Length Low-density Parity-check Codes Research Articles

Free Ride on LDPC Coded Transmission

In this paper, we formulate the problem to cope with the transmission of extra bits over an existing coded transmission link (referred to as coded payload link) without any cost of extra transmission energy or extra bandwidth. This is possible since a gap to the channel capacity typically exists for a practical code. A new concept, termed as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">accessible capacity</i> , is introduced to specify the maximum rate at which the superposition transmission of extra bits is reliable and has a negligible effect on the performance of the coded payload link. For a binary-input output-symmetric (BIOS) memoryless channel, the accessible capacity can be characterized as the difference between the channel capacity and the mutual information rate of the coded payload link, which can be numerically evaluated for very short payload codes. For a general payload code, we present a simple lower bound on the accessible capacity, given by the channel capacity minus the coding rate of the payload code. We then focus on the scenarios where low-density parity-check (LDPC) codes are implemented for the payload link. We propose to transmit extra bits by random superposition for encoding, and exhaustive search (with the aid of statistical learning) for decoding. We further propose, by establishing an auxiliary channel (called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">syndrome channel</i> ) induced from “zero-forcing” over the binary field, to transmit extra bits with structured codes such as repetition codes and first-order Reed-Muller (RM) codes. Numerical results show that up to 60 extra bits can be reliably transmitted along with a rate-1/2 LDPC code of length 8064.

Transmitting Extra Bits With Cyclically Shifted LDPC Codes

In this letter, we propose a simple yet efficient coding scheme to transmit extra bits by cyclically shifting a low-density parity-check (LDPC) coded sequence without any additional consumption in bandwidth or transmission energy. At the transmitter, the payload data is encoded into an LDPC coded sequence and then cyclically shifted under a pattern specified by the extra bits. At the decoder, the extra bits are first retrieved by estimating the number of cyclic shifts based on the hard or soft syndromes. After removing the effect of cyclic shift, the payload data is recovered by the LDPC decoder. Simulation results show that, with a 5G LDPC code of length 1910, up to 10 extra bits can be reliably transmitted with negligible influence on the reliability of the LDPC coded data.

Modified Min Sum Decoding Algorithm for Low Density Parity Check Codes

Abstract The Sum Product Algorithm (SPA) is known to offer the best performance in decoding large block length Low Density Parity Check (LDPC) codes. However, modifications in SPA to decrease the computation complexity have resulted in performance degradation. The Min Sum Algorithm (MSA) is one such variant, where the computation of the check-to-variable message is simplified to a minimum operation, instead of a hyperbolic tan calculation. Several modifications have been proposed in order to recover the performance loss of the MSA with respect to SPA. This paper proposes a modification to the MSA, based on the Wiener and Linear Minimum Mean Square Error (LMMSE) estimators for LDPC codes. The golden section search algorithm has been used to optimize the computations of the parameter estimation. Simulation results with these estimated parameters show that for low values of SNR, the modified algorithm outperforms MSA and provides performance close to that of SPA with a gain of 0.1 dB. For higher values of SNR, it outperforms MSA with a gain of 1.75 dB while providing an acceptable performance degradation with respect to SPA.

Transmitting extra bits by rotating signal constellations

In this Letter, the authors propose a novel low-density parity-check (LDPC) coding scheme to transmit extra bits aided by rotated signal constellations without any additional cost in transmission power or bandwidth. In the proposed scheme, the LDPC coded data are modulated by a rotated two-dimensional signal constellation, in which the rotation angle is specified by the given extra bits. At the receiver, the rotation angle is estimated with the aid of the statistical learning of the syndrome of the LDPC code. After recovering the rotation angle, the coded payload data can be decoded by the LDPC decoder. The simulation results show that, for an LDPC code of length 2304, up to four extra bits can be transmitted with negligible influence on the reliability of the LDPC coded data.

Packing additional bits into LDPC coded data

In this Letter, the authors proposed a novel scheme, with neither extra transmission energy nor extra bandwidth, to transmit additional bits along with low-density parity-check (LDPC) coded payload data. At the transmitter, additional bits are first transformed into a random-like sequence, which is then superimposed onto the LDPC coded data, resulting in the transmitted sequence. At the receiver, a statistical learning-based algorithm is employed to detect the additional bits. Then the interference of the additional bits is removed and the payload data is recovered by the conventional LDPC decoder. Simulation results show that, for a (3,6)-regular LDPC code of length 8064, ten additional bits can be transmitted reliably with negligible effect on the reliability of the payload data.

