Component Codes Research Articles

DC-free error-correcting codes based on convolutional codes

A new construction of direct current (DC)-free error-correcting codes based on convolutional codes is proposed. The new code is constructed by selecting a proper subcode from a convolutional code composed of two different component codes. The encoder employs a Viterbi algorithm as the codeword selector so that the selected code sequences satisfy the DC constraint. A lower bound on the free distance of such codes is proposed, and a procedure for obtaining this bound is presented. A sufficient condition for these codes to have a bounded running digital sum (RDS) is proposed. Under the assumption of a simplified codeword selection algorithm, we present an upper bound on the maximum absolute value of the RDS and derive the sum variance for a given code. A new construction of standard DC-free codes, i.e., DC-free codes without error-correcting capability, is also proposed. These codes have the property that the decoder can be implemented by simple symbol-by-symbol hard decisions. Finally, under the new construction, we propose several codes that are suitable for the systems that require small sum variance and good error-correction capability.

Comparison of convolutional and turbo codes for OFDM with antenna diversity in high-bit-rate wireless applications

This letter presents simulation results for the application of turbo coding to an OFDM system with diversity. First, results are presented for convolutional and Reed-Solomon codes. It is shown that even with short constraint lengths, convolutional codes have the potential to outperform Reed-Solomon codes, provided that sufficient precision is used in the soft decoder. We then evaluate the performance of turbo codes under slow fading conditions and study the effects of varying codeword size. Increasing codeword size theoretically provides better interleaving between the two component codes. However, this advantage is less clear when the fading rate is significantly lower than the symbol rate, which is typical of the high-data-rate systems considered here. Under such conditions, the advantage of using two component convolutional codes in turbo codes is limited. A single convolutional code with a long constraint length may be a better choice.

The weight hierarchies of outer product codes

The weight hierarchy of the outer product of two self-complementary codes is expressed as a function of the weight hierarchies of the component codes. An alternative method to calculate the rth generalized Hamming weight (and so the weight hierarchy) of the first-order Reed–Muller code is given. The outer product of a self-complementary code and a nonself-complementary code is also discussed.

Correct Coding in Interventional Pain Management

The National Correct Coding Council (NCCC) was created by the Health Care Financing Administration (HCFA) in 1996 to help ensure that providers across various jurisdictions receive like payment for the same services, use the same codes and provide similar documentation for services performed. The Correct Coding Initiative (CCI) was a direct outgrowth of the NCCC's, the purpose of which was to identify and isolate inappropriate coding, unbundling, and other irregularities in coding. To avoid inappropriate or incorrect coding and billing in interventional pain management, it is imperative that interventional pain physicians and their staff be familiar with correct coding policies, as well as understand the meaning of Current Procedural Terminology, along with comprehensive codes, component codes, and mutually exclusive codes. This review describes CCI and various correct coding policies specifically relevant to interventional pain medicine. In addition, certain commonly used codes of interventional techniques are also described with implications of comprehensive, component, and mutually exclusive coding terminology.

Performance evaluation of error-forecasting decoding for interleaved codes on Gilbert channels

Interleaved codes are constructed from error correcting codes with short code length (component code) as the basic elements and are known to be an effective means for handling burst error. The decoding method for the interleaved codes is usually one in which decoding is carried out independently for the component codes. However, if the error correlation is taken into account, it is clear that error-forecasting decoding can improve the characteristics because the error location of the adjacent component code is considered as a lost location to provide lost error decoding. In this article, the error rate characteristics (the probability of not carrying out the correct decoding for the component codes and the interleaved codes) in a Gilbert channel are theoretically derived for a system in which error-forecasting decoding is used for the interleaved codes. In the analysis, several related Markov chains are derived step by step from a two-state Markov chain for the Gilbert channel, and the generating functions for the probability corresponding to the number of errors and the success in decoding of the component codes are derived. It is shown that the error rate (word error rate) for the component codes and the error rate (frame error rate) for the interleaved codes can be evaluated from the identical generating function. © 1999 Scripta Technica, Electron Comm Jpn Pt 1, 83(4): 63–72, 2000

Improved upper bounds on the ML decoding error probability of parallel and serial concatenated turbo codes via their ensemble distance spectrum

The ensemble performance of parallel and serial concatenated turbo codes is considered, where the ensemble is generated by a uniform choice of the interleaver and of the component codes taken from the set of time-varying recursive systematic convolutional codes. Following the derivation of the input-output weight enumeration functions of the ensembles of random parallel and serial concatenated turbo codes, the tangential sphere upper bound is employed to provide improved upper bounds on the block and bit error probabilities of these ensembles of codes for the binary-input additive white Gaussian noise (AWGN) channel, based on coherent detection of equi-energy antipodal signals and maximum-likelihood decoding. The influence of the interleaver length and the memory length of the component codes is investigated. The improved bounding technique proposed here is compared to the conventional union bound and to a alternative bounding technique by Duman and Salehi (1998) which incorporates modified Gallager bounds. The advantage of the derived bounds is demonstrated for a variety of parallel and serial concatenated coding schemes with either fixed or random recursive systematic convolutional component codes, and it is especially pronounced in the region exceeding the cutoff rate, where the performance of turbo codes is most appealing. These upper bounds are also compared to simulation results of the iterative decoding algorithm.

