LDPC Convolutional Codes Research Articles

An Approach to Constructing Convolutional Codes With Moderate Density and Quasi-Cyclic Structure

There exist in the literature two natural generalizations of low density parity check (LDPC) codes: 1) LDPC convolutional codes or sometimes also called spatially coupled LDPC codes which have shown to be able to reach Shannon capacity in a practical way. 2) Moderate density parity check (MDPC) codes which are linear codes possessing a parity check matrix whose row weight is not more than O(√n), still allowing efficient decoding with high probability. MDPC codes became a highly interesting class of block codes for the purpose of doing code-based cryptography.In this paper we study MDPC convolutional codes and some of their basic properties. At the end of the paper we explain how one can construct a new code-based cryptographic system. This new system can be seen as a convolutional version of the famous BIKE system currently evaluated for possible standardization by the National Institute for Standards and Technology (NIST).

An Efficient Reconfigurable Encoder for the IEEE 1901 Standard

The IEEE 1901 standard for power line communication (PLC) enables simple connection among Internet of Things devices. The forward error correction (FEC) codes specified in the IEEE 1901 standard include low-density parity-check convolutional codes (LDPC-CCs) and Reed-Solomon convolutional concatenated (RSCC) codes. This work introduces an efficient reconfigurable encoder in full compliance with the IEEE 1901 standard. First, we propose a reconfigurable LDPC-CC encoder to fulfill the multirate requirement and improve the architecture by fine-tuned parallelization, which takes full advantage of the characteristics of the codeword structure. Then, for area reduction, the optimization regarding the RSCC encoder is extensively exploited. Moreover, the commonality between the encoders is discovered, and some circuitries are shared to reduce the hardware complexity. Equipped with these techniques, an efficient reconfigurable encoder for the IEEE 1901 standard is developed and implemented with 28-nm technology. Implementation results demonstrate that the proposed encoder can meet the throughput requirement of the IEEE 1901 standard and is both power- and area-efficient.

Finite field construction for quasi-cyclic LDPC convolutional codes with cyclic 2-D MDS codes

Finite field construction for quasi-cyclic LDPC convolutional codes with cyclic 2-D MDS codes

Construction of LDPC convolutional codes via difference triangle sets

In this paper, a construction of (n,k,delta ) LDPC convolutional codes over arbitrary finite fields, which generalizes the work of Robinson and Bernstein and the later work of Tong is provided. The sets of integers forming a (k, w)-(weak) difference triangle set are used as supports of some columns of the sliding parity-check matrix of an (n,k,delta ) convolutional code, where nin {mathbb {N}}, n>k. The parameters of the convolutional code are related to the parameters of the underlying difference triangle set. In particular, a relation between the free distance of the code and w is established as well as a relation between the degree of the code and the scope of the difference triangle set. Moreover, we show that some conditions on the weak difference triangle set ensure that the Tanner graph associated to the sliding parity-check matrix of the convolutional code is free from 2ell -cycles not satisfying the full rank condition over any finite field. Finally, we relax these conditions and provide a lower bound on the field size, depending on the parity of ell , that is sufficient to still avoid 2ell -cycles. This is important for improving the performance of a code and avoiding the presence of low-weight codewords and absorbing sets.

Ensemble of high performance structured binary convolutional LDPC codes with moderate rates

An algebraic construction methodology is proposed to design binary time-invariant convolutional low-density parity-check (LDPC) codes. Assisted by a proposed partial search algorithm, the polynomialform parity-check matrix of the time-invariant convolutional LDPC code is derived by combining some special codewords of an (n, 2, n − 1) code. The achieved convolutional LDPC codes possess the characteristics of comparatively large girth and given syndrome former memory. The objective of our design is to enable the time-invariant convolutional LDPC codes the advantages of excellent error performance and fast encoding. In particular, the error performance of the proposed convolutional LDPC code with small constraint length is superior to most existing convolutional LDPC codes.

Edge-Coloring Technique to Analyze Elementary Trapping Sets of Spatially-Coupled LDPC Convolutional Codes

In this letter, for the first time, an edge-coloring technique is proposed to characterize a certain elementary trapping set (ETS) and to obtain sufficient conditions to avoid small ETSs from occurrence in the Tanner graph of SC-LDPC convolutional codes. This technique is applicable to all protograph-based LDPC codes with different girths whose protographs are single-edge, that is, the entries of their base matrices are 0, 1. To further demonstrate the effectiveness of our proposed technique, we apply it to Time-Invariant SC-LDPC-CCs with girths 6 and 8 and column weights up to 5.

