Construction of quasi‐cyclic low‐density parity‐check codes with low encoding complexity

Seeking good error correcting codes to improve the efficiency of continuous-variable quantum key distribution (CVQKD) reconciliation is a concerning issue. Due to the introduction of multidimensional reconciliation, the error correcting techniques in the classical binary-input additive white Gaussian noise channel are applicable to CVQKD. In this Letter, we apply the quasi-cyclic low-density parity-check (QC-LDPC) codes, which are specified in 5G protocol, to the reconciliation process. Simulation results show that the reconciliation efficiency can reach 92.6% when the code rate is 22/68 and the signal-to-noise ratio is 0.623. Such a new error correcting code points out a new direction for the development of CVQKD.

In this paper, we propose a simple mathematical structure to improve the BER (bit error rate) performance of quasi-cyclic low-density parity-check (QC-LDPC) codes. Since the proposed structure can be defined in-between those of random codes and structured codes, these proposed QC-LDPC codes not only perform as well as random-LDPC codes, but also preserve the advantage of structured LDPC codes which is easier to implement over coding hardware. Moreover, the proposed structure, constructed from balanced incomplete block design (BIBD), can reduce the density of parity-check matrix and hence reduce the number of short cycles, which generally lead to better performance. At the end, we show that the error floor performance of QC-LDPC codes can be improved by proposed hybrid structures and perform well over binary-input additive white Gaussian noise (BI-AWGN) channel.

A novel construction method of quasi-cyclic low-density parity-check (QC-LDPC) code is proposed based on the finite field multiplicative group, which has easier construction, more flexible code-length code-rate adjustment and lower encoding/decoding complexity. Moreover, a regular QC-LDPC(5334,4962) code is constructed. The simulation results show that the constructed QC-LDPC(5334,4962) code can gain better error correction performance under the condition of the additive white Gaussian noise (AWGN) channel with iterative decoding sum-product algorithm (SPA). At the bit error rate (BER) of 10−6, the net coding gain (NCG) of the constructed QC-LDPC(5334,4962) code is 1.8 dB, 0.9 dB and 0.2 dB more than that of the classic RS(255,239) code in ITU-T G.975, the LDPC(32640,30592) code in ITU-T G.975.1 and the SCG-LDPC(3969,3720) code constructed by the random method, respectively. So it is more suitable for optical communication systems.

In this paper, a transmission scheme based on Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes with an invertible quasi-cyclic Matrix Q is proposed over Additive White Gaussian Noise (AWGN) channel in order to obtain faster convergence of hard-decision decoding. Combined with the soft information from channel, another improved Reliability-based Q (RQ) decoding algorithm is proposed. The new design can also be applied to QC-LDPC codes in IEEE 802.16e Standards. The simulation results show that the proposed scheme with Q or RQ decoding algorithm has better error correction performance and lower average number of iterations than Bit-Flipping (BF) or weighted Bit-Flipping (WBF) decoding algorith

Evolutionary optimization algorithms, including particle swarm optimization (PSO), have been successfully applied in oil industry for production planning and control. Such optimization studies are quite challenging due to large number of decision variables, production scenarios, and subsurface uncertainties. In this work, multi-stage, multi-swarm PSO (MS2PSO) algorithm is proposed to fix certain issues with canonical PSO algorithm such as premature convergence, excessive influence of global best solution, and oscillation. Multiple experiments are conducted using Olympus benchmark to compare the efficacy of algorithms. Results from canonical PSO are first compared with two PSO variations in which hyperparameters are tuned to prioritize exploration in early phase and exploitation in late phase. Firstly, linearly decreasing inertia weight (LDIW-PSO) is used to have greater weight of current particle position during initial iterations, and vice versa. Then, time-varying acceleration coefficients (TVAC-PSO) are used to have greater weight of personal best and lesser weight of global best during the initial iterations, and vice versa. Next, a two-stage multi-swarm PSO (2SPSO) is used where multiple-swarms of the first stage collapse into a single swarm in the second stage. Finally, MS2PSO with multiple stages and multiple swarms is used in which swarms recursively collapse after each stage. Multiple swarm strategy ensures that diversity is retained within the population and multiple modes are explored. Staging ensures that local optima found during initial stage does not lead to premature convergence. Optimization test case comprises of 90 control variables of which 72 are well control related and 18 are well placement related. Swarm intelligence refers to global patterns emerging from simple interactions among population. Algorithmic rules at micro level lead to social interaction at meso level, which then further leads to collective behavior at macro level. It is observed that different algorithm designs have their own benefits and drawbacks. Decreasing inertia weight in LDIW-PSO enables exploration in early stages and convergence around global best in the later stages. TVAC-PSO on the other hand restricts social learning and aids exploration in the early iterations. Social learning component is increased as run progresses, and population moves towards global best. 2SPSO aids in exploring multi-modal objective space, thus preventing premature convergence to a local optima. Swarms collapse into one group in the second stage, and run finally converges towards global best. Multiple swarms and stages in MS2PSO ensure that diversity in population is maintained throughout the run which enables continuous learning, and thus mitigates premature convergence. Both 2SPSO and MS2PSO are found to be helpful for problems with high dimensions and multiple modes where greater degree of exploration is desired. Commercial cloud computing and parallel programming were used to handle high computational workload and reduce run-time from weeks to days. Coefficients of canonical PSO are tuned in LDIW-PSO and TVAC-PSO, which helps in mitigating issues like premature convergence and oscillation. Two-stage PSO (2SPSO) where multiple swarms of first stage collapse into one in second stage, and multi-stage multi-swarm PSO (MS2PSO) where swarms recursively collapse into one are proposed. These algorithms modify the social behavior at meso scale based on number of swarms, number of stages and iterations in each stage.