Parallel concatenated Gallager codes

A class of concatenated codes called parallel concatenated Gallager codes (PCGCs) in which low density parity check (LDPC) codes are used as the component codes is presented. Simulation results for PCGCs that outperform comparable LDPC codes in an AWGN channel are also presented.

DMT scheme with multidimensional turbo trellis code

Turbo codes can be naturally extended to bandwidth-efficient applications. A turbo code with powerful bandwidth-efficient component codes is investigated and its application in the discrete multitone environment is considered. Simulation results show that the multidimensional turbo trellis-coded modulation scheme can achieve outstanding performance even at relatively low signal-to-noise ratio.

The Impact of National Correct Coding Policy on Interventional Pain Management

Health Care Financing Administration ("HCFA") created the National Correct Coding Council ("NCCC") to help ensure that providers across various jurisdictions received like payment for the same services and use the same codes and provide similar documentation for services performed. As a direct out growth of NCCC's work, HCFA established the National Correct Coding Policy in 1996 and eventually implemented the Medicare "Correct Coding Initiative" to identify and isolate inappropriate coding, unbundling, and other irregularities in coding. To appropriately implement National Correct Coding Policy in interventional pain management an interventional pain management specialist and their staff must be familiar with correct coding policies as well as understand the physicians current procedural terminology (CPT, medical surgical practice and packages, modifiers, separate procedures, comprehensive and component services, incorrect coding/unbundling and various specific issues relevant to practice of interventional pain management) Comprehensive codes include certain defined services that are separately identifiable by other codes known as component codes. Because component codes are captured by comprehensive codes they man not be listed separately when the complete procedure is done. For example, in interventional pain management, 62279, which is continuous lumbar epidural, is considered a comprehensive code. Various component codes include 62270, 62272 - 62274, 62276 - 62278, 62288 and 62289, among others. This review describes National Correct Coding Policy, correct procedural terminology, medical and surgical practice and packages, evaluation and management services along with description of most codes used in interventional pain management with correcting coding edits for comprehensive codes and for mutually exclusive codes.

On generalized low-density parity-check codes based on Hamming component codes

In this paper we investigate a generalization of Gallager's (1963) low-density (LD) parity-check codes, where as component codes single error correcting Hamming codes are used instead of single error detecting parity-check codes. It is proved that there exist such generalized low-density (GLD) codes for which the minimum distance is growing linearly with the block length, and a lower bound of the minimum distance is given. We also study iterative decoding of GLD codes for the communication over an additive white Gaussian noise channel. The performance in terms of the bit error rate, obtained by computer simulations, is presented for GLD codes of different lengths.

Combined array processing and space-time coding

The information capacity of wireless communication systems may be increased dramatically by employing multiple transmit and receive antennas. The goal of system design is to exploit this capacity in a practical way. An effective approach to increasing data rate over wireless channels is to employ space-time coding techniques appropriate to multiple transmit antennas. These space-time codes introduce temporal and spatial correlation into signals transmitted from different antennas, so as to provide diversity at the receiver, and coding gain over an uncoded system. For large number of transmit antennas and at high bandwidth efficiencies, the receiver may become too complex whenever correlation across transmit antennas is introduced. This paper dramatically reduces encoding and decoding complexity by partitioning antennas at the transmitter into small groups, and using individual space-time codes, called the component codes, to transmit information from each group of antennas. At the receiver, an individual space-time code is decoded by a novel linear processing technique that suppresses signals transmitted by other groups of antennas by treating them as interference. A simple receiver structure is derived that provides diversity and coding gain over uncoded systems. This combination of array processing at the receiver and coding techniques for multiple transmit antennas can provide reliable and very high data rate communication over narrowband wireless channels. A refinement of this basic structure gives rise to a multilayered space-time architecture that both generalizes and improves upon the layered space-time architecture proposed by Foschini (see Bell Labs Tech. J., vol.1, no.2, 1996).

Performance bounds for turbo-coded modulation systems

We apply the standard union bound to turbo-coded modulation systems with maximum-likelihood decoding. To illustrate the methodology, we explicitly derive the bounds for the 2-bits/s/Hz 16 QAM system. Generalization of this bound to other turbo-coded modulation systems is straightforward. As in the case of the standard union bound for turbo codes, we expect these bounds to be useful for rather large values of signal-to-noise ratios, i.e., signal-to-noise ratios for which the code rate is smaller than the corresponding cutoff rate. The bound is based on "uniform interleaving" just as its counterpart for standard turbo coding. The derived bound provides a tool for comparing coded modulation schemes having different component codes, interleaver lengths, mappings, etc., using maximum-likelihood decoding. It is also useful in studying the effectiveness of various suboptimal decoding algorithms. The bounding technique is also applicable to other coded-modulation schemes such as serially concatenated coded modulation.