Rate-compatible LDPC convolutional codes over non-gaussian noise channel

This paper is aimed to study the characteristics of the underwater acoustic channel with non-Gaussian noise channel. And Gaussian mixture model (GMM) is utilized to fit the background noise over the non-Gaussian noise channel. Furthermore, coding techniques which use a sequence of rate-compatible low-density parity-check (RC-LDPC) convolutional codes with separate rates are constructed based on graph extension method. The performance study of RC-LDPC convolutional codes over non-Gaussian noise channel and the additive white Gaussian noise (AWGN) channel is performed. Study implementation of simulation is that modulation with binary phase shift keying (BPSK), and iterative decoding based on pipeline log-likelihood rate belief propagation (LLRBP) algorithm. Finally, it is shown that RC-LDPC convolutional codes have good bit-rate-error (BER) performance and can effectively reduce the impact of noise.

Optimal Search for Girth-8 Quasi Cyclic and Spatially Coupled Multiple-Edge LDPC Codes

Two essential categories of LDPC codes that are more preferable to other types are quasi-cyclic LDPC (QC-LDPC) codes and spatially coupled LDPC convolutional codes (SC-LDPC-CCs) because of their excellent performance curves in waterfall and error floor regions. An efficient approach to construct these codes is protograph-based method that is categorized into two classes: single-edge (SE) and multiple-edge (ME) protographs. We, for the first time, provide a necessary and sufficient condition for exponent matrices of these codes with girth-8 and based on the ME-protographs. As a result, a lower bound on the lifting degree of girth-8 ME-QC-LDPC codes and a lower bound on the syndrome former memory order of girth-8 ME-SC-LDPC-CCs are obtained, which are tighter than the existing bounds in the literature.

Serial Decoding Algorithm with Continuous Backtracking for LDPC Convolutional Codes

Two main decoding algorithms are usually used for low-density parity-check convolutional (LDPC-C) codes, belief propagation algorithm and on-demand variable node activation algorithm. However like the decoding for block codes, the iterations of these algorithms cannot utilize the messages of newly joined variable nodes within the next iteration windows continuously, limiting the decoding performance. Thus a serial decoding algorithm with continuous backtracking for LDPC-C codes is proposed. Based on the iteration windows, the proposed algorithm performs advance iteration with two adjacent iteration windows and then backtracks to the preceding iteration window. During each iteration the latest updated and newly joined messages are utilized. The whole iteration uses the backtracking continuously until all the results output. Different LDPC-C codes are used to test the performance of proposed algorithm. Simulation results show that the proposed algorithm outperforms other algorithms and requires less number of iterations, improving the decoding convergence rate.

Optimization–based decoding algorithms for LDPC convolutional codes in communication systems

In a digital communication system, information is sent from one place to another over a noisy communication channel. It may be possible to detect and correct errors that occur during the transmission if one encodes the original information by adding redundant bits. Low–Density Parity–Check (LDPC) convolutional codes, a member of the LDPC code family, encode the original information to improve error correction capability. In practice these codes are used to decode very long information sequences, where the information arrives in subsequent packets over time, such as video streams. We consider the problem of decoding the received information with minimum error from an optimization point of view and investigate integer programming–based exact and heuristic decoding algorithms for its solution. In particular, we consider relax–and–fix heuristics that decode information in small windows. Computational results indicate that our approaches identify near–optimal solutions significantly faster than a commercial solver in high channel error rates. Our proposed algorithms can find higher quality solutions compared with the state of the art iterative decoding heuristic.