In practical communication applications, the channels for intended users and eavesdroppers are not error-free and Wyner’s wiretap channel model deals with the scenario. Using this model, the security of a stand-alone stream cipher can be strengthened by exploiting the properties of physical layer. In this paper, a joint channel coding and light weight cryptography for setting a Gaussian wiretap channel is proposed. The scheme is based on a keyed Quasi Cyclic Low Density Parity Check (QCLDPC) encoder and light weight stream cipher based on Linear Feedback Shift Register (LFSR). The significant contribution is that, highly complex non-linear function that provides security in a Non-Linear Filter Generator (NLFG) is replaced by a simple non-linear function without compromising security. Enhanced security with lesser complexity is achieved by embedding security in channel encoder. Results show that attacker cannot extract the secret key because of the errors introduced in the physical layer due to unknown structure of the channel encoder.

In view of the problems that the encoding complexity of quasi-cyclic low-density parity-check (QC-LDPC) codes is high and the minimum distance is not large enough which leads to the degradation of the error-correction performance, the new irregular type-II QC-LDPC codes based on perfect cyclic difference sets (CDSs) are constructed. The parity check matrices of these type-II QC-LDPC codes consist of the zero matrices with weight of 0, the circulant permutation matrices (CPMs) with weight of 1 and the circulant matrices with weight of 2 (W2CMs). The introduction of W2CMs in parity check matrices makes it possible to achieve the larger minimum distance which can improve the error- correction performance of the codes. The Tanner graphs of these codes have no girth-4, thus they have the excellent decoding convergence characteristics. In addition, because the parity check matrices have the quasi-dual diagonal structure, the fast encoding algorithm can reduce the encoding complexity effectively. Simulation results show that the new type-II QC-LDPC codes can achieve a more excellent error-correction performance and have no error floor phenomenon over the additive white Gaussian noise (AWGN) channel with sum-product algorithm (SPA) iterative decoding.

This paper proposes a new method of constructing compact fully-connected Quasi-Cyclic Low Density Parity Check (QC-LDPC) codes with girth <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula> = 8, 10, and 12. The originality of the proposed method is to impose constraints on the exponent matrix P to reduce the search space drastically. For a targeted lifting degree of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> , the first step of the method is to sieve the integer ring <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {Z}_{N}$ </tex-math></inline-formula> to make a particular sub-group with specific properties to construct the second column of P (the first column being filled with zeros). The remaining columns of P are determined recursively as multiples of the second column by adapting the sequentially multiplied column (SMC) method whereby a controlled greedy search is applied at each step. The codes constructed with the proposed semi-algebraic method show lengths that can be significantly shorter than their best counterparts in the literature.

In this letter, we study the construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes by array dispersion and masking. We first propose a class of arrays of circulant permutation matrices (CPMs) of arbitrary size obtained by dispersion. Based on this class of arrays of CPMs, a class of QC-LDPC codes is constructed. For these resulting codes, we analyze their cycle structure and derive the numbers, which indicate how many cycles pass through each CPM in their parity-check matrices. Following the indication numbers with the order from largest to smallest, we replace the corresponding CPMs with zero matrices (ZMs) of the same size one by one until a targeted masking termination condition is satisfied. Then, the resultant arrays of CPMs and ZMs give a large class of QC-LDPC codes. Numerical results show that the codes constructed perform well over the additive white Gaussian noise (AWGN) channel when decoded with the sum-product algorithm.

A simple and efficient search method based on the coset concept is proposed to construct (3, L) quasi-cyclic low-density parity-check (QC-LDPC) codes without small cycles. Compared with the random search method, the chance to find large-girth codes is remarkably increased for the new method. Compared with Tanner's construction, the proposed method is much more flexible in the selection of row weight L. Some new codes are constructed with the shortest lengths compared with those constructed by the existing well-known methods. Simulation results show that the novel codes with moderate length perform well in the additive white Gaussian noise channel.

