Coset codes for partial response channels; or, coset codes withspectral nulls

Known coset codes are adapted for use on partial response channels or to generate signals with spectral nulls. By using coset precoding and running digital sum feedback, any desired tradeoff can be achieved between the power and spectra of the relevant sequences, up to the optimum tradeoff possible. A fundamental theorem specifying this optimum tradeoff is given. A maximum-likelihood-sequence-estimation (MLSE) decoder for the original code may be used for the adapted code, and such a decoder then attains the minimum squared distance of the original code. These methods sometimes generate codes with greater minimum squared distance than that of the original code; this distance can be attained by augmented decoders, although such decoders inherently require long decoding delays and may be subjected to quasi-catastrophic error propagation. The authors conclude that, at least for sequences supporting large numbers of bits per symbol, coset codes can be adapted to achieve effectively the same performance and complexity on partial response channels, or for sequences with spectral nulls, as they do in the ordinary memoryless case     

Binary convolutional codes with application to magnetic recording

Calderbank, Heegard, and Ozarow [1] have suggested a method of designing codes for channels with intersymbol interference, such as the magnetic recording channel. These codes are designed to exploit intersymbol interference. The standard method is to minimize intersymbol interference by constraining the input to the channel using run-length limited sequences. Calderbank, Heegard, and Ozarow considered an idealized model of an intersymbol interference channel that leads to the problem of designing codes for a partial response channel with transfer function(1 - D^{N}) /2, where the channel inputs are constrained to bepm 1. This problem is considered here. Channel inputs are generated using a nontrivial coset of a binary convolutional code. The coset is chosen to limit the zero-run length of the output of the channel and so maintain clock synchronization. The minimum squared Euclidean distance between outputs corresponding to distinct inputs is bounded below by the free distance of a second convolutional code which we call the magnitude code. An interesting feature of the analysis is that magnitude codes that are catastrophic may perform better than those that are noncatastrophic.     

DC-free coset codes

A class of block coset codes with disparity and run-length constraints are studied. They are particularly well suited for high-speed optical fiber links and similar channels, where DC-free pulse formats, channel error control, and low-complexity encoder-decoder implementations are required. The codes are derived by partitioning linear block codes. The encoder and decoder structures are the same as those of linear block codes with only slight modifications. A special class of DC-free coset block codes are derived from BCH codes with specified bounds on minimum distance, disparity, and run length. The codes have low disparity levels (a small running digital sum) and good error-correcting capabilities     

A pragmatic approach to trellis-coded modulation

Since the early 1970s, for power-limited applications, the convolutional code constraint length K=7 and rate 1/2, optimum in the sense of maximum free distance and minimum number of bit errors caused by remerging paths at the free distance, has become the de facto standard for coded digital communication. This was reinforced when punctured versions of this code became the standard for rate 3/4 and 7/8 codes for moderately bandlimited channels. Methods are described for using the same K=7, rate 1/2 convolutional code with signal phase constellations of 8-PSK and 160PSK and quadrature amplitude constellations of 16-QASK, 64-QASK, and 256-QASK to achieve, respectively, 2 and 3, and 2, 4, and 6 b/s/Hz bandwidth efficiencies while providing power efficiency that in most cases is virtually equivalent to that of the best Ungerboeck codes for constraint length 7 or 64 states. This pragmatic approach to all coding applications permits the use of a single basic coder and decoder to achieve respectable coding (power) gains for bandwidth efficiencies from 1 b/s/Hz to 6 b/s/Hz     

Free distance bounds for convolutional codes

The best asymptotic bounds presently known on free distance for convolutional codes are presented from a unified point of view. Upper and lower bounds for both time-varying and fixed codes are obtained. A comparison is made between bounds for nonsystematic and systematic codes which shows that more free distance is available with nonsystematic codes. This result is important when selecting codes for use with sequential or maximum-likelihood (Viterbi) decoding since the probability of decoding error is closely related to the free distance of the code. An ancillary result, used in proving the lower bound on free distance for time-varying nonsystematic codes, furnishes a generalization of two earlier bounds on the definite decoding minimum distance of convolutional codes.     

衰落信道下的格形码设计

衰落信道下ＴＣＭ好码的设计准则是使有效码长度最长，同时使其对应路径的欧几里德距离乘积最大。本文首先从理论上得到有效码长度与卷积编码器的状态数，并行输入数之间的关系。提出了一种能达到最大自由长的状态转移图－标准拓扑篱笆图的概念，在此基础上，对衰落信道下采用速率为（２／３）８ＰＳＫ信号集合时的ＴＣＭ好码进行搜索，与文献中已有码相比，利用准则判别和进行蒙特－卡洛模拟都说明了新码在抗衰落方面的良好性能。     