Low-complexity LDPC-convolutional codes based on cyclically shifted identity matrices

In this study, a construction methodology for low-density parity-check convolutional codes (LDPC-CCs) ensembles based on cyclically shifted identity matrices is proposed. The proposed method directly generates the syndrome former matrices according to the specified code parameters and constraints i.e., code-rate, degree-distribution, constraint length, period and memory, in contrast to the majority of the available approaches that produce relevant error-correcting codes based on either block ones, protographs or spatially-coupled type of codes. Simulation results show that the constructed ensembles demonstrate advanced error-correcting capability of up to 0.2 dB in terms of frame-error and bit-error rates at the convergence region, when compared with the performance of error-correcting schemes adopted by various communication standards, with equivalent hardware complexity even at short codeword-lengths. Specifically, the constructed LDPC-CCs have been assessed against the corresponding error-correcting codes used in WiMAX and G.hn standards for wireless and wireline telecommunications, respectively.

Low-complexity LDPC-convolutional codes based on cyclically shifted identity matrices

In this study, a construction methodology for low-density parity-check convolutional codes (LDPC-CCs) ensembles based on cyclically shifted identity matrices is proposed. The proposed method directly generates the syndrome former matrices according to the specified code parameters and constraints i.e., code-rate, degree-distribution, constraint length, period and memory, in contrast to the majority of the available approaches that produce relevant error-correcting codes based on either block ones, protographs or spatially-coupled type of codes. Simulation results show that the constructed ensembles demonstrate advanced error-correcting capability of up to 0.2 dB in terms of frame-error and bit-error rates at the convergence region, when compared with the performance of error-correcting schemes adopted by various communication standards, with equivalent hardware complexity even at short codeword-lengths. Specifically, the constructed LDPC-CCs have been assessed against the corresponding error-correcting codes used in WiMAX and G.hn standards for wireless and wireline telecommunications, respectively.

Reduced Complexity Window Decoding of Spatially Coupled LDPC Codes for Magnetic Recording Systems

In channel coding theory, the performance of error correcting codes (ECCs) approaching the Shannon limit can be achieved through increasing code lengths. Unfortunately, the complexity of ECCs will be increased as the code length increases. Nowadays, the magnetic recording (MR) system takes advantage of powerful ECCs by using 4 Kbytes sector. Among the advanced ECCs, the spatially coupled LDPC (SC-LDPC) codes (also known as a LDPC convolutional code) [1]are shown to have the decoding latency and complexity lower than those of the underlying LDPC block codes (LDPC-BC). Moreover, the SC-LDPC codes with threshold decoding outperform the LDPC-BC codes [2]. Hence, the SC-LDPC codes are the strong candidate for the future MR systems, when the sector size is increased beyond 4 Kbytes. An SC-LDPC decoder can use sliding window decoding [3]whereby the received signals are decoded by sliding window along the bit sequence. The window decoder is called “uniform window decoding (U-WD),” when all variable nodes (VNs) within a window are updated. In order to reduce the complexity of window decoding, some researchers proposed the non-uniform window decoding (N-WD) [4], which do not update the VNs with no improvement in the bit error rate (BER). This approach provides about 35-50% reduction in complexity compared to U-WD. In this work, we consider the application of SC-LDPC codes in MR systems, whereby SC-LDPC decoder cooperates with BCJR detector to encounter inter-symbol interference (ISI). We propose the dynamic shifting of window decoding (DS-WD) to reduce the complexity of SC-LDPC codes. Herein, the number of shifted bits is defined according to their soft BERs which are estimated at each decoding position. In addition, we modify the N-WD [4]to reinforce our proposed algorithm called “dynamic-shifting non-uniform window decoding (DS-N-WD).” The DS-WD and DS-N-WD achieve the complexity reduction of 7% and 25% without any loss in performance compared to the N-WD algorithms.

Connections Between Low-Weight Codewords and Cycles in Spatially Coupled LDPC Convolutional Codes

In this paper, time-invariant spatially coupled low-density parity-check convolutional codes (SC-LDPC-CCs) are considered, and the connections existing between their low-weight codewords and cycles in their Tanner graphs are studied. Using the polynomial representation of these codes, we show that parity-check matrices having columns with weight ≥2 can be analyzed considering a certain number of parity-check sub-matrices having regular columns with weight 2. These sub-matrices are associated to cycles in the code Tanner graph and define as many codes we denote as component codes . Based on this observation, we find that codewords of the main code can be expressed as combinations of codewords of the component codes. The design of codes free of codewords up to a certain weight is also addressed. We show that low-weight codewords in the main code can be avoided by removing some types of cycles in its Tanner graph. Our design approach is applied to some well-known ensembles of SC-LDPC-CCs to prove its effectiveness.

