Constituent Codes Research Articles

On Achieving an Asymptotically Error-Free Fixed-Point of Iterative Decoding for Perfect A Priori Information

In this paper we provide necessary and sufficient conditions for constituent codes in (multiple) concatenated and graph-based coding schemes to achieve an asymptotically error-free iterative decoding fixed-point if the maximum possible a priori information is available. At least one constituent code in an iterative decoding scheme must satisfy these conditions in order to ensure an asymptotically vanishing bit error probability at the convergence point of the decoder. Our results are proved for arbitrary binary-input symmetric memoryless channels (BISMCs) and thus can be universally applied to many transmission scenarios. Specifically, using a factor graph framework, it is shown that non-inner codes in a serial concatenation or check nodes in generalized LDPC codes achieve perfect extrinsic information if and only if the minimum Hamming distance between codewords is two or greater. For inner codes in a serial concatenation, constituent codes in a parallel concatenation, or variable nodes in doubly-generalized LDPC codes the corresponding encoder condition for acquiring perfect extrinsic information is an infinite codeword weight for a weight-one input sequence. For this case we provide a general proof which holds for all linear encoders and BISMCs. We also show that these results can improve the performance of concatenated coding schemes.

Randomness of finite‐state sequence machine over GF(4) and quality of hopping turbo codes

In this study, the authors study a turbo-coding (TC) scheme, whose constituent codes are designed using convolutional encoders. These encoders are finite-state sequence machines (FSSMs) operating over the Galois Field, GF(4). The scheme includes encryption polynomials whose coefficients are selected every L steps, from the set of optimal polynomials of GF(4). Two cases are considered for the polynomial selection: periodical and random. This kind of encoder was studied in a previous study and a correspondence between the randomness of the encoded sequence and performance of the TC was conjectured. The main contribution of this study is to systematically confirm this correspondence, by analysing the randomness of the output and performance of the TC using several randomness quantifiers. Three of the quantifiers are defined on the basis of recurrence plots. Other two quantifiers are defined on the basis of the information theory. All the quantifiers allow one to justify why the proposed TC works better with random selection of the optimal polynomials and with small values of L. In summary, it is shown that a random selection of polynomials and a small L produce FSSMs with enhanced randomness properties and it is also shown that they produce the best quality of the TC, measured by means of the corresponding bit error rate.

The Concatenated Structure of Quasi-Cyclic Codes and an Improvement of Jensen's Bound

Following Jensen's work from 1985, a quasi-cyclic code can be written as a direct sum of concatenated codes, where the inner codes are minimal cyclic codes and the outer codes are linear codes. We observe that the outer codes are nothing but the constituents of the quasi-cyclic code in the sense of Ling-Solé. This concatenated structure enables us to recover some earlier results on quasi-cyclic codes in a simple way, including one of our recent results which says that a quasi-cyclic code with cyclic constituent codes are 2-D cyclic codes. In fact, we obtain a generalization of this result to multidimensional cyclic codes. The concatenated structure also yields a lower bound on the minimum distance of quasi-cyclic codes, as noted by Jensen, which we call Jensen's bound. We show that a recent lower bound on the minimum distance of quasi-cyclic codes that we obtained is in general better than Jensen's lower bound.

Comparison of Turbo Codes and Low Density Parity Check Codes

The most powerful channel coding schemes, namely, those based on turbo codes and LPDC (Low density parity check) codes have in common principle of iterative decoding. Shannon's predictions for optimal codes would imply random like codes, intuitively implying that the decoding operation on these codes would be prohibitively complex. A brief comparison of Turbo codes and LDPC codes will be given in this section, both in term of performance and complexity. In order to give a fair comparison of the codes, we use codes of the same input word length when comparing. The rate of both codes is R = 1/2. However, the Berrou's coding scheme could be constructed by combining two or more simple codes. These codes could then be decoded separately, whilst exchanging probabilistic, or uncertainty, information about the quality of the decoding of each bit to each other. This implied that complex codes had now become practical. This discovery triggered a series of new, focused research programmes, and prominent researchers devoted their time to this new area.. Leading on from the work from Turbo codes, MacKay at the University of Cambridge revisited some 35 year old work originally undertaken by Gallagher (5), who had constructed a class of codes dubbed Low Density Parity Check (LDPC) codes. Building on the increased understanding on iterative decoding and probability propagation on graphs that led on from the work on Turbo codes, MacKay could now show that Low Density Parity Check (LDPC) codes could be decoded in a similar manner to Turbo codes, and may actually be able to beat the Turbo codes (6). As a review, this paper will consider both these classes of codes, and compare the performance and the complexity of these codes. A description of both classes of codes will be given. I. Turbo Codes There are different ways of constructing Turbo codes, but the commonality is that they use two or more codes in the encoding process. The constituent encoders are typically recursive, systematic convolutional (RSC) codes (3). Figure 1 shows a general set-up of the encoder. The number of bits in the code word varies with the rate of the code. For an overall code rate of R = 1/2, it is normal to use two R = 1/2 RSC codes, where every systematic bit is transmitted, but for the coded bits a puncturing matrix is used where only every second code bit from each of the constituent codes is included in the transmitted symbol. The permutation matrix Π permutes the systematic bits, so that the two constituent codes are excited by the same set of information bits, albeit in a different order. The permutation matrix is a critical design factor for designing good codes. The code word consists o f Ci as the systematic bit (di) and Ci and Ci which are vectors of encoded bits from the two convolution codes. These vectors can contain one or more bits and di is the information bit, Π is the permutation matrix.