BP-LED Decoding Algorithm for LDPC Codes Over AWGN Channels

A new method is presented for low-complexity near-maximum-likelihood (ML) decoding of low-density parity-check (LDPC) codes over the additive white Gaussian noise channel. The proposed method termed belief-propagation–list erasure decoding (BP-LED) is based on erasing carefully chosen unreliable bits performed in case of BP decoding failure. A strategy of introducing erasures into the received vector and a new erasure decoding algorithm are proposed. The new erasure decoding algorithm, called list erasure decoding, combines ML decoding over the BEC with list decoding applied if the ML decoder fails to find a unique solution. The asymptotic exponent of the average list size for random regular LDPC codes from the Gallager ensemble is analyzed. Furthermore, a few examples of irregular quasi-cyclic LDPC as well as randomly constructed regular LDPC codes of short and moderate lengths are studied by simulations and their performance is compared to the tightened upper bound on the LDPC ensemble-average performance and the upper bound on the average performance of random linear codes under ML decoding. A comparison of the BP decoding and BP-LED performance of the WiMAX standard codes and performance of the near-ML BEAST decoding are presented. The new algorithm is applied to decoding a short nonbinary (NB) LDPC code over extensions of the binary Galois field. The obtained simulation results are compared to the tightened upper bound on the ensemble-average performance of the binary image of regular NB LDPC codes.

Multichannel Cooperative Spectrum Sensing That Integrates Channel Decoding With Fusion-Based Decision

This paper considers coded multichannel cooperative spectrum sensing (MC-CSS) employing local Neyman–Pearson testing at each sensor and channel decoding integrated with fusion-based decision. A joint multichannel decoding and decision fusion (JMCDDF) algorithm for performing a likelihood ratio test at the fusion center is derived based on the belief propagation technique. For benchmark comparison, we also derive the analytical performance of MC-CSS with error-free reporting channels that do not require channel coding, employing equal decision thresholds at each sensor. Using the JMCDDF algorithm, we compare the performance between uncoded and coded MC-CSS schemes when applying low-density parity-check (LDPC) codes in the presence of reporting channel errors. Monte–Carlo simulation results show considerable performance gains when using the proposed coded MC-CSS schemes. A 3-dB saving in link budget can be achieved by such coded MC-CSS schemes with a short (3, 6) regular LDPC code of codeword length $n_c=\text{200}$ , over uncoded MC-CSS. Finally, it is shown that in some cases, protecting the primary user (PU) channels by using variable nodes of higher degrees improves the performance. To equalize the performance for all sensed PU channels, we introduce a simple permutation technique, where, in each transmission, different variable nodes are used to protect local decisions.

Bit Mapping Design for LDPC Coded BICM Schemes With Multi-Edge Type EXIT Chart

In this letter, we consider the bit mapping (BM) design for regular low-density parity-check (LDPC) coded bit-interleaved coded modulation (BICM) schemes. We propose a check-node type distribution and develop multi-edge type extrinsic information transfer (EXIT) functions for the regular LDPC coded BICM schemes. Based on the EXIT functions, we propose a two-phase design method to generate the optimized practical BM in terms of the lowest decoding threshold for a given finite length LDPC code. Simulation results show that the proposed BM has a better decoding threshold compared with those of the existing works.

CS image reconstruction by novel small length LDPC code

Compressed sensing (CS) is capable of reconstructing the data signal from very small number of measurements. However the reconstructed image quality gets highly affected in the case of far end reconstruction involving transmission through a channel. In this work, a novel small length binary low density parity check (LDPC) code for far-end transmission of compressively sampled signals is proposed. Smaller length of LDPC code means faster computation. Also the code rate of proposed LDPC code is constructed to be 0.6 as in an earlier work by the same group it has been demonstrated that quasi-cyclic LDPC codes, with code rate 0.67 gives optimum performance with CS. The proposed code has a base matrix of size 4 × 10. The length of the LDPC code constructed, can be adjusted by a base matrix multiplying factor. The multiplying factor varies from 10 to 30 in steps of 4. The constructed code has a variable length between 100 and 300, in steps of 40. Simulation results of the proposed LDPC code show good performance when decoded iteratively by min-sum decoding algorithm. The proposed LDPC code also gives improved CS reconstruction for far end reconstruction.

