevelconcatenation approach. I. Code design

The first part of this paper presents a simple and systematic technique for constructing multidimensional M-ary phase shift keying (MPSK) trellis coded modulation (TCM) codes. The construction is based on a multilevel concatenation approach. In which binary convolutional codes with good free branch distances are used as the outer codes and block MPSK modulation codes are used as the inner codes (or the signal spaces). Conditions on phase invariance of these codes are derived and a multistage decoding scheme for these codes is proposed. The proposed technique can be used to construct good codes for both the additive white Gaussian noise (AWGN) and fading channels as is shown in the second part of this paper     

Nonlinear Codes: The Product Construction

The standard product construction is discussed with respect to nonlinear codes. Thus, so-called nonlinear product codes are obtained that are better than linear product codes of similar length and code rate, and at the same time, amenable for encoding/decoding. On the other hand, it is shown that certain notorious nonlinear codes have an augmented product construction, namely, they can be constructed by taking the union of a product code and certain of its cosets. The binary Hamming codes are shown to have similar construction. A simple two-stage decoder is proposed for nonlinear product (NLP) codes. The decoder is shown to be a bounded-distance (BD) information decoder that is the nonlinear equivalent of the BD decoder employed for linear codes. A list-based maximum-likelihood decoder is also discussed.     

Double-differential orthogonal space-time block codes for arbitrarily correlated Rayleigh channels with carrier offsets

The presence of carrier offsets in the multiple-input multiple-output (MIMO) channels is an important practical and theoretical problem. Double-differential coding is a technique which allows the receiver to decode the data without any channel or carrier offset knowledge. We propose a double-differential (DD) coding scheme which is applicable to square orthogonal space-time block codes (OSTBC) using M-PSK constellation. The main advantages of our proposed DD coding scheme are: 1) The previously proposed DD codes are applicable only to the specific class of space-time block codes which follow the diagonal unitary group property, whereas our DD coding is applicable to any square OSTBC. 2) We propose a suboptimal decoder which preserves the linear decoding property of the OSTBC. 3) A theoretical analysis is performed to find a pairwise error probability (PEP) upper bound of the proposed doubledifferential orthogonal space-time block codes (DDOSTBC). 4) In order to improve the performance of DDOSTBC over the arbitrarily correlated Rayleigh channels we propose a precoder which minimizes an upper bound of the PEP. The proposed DDOSTBC are able to achieve higher coding gain than the similar rate existing DD coding scheme. In addition, the proposed precoded DDOSTBC achieves performance gain for correlated channels as compared to the unprecoded DDOSTBC.     

分层空时码的模型、泽码接收及其性能仿真

本文讨论分层空时编码技术,给出了种基于分层空时编码技术的发送和接收模型,分析了分层空时编码方案,译码与接收技术,分层空时编码的译码与接收技术是分层空时编码应用的关键,对此,我们讨论了具有线性复杂度的迫零判决反馈均衡器结合分层空时译码方法,仿真结果显示了良好的性能和更高的频谱利用率。     

A lower bound on the undetected error probability and strictlyoptimal codes

Error detection is a simple technique used in various communication and memory systems to enhance reliability. We study the probability that a q-ary (linear or nonlinear) block code of length n and size M fails to detect an error. A lower bound on this undetected error probability is derived in terms of q, n, and M. The new bound improves upon other bounds mentioned in the literature, even those that hold only for linear codes. Block codes whose undetected error probability equals the new lower bound are investigated. We call these codes strictly optimal codes and give a combinatorial characterization of them. We also present necessary and sufficient conditions for their existence. In particular, we find all values of n and M for which strictly optimal binary codes exist, and determine the structure of all of them. For example, we construct strictly optimal binary-coded decimal codes of length four and five, and we show that these are the only possible lengths of such codes     

Nonlinear Product Codes and Their Low Complexity Iterative Decoding

This paper proposes encoding and decoding for nonlinear product codes and investigates the performance of nonlinear product codes. The proposed nonlinear product codes are constructed as N‐dimensional product codes where the constituent codes are nonlinear binary codes derived from the linear codes over higher order alphabets, for example, Preparata or Kerdock codes. The performance and the complexity of the proposed construction are evaluated using the well‐known nonlinear Nordstrom‐Robinson code, which is presented in the generalized array code format with a low complexity trellis. The proposed construction shows the additional coding gain, reduced error floor, and lower implementation complexity. The (64, 24, 12) nonlinear binary product code has an effective gain of about 2.5 dB and 1 dB gain at a BER of 10^?6 when compared to the (64, 15, 16) linear product code and the (64, 24, 10) linear product code, respectively. The (256, 64, 36) nonlinear binary product code composed of two Nordstrom‐Robinson codes has an effective gain of about 0.7 dB at a BER of 10^?5 when compared to the (256, 64, 25) linear product code composed of two (16, 8, 5) quasi‐cyclic codes.