EXIT-constrained BICM-ID design using extended mapping

This article proposes a novel design framework, EXIT-constrained binary switching algorithm (EBSA), for achieving near Shannon limit performance with single parity check and irregular repetition coded bit interleaved coded modulation and iterative detection with extended mapping (SI-BICM-ID-EM). EBSA is composed of node degree allocation optimization using linear programming (LP) and labeling optimization based on adaptive binary switching algorithm jointly. This technique achieves exact matching between the Demapper (Dem) and decoder's extrinsic information transfer (EXIT) curves while the convergence tunnel opens until the desired mutual information (MI) point. Moreover, this article proposes a combined use of SI-BICM-ID-EM with Doped-ACCumulator (D-ACC) and modulation doping (MD) to further improve the performance. In fact, the use of D-ACC and SI-BICM-ID (noted as DSI-BICM-ID-EM) enables the right-most point of the EXIT curve of the combined demapper and D-ACC decoder (Ddacc), denoted as DemDdacc, to reach a point very close to the (1.0, 1.0) MI point. Furthermore, MD provides us with additional degree-of-freedom in "bending" the shape of the demapper EXIT curve by choosing the mixing ratio of modulation formats, and hence the left most point of the demapper EXIT curve can flexibly be lifted up/pushed down with MD aided DSI-BICM-ID-EM (referred to as MDSI-BICM-ID-EM). Results of the simulations show that near-Shannon limit performance can be achieved with the proposed technique; with a parameter set obtained by EBSA for MDSI-BICM-ID-EM, the threshold signal-to-noise power ratio (SNR) is only roughly 0.5 dB away from the Shannon limit, for which the required computational complexity per iteration is at the same order as a Turbo code with only memory-2 convolutional constituent codes.

List decoding of matrix-product codes from nested codes: An application to quasi-cyclic codes

A list decoding algorithm for matrix-product codes is provided when $C_1,..., C_s$ are nested linear codes and $A$ is a non-singular by columns matrix. We estimate the probability of getting more than one codeword as output when the constituent codes are Reed-Solomon codes. We extend this list decoding algorithm for matrix-product codes with polynomial units, which are quasi-cyclic codes. Furthermore, it allows us to consider unique decoding for matrix-product codes with polynomial units.

A Bound on the Minimum Distance of Quasi-cyclic Codes

We give a general lower bound for the minimum distance of $q$-ary quasi-cyclic codes of length $m\ell$ and index $\ell$, where $m$ is relatively prime to $q$. The bound involves the minimum distances of constituent codes of length $\ell$ as well as the minimum distances of certain cyclic codes of length $m$ which are related to the fields over which the constituents are defined. We present examples which show that the bound is sharp in many instances. We also compare the performance of our bound against the bounds of Lally and Esmaeili-Yari.

New Codes from Dual BCH Codes with Applications in Low PAPR OFDM

Dual Bose-Ray-Chaudhuri (BCH) codes, despite their favorable peak to average power ratio (PAPR) properties, have not been used in coded orthogonal frequency division multiplexing (OFDM) systems. This is due to unavailability of a practical decoder and large performance gap to the Shannon limit. In this paper, we propose an advanced approach to solve the aforementioned problems. We construct a new code with favorable PAPR properties based on dual BCH codes and develop the associated maximum a posteriori (MAP) decoding algorithm. By exploiting this code as the frequency domain constituent code in a time-frequency turbo structure, we reduce the gap between the performance and Shannon limit while the bounded PAPR of OFDM symbols is guaranteed. By comparative performance evaluation we illustrate that the performance of this system is comparable with that of capacity approaching codes while it has 7 dB lower PAPR.