Efficient Search of Compact QC-LDPC and SC-LDPC Convolutional Codes With Large Girth

We propose a low-complexity method to find quasi-cyclic low-density parity-check block codes with girth 10 or 12 and length shorter than those designed through classical approaches. The method is extended to time-invariant spatially coupled low-density parity-check convolutional codes, permitting to achieve small syndrome former constraint lengths. Several numerical examples are given to show its effectiveness.

Design and Analysis of Time-Invariant SC-LDPC Convolutional Codes With Small Constraint Length

In this paper, we deal with time-invariant spatially coupled low-density parity-check convolutional codes (SC-LDPC-CCs). Classic design approaches usually start from quasi-cyclic low-density parity-check block codes and exploit suitable unwrapping procedures to obtain SC-LDPC-CCs. We show that the direct design of the SC-LDPC-CCs syndrome former matrix or, equivalently, the symbolic parity-check matrix, leads to codes with smaller syndrome former constraint lengths with respect to the best solutions available in the literature. We provide theoretical lower bounds on the syndrome former constraint length for the most relevant families of SC-LDPC-CCs, under constraints on the minimum length of cycles in their Tanner graphs. We also propose new code design techniques that approach or achieve such theoretical limits.

Non-binary Tail-Biting LDPC Convolutional Code Encoding for Image Transmission

In current image transmission, data channel encoding is needed to consider noise and bit-error protection in order to enhance and guarantee image-data quality. This paper focuses on a problem of image transmission with high bit error rate (BER) and additive white Gaussian noise (AWGN) channel, and proposes a method of non-binary tail-biting low-density parity-check convolutional (NB TB-LDPCC) code encoding for image transmission. In this method, image data compressed by JPEG are used to construct finite fields in non-binary, then construct a base matrix in column weight two, and finally create tail biting LDPC convolution code for modulation in the next process in image transmission. The performance of the proposed method is evaluated by 30 images on USC-SIPI Image Database, and the comparison results with conventional method reveal BER is improved at most 1 dB, and PSNR is increased 2.01 dB approximately.

Construction of rate-compatible (RC) low-density parity-check (LDPC) convolutional codes based on RC-LDPC block codes

In this paper, a family of rate-compatible (RC) low-density parity-check (LDPC) convolutional codes can be obtained from RC-LDPC block codes by graph extension method. The resulted RC-LDPC convolutional codes, which are derived by permuting the matrices of the corresponding RC-LDPC block codes, are systematic and have maximum encoding memory. Simulation results show that the proposed RC-LDPC convolutional codes with belief propagation (BP) decoding collectively offer a steady improvement on performance compared with the block counterparts over the binary-input additive white Gaussian noise channels (BI-AWGNCs).

Time-Invariant Quasi-Cyclic Spatially Coupled LDPC Codes Based on Packings

This paper presents two packings derived from balanced incomplete block designs to construct quasi-cyclic spatially coupled LDPC convolutional codes (SC-LDPC-CCs). The construction gives time-invariant codes, since the sub-blocks corresponding to each time instant of the parity-check matrix are identical. Moreover, it provides flexible design rates and constraint lengths. Simulation results show that the proposed packing-based SC-LDPC-CCs outperform the existing time-invariant codes and perform closely to the time-varying protograph-based codes.

Construction of regular rate-compatible LDPC convolutional codes

In this paper, we propose a new method to derive a family of regular rate-compatible low-density parity-check (RC-LDPC) convolutional codes from RC-LDPC block codes. In the RC-LDPC convolutional family, each extended sub-matrix of each extended code is obtained by choosing specified elements from two fixed matrices Hk E1 and Hk E2 , which are derived by modifying the extended matrices H E1 and H E2 of a systematic RC-LDPC block code. The proposed method which is based on graph extension simplifies the design, and prevent the defects caused by the puncturing method. It can be used to generate both regular and irregular RC-LDPC convolutional codes. All resulted codes in the family are systematic which simplify the encoder structure and have maximum encoding memories which ensure the property. Simulation results show the family collectively offer a steady improvement in performance with code compatibility over binary-input additive white Gaussian noise channel (BI-AW-GNC).