Combined turbo codes and interleaver design

The impact of the distance spectrum and interleaver structure on the bit error probability of turbo codes is considered. A new turbo code design method for Gaussian channels is presented. The proposed method combines a search for good component codes with interleaver design. The optimal distance spectrum is used as the design criterion to construct good turbo component codes at low signal-to-noise ratios (SNRs). In addition, an interleaver design method is proposed. This design improves the code performance at high SNR. Search for good component codes at low SNR is combined with a code matched interleaver design. This results in new turbo codes with a superior error performance relative to the best known codes at both low and high SNR. The performance is verified by both analysis and simulation.

Selection of component codes for turbo coding based on convergence properties

The turbo decoding is a sub-optimal decoding, i.e. it is not a maximum likelihood decoding. It is important to be aware of this fact when the parameters for the scheme are chosen. This goes especially for the selection of component codes, where the selection often has been based solely on the performance at highSnr’s. We will show that it is important to base the choice on the performance at lowSnr’s. i.e. the convergence properties, as well. Further, the study of the performance with different component codes may lead to an understanding of the convergence process in the turbo decoder.

A note on asymmetric turbo-codes

Little attention has been devoted in the literature to turbo-codes with nonidentical component codes. However, in order to optimize the performance of parallel concatenated schemes with respect to both the so called error-floor and the waterfall regions, the idea has certain advantages.

Block-coded modulation using two-level group codes over generalized quaternion groups

A length n group code over a group G is a subgroup of G/sup n/ under component-wise group operation. Two-level group codes over the class of generalized quaternion groups, Q(2/sup m/), m/spl ges/3, are constructed using a binary code and a code over Z(2/sup m-1/), the ring of integers modulo 2/sup m-1/ as component codes and a mapping f from Z/sub 2//spl times/Z(2/sup m-1/)to Q(2/sup m/). A set of necessary and sufficient conditions on the component codes is derived which will give group codes over Q(2/sup m/). Given the generator matrices of the component codes, the computational effort involved in checking the necessary and sufficient conditions is discussed. Starting from a four-dimensional signal set matched to Q(2/sup m/), it is shown that the Euclidean space codes obtained from the group codes over Q(2/sup m/) have Euclidean distance profiles which are independent of the coset representative selection involved in f. A closed-form expression for the minimum Euclidean distance of the resulting group codes over Q(2/sup m/) is obtained in terms of the Euclidean distances of the component codes. Finally, it is shown that all four-dimensional signal sets matched to Q(2/sup m/) have the same Euclidean distance profile and hence the Euclidean space codes corresponding to each signal set for a given group code over Q(2/sup m/) are automorphic Euclidean-distance equivalent.

A multilevel approach to constructing trellis-matched codes for binary-input partial-response channels

The multilevel coding approach of Imai and Hirakawa (1977) is used to construct trellis-matched codes for binary-input partial-response channels. For the codes to be trellis matched, the signal constellations are selected according to certain constraints, but no conditions are imposed on the component codes. New codes for the (1-D)(1+D)/sup n/ channel compare favorably to existing codes.

The Golay convolutional code-some application aspects

The Golay convolutional code and related block codes are investigated from an application point of view. We calculate the distance spectrum of the Golay convolutional code and the weight distributions of the block codes using the transfer-matrix method. The performance of the considered codes is shown and discussed. Moreover, the codes are applied as component codes of parallel concatenated codes which are iteratively decoded.

Design of efficient codes for the AWGN channel based on decomposable binary lattices

This work is concerned with the use of binary decomposable lattice codes over the QAM Gaussian channel. First, we investigate the structure of such class of lattices: we derive consistency conditions for the binary codes appearing in their decomposition and express their nominal coding gain and some bounds for their error coefficient in terms of the parameters of the component codes. Then we describe a general multistage bounded‐distance decoding algorithm with low complexity and we evaluate its performance. Finally, we develop a design example and report the corresponding simulation results; as a reference some comparisons with standard TCM codes are also presented.

Block-coded PSK modulation using two-level group codes over dihedral groups

A length n group code over a group G is a subgroup of G/sup n/ under component-wise group operation. Group codes over dihedral groups D/sub M/, with 2M elements, that are two-level constructible using a binary code and a code over Z/sub M/ residue class integer ring modulo M, as component codes are studied for arbitrary M. A set of necessary and sufficient conditions on the component codes for the two-level construction to result in a group code over D/sub M/ are obtained. The conditions differ for M odd and even. Using two-level group codes over D/sub M/ as label codes, the performance of a block-coded modulation scheme is discussed under all possible matched labelings of 2M-APSK and 2M-SPSK (asymmetric and symmetric PSK) signal sets in terms of the minimum squared Euclidean distance. Matched labelings that lead to automorphic Euclidean distance equivalent codes are identified. It is shown that depending upon the ratio of Hamming distances of the component codes some labelings perform better than others. The best labeling is identified under a set of restrictive conditions. Finally, conditions on the component codes for phase rotational invariance properties of the signal space codes are discussed.