A Simplified Successive-Cancellation Decoder for Polar Codes

A modification is introduced of the successive-cancellation decoder for polar codes, in which local decoders for rate-one constituent codes are simplified. This modification reduces the decoding latency and algorithmic complexity of the conventional decoder, while preserving the bit and block error rate. Significant latency and complexity reductions are achieved over a wide range of code rates.

Bounds on the minimum code distance for nonbinary codes based on bipartite graphs

The minimum distance of codes on bipartite graphs (BG codes) over GF(q) is studied. A new upper bound on the minimum distance of BG codes is derived. The bound is shown to lie below the Gilbert-Varshamov bound when q ? 32. Since the codes based on bipartite expander graphs (BEG codes) are a special case of BG codes and the resulting bound is valid for any BG code, it is also valid for BEG codes. Thus, nonbinary (q ? 32) BG codes are worse than the best known linear codes. This is the key result of the work. We also obtain a lower bound on the minimum distance of BG codes with a Reed-Solomon constituent code and a lower bound on the minimum distance of low-density parity-check (LDPC) codes with a Reed-Solomon constituent code. The bound for LDPC codes is very close to the Gilbert-Varshamov bound and lies above the upper bound for BG codes.

Joint iterative decoding for pragmatic irregular LDPC-coded multi-relay cooperations

In this article, a new kind of pragmatic simple-encoding irregular systematic low-density parity-check (LDPC) code for multi-relay coded cooperation is designed, where the introduced joint iterative decoding is performed in the destination based on a proposed joint Tanner graph for all the constituent LDPC codes used by the source and relays in multi-relay cooperation. The theoretical analysis and numerical results show that the coded cooperations outperform the coded non-cooperation under the same code rate, and also achieve a good trade-off between the performance and the decoding complexity associated with the number of relays. This performance gain can be credited to the additional exchange of extrinsic information from the LDPC codes used by the source and the relays in both ideal and non-ideal cooperations.

Multiple LDPC-Encoder Layered Space-Time-Frequency Architectures for OFDM MIMO Multiplexing

This paper is focused on the study of layered space-time-frequency (LSTF) architectures with channel coding for orthogonal frequency division multiplexing (OFDM) multiple-input multiple-output (MIMO) multiplexing systems for high speed wireless communications over a frequency-selective fading channel. In order to achieve the available spatial, temporal and frequency diversities, and also make the system implementation feasible for high speed OFDM MIMO multiplexing, a novel LSTF architecture with multiple channel encoders is proposed with each independently coded layer being threaded in the three-dimensional space-time-frequency transmission resource array. Non-iterative receiver is adopted which consists of list sphere detector and irregular low-density parity-check codes as the constituent codes. Simulation results show that the performance of the proposed multiple-encoder LSTF architecture is very close to that of the conventional single-encoder LSTF where coding is applied across the whole information stream. However, due to the use of multiple parallel encoders/decoders with a shorter codeword length, the proposed LSTF architecture has much lower hardware processing speed and complexity than the conventional LSTF.

Correcting a Fraction of Errors in Nonbinary Expander Codes With Linear Programming

A linear-programming decoder for \emph{nonbinary} expander codes is presented. It is shown that the proposed decoder has the maximum-likelihood certificate properties. It is also shown that this decoder corrects any pattern of errors of a relative weight up to approximately 1/4 \delta_A \delta_B (where \delta_A and \delta_B are the relative minimum distances of the constituent 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

Asymptotic estimation of the fraction of errors correctable by q-ary LDPC codes

We consider an ensemble of random q-ary LDPC codes. As constituent codes, we use q-ary single-parity-check codes with d = 2 and Reed-Solomon codes with d = 3. We propose a hard-decision iterative decoding algorithm with the number of iterations of the order of the logarithm of the code length. We show that under this decoding algorithm there are codes in the ensemble with the number of correctable errors linearly growing with the code length. We weaken a condition on the vertex expansion of the Tanner graph corresponding to the code.

Near-capacity iterative decoding of binary self-concatenated codes using soft decision demapping and 3-D EXIT charts

In this paper 3-D Extrinsic Information Transfer (EXIT) charts are used to design binary Self-Concatenated Convolutional Codes employing Iterative Decoding (SECCC-ID), exchanging extrinsic information with the soft-decision demapper to approach the channel capacity. Recursive Systematic Convolutional (RSC) codes are selected as constituent codes, an interleaver is used for randomising the extrinsic information exchange of the constituent codes, while a puncturer helps to increase the achievable bandwidth efficiency. The convergence behaviour of the decoder is analysed with the aid of bit-based 3-D EXIT charts, for accurately calculating the operating EbN0 threshold, especially when SP based soft demapper is employed. Finally, we propose an attractive system configuration, which is capable of operating within about 1 dB from the channel capacity.