Simulation of error sources in digital channels

A general approach to error-source simulation in digital channels is discussed. It is shown that the stochastic sequential machine (SSM) model is general enough to describe error-source statistics in digital communication channels. The SSM model is capable of describing both the correlation between errors in a channel (bursts of errors) and the correlation between errors in different channels (crosstalk). Physical causes of error burstiness in satellite channels are considered, and it is shown that the SSM model can account for differential encoding, scrambling, forward error correction, intersymbol interference, nonlinear signal distortion, adjacent channel interference, modem defects, etc. SSM models of the satellite channel with differential and convolutional codes are simulated. The results of simulation agree with the experimental data for the actual channels. Analytical methods for determining the performance characteristics of SSM-modeled systems are developed     

Systematic redundant residue number system codes: analytical upper bound and iterative decoding performance over AWGN and Rayleigh channels

The novel family of redundant residue number system (RRNS) codes is studied. RRNS codes constitute maximum-minimum distance block codes, exhibiting identical distance properties to Reed-Solomon codes. Binary to RRNS symbol-mapping methods are proposed, in order to implement both systematic and nonsystematic RRNS codes. Furthermore, the upper-bound performance of systematic RRNS codes is investigated, when maximum-likelihood (ML) soft decoding is invoked. The classic Chase algorithm achieving near-ML soft decoding is introduced for the first time for RRNS codes, in order to decrease the complexity of the ML soft decoding. Furthermore, the modified Chase algorithm is employed to accept soft inputs, as well as to provide soft outputs, assisting in the turbo decoding of RRNS codes by using the soft-input/soft-output Chase algorithm.     

Lower bound on minimum lee distance of algebraic?geometric codes over finite fields

Algebraic-geometric (AG) codes over finite fields with respect to the Lee metric have been studied. A lower bound on the minimum Lee distance is derived, which is a Lee-metric version of the well-known Goppa bound on the minimum Hamming distance of AG codes. The bound generalises a lower bound on the minimum Lee distance of Lee-metric BCH and Reed-Solomon codes, which have been successfully used for protecting against bitshift and synchronisation errors in constrained channels and for error control in partial-response channels.     

New short constraint length convolutional code constructions for selected rational rates (Corresp.)

New short constraint length convolutional code constructions are tabulated for ratesR=(n-k)/n, k=1,2, cdots ,n-1withn=2, 3,cdots ,8, and for constraint lengthsK=3,4, cdots,8. These codes have been determined by iterative search based upon a criterion of optimizing the free distance profile. Specifically, these codes maximize the free distanced_{f}while minimizing the number of adversaries in the distance, or weight, spectrum. In several instances we demonstrate the superiority of these codes over previously published code constructions at the same rate and constraint length. These codes are expected to have a number of applications, including combined source-channel coding schemes as well as coding for burst or impulsive noise channels.     

Further Comments on the Characteristics of the Hedeman H-1, H-2, and H-3 Codes

This letter clarifies some of the properties of the Hedeman codes and examines their application to band-limited channels. Attention is drawn to the equivalence of H-2 and H-3 to simple linear modulation schemes. All three codes are dc free by virtue of a charge constraint which additionally ensures a minimum distance two (considered as rate one-half codes). The asymptotic error performance at high signal-to-noise ratio can thus be similar to NRZ or biphase signaling, although the Viterbi algorithm is required to achieve this performance in the case of H-1 code. Like Miller code, both H-1 and H-2 will pass through unequalized low-pass channels with bandwidths little greater than required for NRZ signaling.     

Woven codes with outer warp: variations, design, and distanceproperties

We consider convolutional and block encoding schemes which are variations of woven codes with outer warp. We propose methods to evaluate the distance characteristics of the considered codes on the basis of the active distances of the component codes. With this analytical bounding technique, we derived lower bounds on the minimum (or free) distance of woven convolutional codes, woven block codes, serially concatenated codes, and woven turbo codes. Next, we show that the lower bound on the minimum distance can be improved if we use designed interleaving with unique permutation functions in each row of the warp of the woven encoder. Finally, with the help of simulations, we get upper bounds on the minimum distance for some particular codes and then investigate their performance in the Gaussian channel. Throughout this paper, we compare all considered encoding schemes by means of examples, which illustrate their distance properties     

Trellis-coded quadrature amplitude modulation with 2N-dimensionalconstellations for mobile radio channels

Using a modified Wei method, originally designed for additive white Gaussian noise (AWGN) channels, we have constructed four-dimensional (4-D) and six-dimensional (6-D) trellis codes with rectangular signal constellations for frequency-nonselective mobile radio channels. Applying a novel way of partitioning the two-dimensional (2-D) constituent constellations, both into subsets with enlarged minimum Euclidean distance and subrings including equal energy signal points, we have obtained partitions of the 2N-D signal sets into subsets with a Hamming distance between signal points which equals N. This is fundamental for constructing good trellis codes to transmit data over flat fading channels     

Analog error-correcting codes based on chaotic dynamical systems

The properties of chaotic dynamical systems make them useful for channel coding in a variety of practical communication applications. To illustrate this, a novel analog code based on tent map dynamics and having a fast decoding algorithm is developed for use on unknown, multiple, and time-varying signal-to-noise ratio (SNR) channels. This code is shown to be an attractive alternative to both digital codes and linear modulation in such scenarios. Several properties and interpretations of the codes are developed, along with some methods for their optimization     