Searching for Binary and Nonbinary Block and Convolutional LDPC Codes

A unified approach to search for and optimize codes determined by their sparse parity-check matrices is presented. Replacing the nonzero elements of a binary parity-check matrix (the base or parent matrix) either by circulants or by companion matrices of elements from a finite field GF $(2^{m})$ , we obtain quasi-cyclic low-density parity-check (LDPC) block codes and binary images of nonbinary LDPC block codes, respectively. By substituting monomials of a formal variable $D$ , we obtain the polynomial description of an LDPC convolutional code. A set of performance measures applicable to different classes of LDPC codes is considered, and a greedy algorithm for code performance optimization is presented. The heart of the new optimization algorithm is a fast procedure for searching for LDPC codes with large girth of their Tanner graphs. For a few classes of LDPC codes, examples of codes combining good error-correcting performance with compact representation are obtained. In particular, we present optimized convolutional LDPC codes and conclude that the LDPC block codes are still superior to their convolutional counterparts if both decoding complexity and coding delay are considered. Moreover, a specific channel model can easily be embedded into the optimization loop. Thereby, the code can be optimized for a specific channel. The efficiency of such an optimization is demonstrated via an example of faster than Nyquist (FTN) signaling using LDPC codes. The FTN strategy combined with a rate $R=1/2$ LDPC code of length 64 800 optimized for effective data rate $R=3/4$ gains more than 0.5 dB compared with the standard LDPC codes of the same rate and length. The obtained gain corresponds to transmission at the capacity of the binary input additive white Gaussian noise channel. In most numerical examples, we consider codes with bidiagonal structure of the parity-check matrix. This restriction preserves low encoding complexity and allows fair comparison with codes selected for communication standards.

High-efficiency Gaussian key reconciliation in continuous variable quantum key distribution

Efficient reconciliation is a crucial step in continuous variable quantum key distribution. The progressive-edge-growth (PEG) algorithm is an efficient method to construct relatively short block length low-density parity-check (LDPC) codes. The qua-sicyclic construction method can extend short block length codes and further eliminate the shortest cycle. In this paper, by combining the PEG algorithm and qua-si-cyclic construction method, we design long block length irregular LDPC codes with high error-correcting capacity. Based on these LDPC codes, we achieve high-efficiency Gaussian key reconciliation with slice recon-ciliation based on multilevel coding/multistage decoding with an efficiency of 93.7%.

Comparison of Convolutional and Block Codes for Low Structural Delay

The performance of short block length low-density parity-check (LDPC) codes (both binary and nonbinary) and convolutional codes is compared under the constraint of tight structural delay constraints. Additionally, we use fundamental bounds on block codes and low rate turbo codes to evaluate our results in a broader context. It turns out that—depending on the code rate and given delay—convolutional codes are able to outperform fundamental lower bounds for block codes, yielding a definite result on the question, which codes are superior in this regime. From a break-even point onward, convolutional codes cannot compete with block codes anymore and nonbinary LDPC codes show the best performance. Turbo codes with a short interleaver length show competitive results.

Construction of High-Rate Regular Quasi-Cyclic LDPC Codes Based on Cyclic Difference Families

For a high-rate case, it is difficult to randomly construct good low-density parity-check (LDPC) codes of short and moderate lengths because their Tanner graphs are prone to making short cycles. Also, the existing high-rate quasi-cyclic (QC) LDPC codes can be constructed only for very restricted code parameters. In this paper, a new construction method of high-rate regular QC LDPC codes with parity-check matrices consisting of a single row of circulants with the column-weight 3 or 4 is proposed based on special classes of cyclic difference families. The proposed QC LDPC codes can be constructed for various code rates and lengths including the minimum achievable length for a given design rate, which cannot be achieved by the existing high-rate QC LDPC codes. It is observed that the parity-check matrices of the proposed QC LDPC codes have full rank. It is shown that the error correcting performance of the proposed QC LDPC codes of short and moderate lengths is almost the same as that of the existing ones through numerical analysis.

Blind reconciliation

Information reconciliation is a crucial procedure in the classical post-processing of quantum key distribution (QKD). Poor reconciliation efficiency, revealing more information than strictly needed, may compromise the maximum attainable distance, while poor performance of the algorithm limits the practical throughput in a QKD device. Historically, reconciliation has been mainly done using close to minimal information disclosure but heavily interactive procedures, like \textit{Cascade}, or using less efficient but also less interactive ---just one message is exchanged--- procedures, like the ones based in low-density parity-check (LDPC) codes. The price to pay in the LDPC case is that good efficiency is only attained for very long codes and in a very narrow range centered around the quantum bit error rate (QBER) that the code was designed to reconcile, thus forcing to have several codes if a broad range of QBER needs to be catered for. Real world implementations of these methods are thus very demanding, either on computational or communication resources or both, to the extent that the last generation of GHz clocked QKD systems are finding a bottleneck in the classical part. In order to produce compact, high performance and reliable QKD systems it would be highly desirable to remove these problems. Here we analyse the use of short-length LDPC codes in the information reconciliation context using a low interactivity, \textit{blind}, protocol that avoids an a priori error rate estimation. We demonstrate that $2 \times 10^3$ bits length LDPC codes are suitable for blind reconciliation. Such codes are of high interest in practice, since they can be used for hardware implementations with very high throughput.