Erasure correction by low-density codes

We generalize the method for computing the number of errors correctable by a low-density parity-check (LDPC) code in a binary symmetric channel, which was proposed by V.V. Zyablov and M.S. Pinsker in 1975. This method is for the first time applied for computing the fraction of guaranteed correctable erasures for an LDPC code with a given constituent code used in an erasure channel. Unlike previously known combinatorial methods for computing the fraction of correctable erasures, this method is based on the theory of generating functions, which allows us to obtain more precise results and unify the computation method for various constituent codes of a regular LDPC code. We also show that there exist an LDPC code with a given constituent code which, when decoded with a low-complexity iterative algorithm, is capable of correcting any erasure pattern with a number of erasures that grows linearly with the code length. The number of decoding iterations, required to correct the erasures, is a logarithmic function of the code length. We make comparative analysis of various numerical results obtained by various computation methods for certain parameters of an LDPC code with a constituent single-parity-check or Hamming code.

Good concatenated code ensembles for the binary erasure channel

In this work, we give good concatenated code ensembles for the binary erasure channel (BEC). In particular, we consider repeat multiple-accumulate (RMA) code ensembles formed by the serial concatenation of a repetition code with multiple accumulators, and the hybrid concatenated code (HCC) ensembles recently introduced by Koller et al. (5th Int. Symp. on Turbo Codes & Rel. Topics, Lausanne, Switzerland) consisting of an outer multiple parallel concatenated code serially concatenated with an inner accumulator. We introduce stopping sets for iterative constituent code oriented decoding using maximum a posteriori erasure correction in the constituent codes. We then analyze the asymptotic stopping set distribution for RMA and HCC ensembles and show that their stopping distance h <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> , defined as the size of the smallest nonempty stopping set, asymptotically grows linearly with the block length. Thus, these code ensembles are good for the BEC. It is shown that for RMA code ensembles, contrary to the asymptotic minimum distance d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> , whose growth rate coefficient increases with the number of accumulate codes, the h <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> growth rate coefficient diminishes with the number of accumulators. We also consider random puncturing of RMA code ensembles and show that for sufficiently high code rates, the asymptotic h <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> does not grow linearly with the block length, contrary to the asymptotic d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> , whose growth rate coefficient approaches the Gilbert-Varshamov bound as the rate increases. Finally, we give iterative decoding thresholds for the different code ensembles to compare the convergence properties.

Design and performance analysis of a new class of rate compatible serially concatenated convolutional codes

In this paper, a novel class of serially concatenated convolutional codes (SCCCs) is addressed. In contrast to standard SCCCs, where high rates are obtained by puncturing the outer code, the heavy puncturing is moved to the inner code, which can be punctured beyond the unitary rate. We derive analytical upper bounds on the error probability of this code structure by considering an equivalent code construction consisting of the parallel concatenation of two codes, and address suitable design guidelines for code optimization. It is shown that the optimal puncturing of the inner code depends on the outer code, i.e., it is interleaver dependent. This dependence cannot be tracked by the analysis for standard SCCCs, which fails in predicting code performance. Based on the considerations arising from the bounds analysis, we construct a family of rate-compatible SCCCs with a high level of flexibility and a good performance over a wide range of code rates, using simple constituent codes. The error rate performance of the proposed codes is found to be better than that of standard SCCCs, especially for high rates, and comparable to the performance of more complex turbo codes.

The augmented state diagram and its application to convolutional and turbo codes

Convolutional block codes, which are commonly used as constituent codes in turbo code configurations, accept a block of information bits as input rather than a continuous stream of bits. In this paper, we propose a technique for the calculation of the transfer function of convolutional block codes, both punctured and nonpunctured. The novelty of our approach lies in the augmentation of the conventional state diagram, which allows the enumeration of all codeword sequences of a convolutional block code. In the case of a turbo code, we can readily calculate an upper bound to its bit error rate performance if the transfer function of each constituent convolutional block code has been obtained. The bound gives an accurate estimate of the error floor of the turbo code and, consequently, our method provides a useful analytical tool for determining constituent codes or identifying puncturing patterns that improve the bit error rate performance of a turbo code, at high signal-to-noise ratios.