QPSK block-modulation codes for unequal error protection

Unequal error protection (UEP) codes find applications in broadcast channels, as well as in other digital communication systems, where messages have different degrees of importance. Binary linear UEP (LUEP) codes combined with a Gray mapped QPSK signal set are used to obtain new efficient QPSK block-modulation codes for unequal error protection. Several examples of QPSK modulation codes that have the same minimum squared Euclidean distance as the best QPSK modulation codes, of the same rate and length, are given. In the new constructions of QPSK block-modulation codes, even-length binary LUEP codes are used. Good even-length binary LUEP codes are obtained when shorter binary linear codes are combined using either the well-known |u¯|u¯+v¯|-construction or the so-called construction X. Both constructions have the advantage of resulting in optimal or near-optimal binary LUEP codes of short to moderate lengths, using very simple linear codes, and may be used as constituent codes in the new constructions. LUEP codes lend themselves quite naturally to multistage decoding up to their minimum distance, using the decoding of component subcodes. A new suboptimal two-stage soft-decision decoding of LUEP codes is presented and its application to QPSK block-modulation codes for UEP illustrated     

On design of error-correcting reversible variable length codes

We propose an algorithm for construction of reversible variable length codes (RVLCs) with good error-correcting properties. The error-correcting properties are evaluated by a metric called the free distance, which is always greater than one in the case of the proposed RVLCs. Since variable length codes (VLCs) typically have free distance equal to one, the proposed RVLCs exhibit significant improvement in symbol error rate relative to VLCs constructed using standard methods.     

On ternary error correcting line codes

The authors present Markov diagrams and tables with the capacities in bits/symbol for input restricted ternary channels with various restrictions on maximum runlengths, digital sum variation, and transitions between extreme signal levels. They derive Gilbert-type lower bounds on the minimum Hamming and Euclidean distances achievable with ternary line codes of rates lower than the capacity of the corresponding input restricted channel. They present some single-symbol-error-correcting ternary line codes, found by computer search methods     

Low-complexity iterative joint source-channel decoding for variable-length encoded Markov sources

In this paper, we present a novel packetized bit-level decoding algorithm for variable-length encoded Markov sources, which calculates reliability information for the decoded bits in the form of a posteriori probabilities (APPs). An interesting feature of the proposed approach is that symbol-based source statistics in the form of the transition probabilities of the Markov source are exploited as a priori information on a bit-level trellis. This method is especially well-suited for long input blocks, since in contrast to other symbol-based APP decoding approaches, the number of trellis states does not depend on the packet length. When additionally the variable-length encoded source data is protected by channel codes, an iterative source-channel decoding scheme can be obtained in the same way as for serially concatenated codes. Furthermore, based on an analysis of the iterative decoder via extrinsic information transfer charts, it can be shown that by using reversible variable-length codes with a free distance of two, in combination with rate-1 channel codes and residual source redundancy, a reliable transmission is possible even for highly corrupted channels. This justifies a new source-channel encoding technique where explicit redundancy for error protection is only added in the source encoder.     

Embedded Rank Distance Codes for ISI Channels

Designs for transmit alphabet constrained space–time codes naturally lead to questions about the design of rank distance codes. Recently, diversity embedded multilevel space–time codes for flat-fading channels have been designed from sets of binary matrices with rank distance guarantees over the binary field by mapping them onto quadrature amplitude modulation (QAM) and phase-shift keying (PSK) constellations. In this paper, we demonstrate that diversity embedded space–time codes for fading intersymbol interference (ISI) channels can be designed with provable rank distance guarantees. As a corollary, we obtain an asymptotic characterization of the fixed transmit alphabet rate–diversity tradeoff for multiple antenna fading ISI channels. The key idea is to construct and analyze properties of binary matrices with a particular structure (Toeplitz structure) induced by ISI channels.      

On the performance of high-rate TPC/SPC codes and LDPC codes overpartial response channels

This paper evaluates two-dimensional turbo product codes based on single-parity check codes (TPC/SPC) and low-density parity check (LDPC) codes for use in digital magnetic recording systems. It is first shown that the combination of a TPC/SPC code and a precoded partial response (PR) channel results in a good distance spectrum due to the interleaving gain. Then, density evolution is used to compute the thresholds for TPC/SPC codes and LDPC codes over PR channels. Analysis shows that TPC/SPC codes have a performance close to that of LDPC codes for large codeword lengths. Simulation results for practical block lengths show that TPC/SPC codes perform as well as LDPC codes in terms of bit error rate, but possess better burst error statistics which is important in the presence of an outer Reed-Solomon code. Further, the encoding complexity of TPC/SPC codes is only linear in the codeword length and the generator matrix does not have to be stored explicitly. Based on. the results in the paper and these advantages, TPC/SPC codes seem like a viable alternative to LDPC codes