In this paper, we propose a construction method of quasi-cyclic low-density parity-check (QC-LDPC) codes with multi-weight circulants (MQC-LDPC codes) which is suitable for high-rate LDPC codes with moderate length. MQC-LDPC codes can achieve better degree distributions, Hamming weight distributions, and stopping set weight distributions than conventional QC-LDPC codes with single-weight circulants (SQC-LDPC codes). Within the same framework, we generalize and simplify the error minimization progressive edge growth (EMPEG) algorithm for supporting the structure with multi-weight circulants. Simulation results show that the constructed MQC-LDPC codes offer better performance at both high and low signal to noise ratio (SNR) regions in additive white Gaussian noise (AWGN) channels than the conventional SQC-LDPC codes defined in 802.11ad and 802.16e.

The combination of multiple-input multiple-output (MIMO) and bit-interleaved coded modulation (BICM)/BICM with iterative demapping (BICM-ID) is one of the key technologies to meet the high-transmission-rate requirement of the next generation broadcasting systems. However, conventional low-density parity check (LDPC) coded MIMO BICM schemes always show poor performance with iterative demapping, and vice versa. In this paper, a subset of spatially coupled codes, finite-length spatially coupled (SC-) quasi-cyclic (QC-) LDPC code, is proposed and the universality of spatially coupled codes is utilized to achieve a good tradeoff between MIMO BICM and BICM-ID schemes. Due to the specific code structure of SC-QC-LDPC, the tool of multi-edge type density evolution (MET-DE) is employed for the performance analysis and code optimization. As an example, a finite-length SC-QC-LDPC code is optimized in binary-input additive white Gaussian noise channel, and then bit error rate simulations are carried out to verify the universality of the designed code in MIMO BICM and BICM-ID schemes.

Bit-interleaved turbo-coded modulations (BITCM) have been shown to achieve bit error rate performance very close to the capacity limit over both additive white Gaussian noise (AWGN) channels and Rayleigh fading channels. So far, most studies have been done on BITCM systems designed using rectangular QAM signal sets. In this paper, we address the problem of finding some constellations that would be more suitable than rectangular QAM for designing power efficient BITCM. When the parameter M is in the form M=2/sup 2p/ where p is an integer, we show that rectangular QAM constellations are surprisingly the most attractive signal sets for the design of power-efficient BITCM schemes over AWGN channels. This is not necessarily true with Rayleigh fading channels.

The replicate-and-mask (R&M) construction of finite-length spatially-coupled (SC) LDPC codes is proposed in this paper. The proposed R&M construction generalizes the conventional matrix unwrapping construction and contains it as a special case. The R&M construction of a class of algebraic spatially coupled (SC) quasi-cyclic (QC) LDPC codes over arbitrary finite fields is demonstrated. The girth, rank, and time-varying periodicity of the proposed R&M SC QC LDPC codes are analyzed. The error rate performance of finite-length nonbinary algebraic SC QC LDPC codes is investigated with window decoding. Compared to the conventional unwrapping construction, it is found through numerical simulations that the R&M construction resulted in SC QC LDPC codes with better block error rate performance and lower error floors. With a flooding schedule decoder, it is shown that the proposed R&M algebraic SC QC LDPC codes have better error performance than the corresponding LDPC block codes and random SC codes. The R&M construction of irregular SC QC LDPC codes is demonstrated. It is shown that low-complexity regular puncturing schemes can be deployed on these codes to construct families of rate-compatible irregular SC QC LDPC codes with good performance.

Recently, the differential detection scheme has attracted considerable attention as a demodulated scheme for digital mobile communications because of its robustness in the frequency nonselective fading channel. A problem of the differential detection scheme is that its bit error rate (BER) performance in the additive white Gaussian noise (AWGN) channel is inferior to that of the coherent detection scheme. To solve this problem, many methods to improve BER performance of the differential detection in the AWGN channel have been proposed. However, those methods require complex signal processings.This paper proposes a novel differential detection scheme as a simpler improvement method which estimates the differential phase of the transmitted signal by a sequence estimation scheme using the Viterbi algorithm. The proposed scheme is simplified in a manner such that its signal processing is composed of only additions and subtractions on the phase information of the received signal. Results of computer simulation show that BER performance of the proposed scheme is close to that of the ideal coherent detection in the AWGN channel. In addition, the trellis diagram of the Viterbi algorithm can be reduced without performance degradation introducing the signal processing similar to the Scarece State Transition (SST) type Viterbi decoding. The proposed scheme also is useful in the case of the fast Rician fading channel and the slow Rayleigh fading channel.