45 Gb/s low complexity optical front-end for soft-decision LDPC decoders

In this paper a low complexity and energy efficient 45 Gb/s soft-decision optical front-end to be used with soft-decision low-density parity-check (LDPC) decoders is demonstrated. The results show that the optical front-end exhibits a net coding gain of 7.06 and 9.62 dB for post forward error correction bit error rate of 10(-7) and 10(-12) for long block length LDPC(32768,26803) code. The performance over a hard decision front-end is 1.9 dB for this code. It is shown that the soft-decision circuit can also be used as a 2-bit flash type analog-to-digital converter (ADC), in conjunction with equalization schemes. At bit rate of 15 Gb/s using RS(255,239), LDPC(672,336), (672, 504), (672, 588), and (1440, 1344) used with a 6-tap finite impulse response (FIR) equalizer will result in optical power savings of 3, 5, 7, 9.5 and 10.5 dB, respectively. The 2-bit flash ADC consumes only 2.71 W at 32 GSamples/s. At 45 GSamples/s the power consumption is estimated to be 4.95 W.

Finding the Exhaustive List of Small Fully Absorbing Sets and Designing the Corresponding Low Error-Floor Decoder

This work provides an efficient exhaustive search algorithm for finding all small fully absorbing sets (FASs) of any arbitrary low-density parity-check (LDPC) code. The proposed algorithm is based on the branch-&-bound principle for solving NP-complete problems. In particular, given any LDPC code, the problem of finding all FASs of size less than t is formulated as an integer programming problem, for which a new branch-&-bound algorithm is devised with new node selection and tree-trimming mechanisms. The resulting algorithm is capable of finding all FASs of size <; 7 for LDPC codes of length <; 1000. When limiting the FASs of interest to those with the number of violated parity-check nodes <; 3, the proposed algorithm is capable of finding all such FASs of size <; 14 for LDPC codes of lengths <; 1000. The resulting exhaustive list of small FASs is then used to devise a new efficient post-processing low-error floor LDPC decoder. The numerical results show that by exploiting the exhaustive list of small FASs, the proposed post-processing decoder can significantly lower the error-floor performance of a given LDPC code. For various example codes of length <; 3000, the proposed post-processing decoder lowers the error floor by a couple of orders of magnitude when compared to the standard belief propagation decoder and by an order of magnitude when compared to other existing low error-floor decoders.

Analytical Solution of Covariance Evolution for Irregular LDPC Codes

The scaling law developed by Amraoui et al. is a powerful technique to estimate the block erasure probabilities of finite- length low-density parity-check (LDPC) codes. Solving a system of differential equations called covariance evolution, one can obtain the scaling parameter. However, the covariance evolution has not been analytically solved. In this paper, we present the analytical solution of the covariance evolution for irregular LDPC code ensembles.

Optimization of Construction and Encoding for Finite Length Irregular LDPC Codes

The main influence for the performance of finite length Low-Density Parity-Check(LDPC) codes is analyzed. According to optimum choice for degree distribution, a check matrix is constructed with improved Progressive-Edge-Growth (PEG) algorithm. A practical efficient encoding algorithm is proposed to optimize the check matrix. A finite length irregular LDPC code with low error-floor performance and approximate linear encoding complexity is obtained. This optimization method can be easily extended to general communication channels.

Novel modified min-sum decoding algorithm for low-density parity-check codes

The problem of improving the performance of min-sum decoding of low-density parity-check (LDPC) codes is considered in this paper. Based on min-sum algorithm, a novel modified min-sum decoding algorithm for LDPC codes is proposed. The proposed algorithm modifies the variable node message in the iteration process by averaging the new message and previous message if their signs are different. Compared with the standard min-sum algorithm, the modification is achieved with only a small increase in complexity, but significantly improves decoding performance for both regular and irregular LDPC codes. Simulation results show that the performance of our modified decoding algorithm is very close to that of the standard sum-product algorithm for moderate length LDPC codes.

Multistep Linear Programming Approaches for Decoding Low-Density Parity-Check Codes

The problem of improving the performance of linear programming (LP) decoding of low-density parity-check (LDPC) codes is considered in this paper. A multistep linear programming (MLP) algorithm was developed for decoding LDPC codes that includes a slight increase in computational complexity. The MLP decoder adaptively adds new constraints which are compatible with a selected check node to refine the results when an error is reported by the original LP decoder. The MLP decoder result is shown to have the maximum-likelihood (ML) certificate property. Simulations with moderate block length LDPC codes suggest that the MLP decoder gives better performance than both the original LP decoder and the conventional sum-product (SP) decoder.