A novel QC-LDPC code based on the finite field multiplicative group for optical communications

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.

DOI: 10.1049/cp:20061337 ISBN: 0 86341 644 6 Location: Hangzhou, China Conference date: 6-9 Nov. 2006 Format: PDF The BER performance of presented quasi-cyclic (QC) low-density parity-check (LDPC) codes is not good as that of randomly constructed LDPC codes, while the lack of structure of randomly constructed LDPC codes implies serious disadvantages in terms of storing and accessing a large parity-check matrix. To solve the problems, this paper proposes a design of good quasi-cyclic (QC) low-density parity-check (LDPC) codes, the obtained algebraically structured codes have large minimum distances and good BER performance. The proposed design is based on index matrices, which determines the shifts of the circulant matrices of the sparse parity-check matrices. Compared with the presented design of QC LDPC codes, the designed QC LDPC codes are free from girth 4, sometimes free from girth 6, and have much larger minimum distances. Based on the algebraic code structure, the conditions of the girth and minimum distance of the codes are found. The BER performance of the designed QC LDPC block codes compares with that of randomly constructed LDPC codes for any block lengths. Better BER performance is obtained by increasing the circulant size of the base QC code. Simulations verify the construction of the QC LDPC codes to be valid. (4 pages) Inspec keywords: AWGN channels; error statistics; channel coding; algebraic codes; parity check codes; sparse matrices Subjects: Algebra; Other topics in statistics; Codes

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.

The generator matrix of quasi-cyclic low-density parity-check (QC-LDPC) codes is not, in general, in systematic form. Therefore, finding the message corresponding to a detected codeword at the receiver is not as straightforward as it is in the case of systematic codes. In this paper, we propose two methods for finding the message from the codeword in a QC-LDPC code. The first method is a general scheme which views the problem as a set of linear equations. This method is not realizable with pure digital circuits and may be implemented by having a processor core in the design. The second method is rather an alternative scheme which makes use of a particular structure in generator matrices of QC-LDPC codes in order to find the message from a non-systematic codeword. We examine a large class of non-systematic Latin squares QC-LDPC codes and one of the IEEE 802.16e standard QC-LDPC codes, and show that they have the required structure to be usable by the algorithm. The latter method is realizable in hardware and can be implemented with a digital circuit consisting of XOR gates.

The short paper proposes a method for constructing regular quasi-cyclic (QC) LDPC codes based on circulant permutation matrices via simple prime field operations. The main advantage is that regular QC LDPC codes with a variety of block lengths and rates can be easily constructed which have no cycles of length four or less. Simulation results show that within only a maximum of ten decoding iterations of sum-product algorithm(SPA) the constructed regular codes of high rates have no error floor down to the bit-error rate of $10^{-7}$.

There exist lots of (3, L)-regular quasi-cyclic (QC) LDPC codes constructed from finite fields, protographs, array codes, and computer search under some design rules. For a given code length and rate, how to select the best one from these codes is considerable. In this paper, we study (3, L)-regular QC LDPC codes from the perspective of graph isomorphism, and non-isomorphic (3, L)-regular QC LDPC codes are determined. By analyzing the cycle structures of the resulting non-isomorphic codes, an efficient algorithm for counting cycles is presented. Also proposed is a simplified exhaustive search of non-isomorphic (3, L)-regular QC LDPC codes free of cycles of length less than g <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> , where g <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> is the estimated optimal girth value for a given code length. Based on these two algorithms, we can easily construct a (3, L)-regular QC LDPC code with optimized cycle distribution for a given L and code length. Numerical results show that the constructed codes have better performance under the iterative decoding algorithms.

SUMMARYLow encoding complexity is very important for quasi‐cyclic low‐density parity‐check (QC‐LDPC) codes used in wireless communication systems. In this paper, a new scheme is presented to construct QC‐LDPC codes with low encoding complexity. This scheme is called two‐stage particle swarm optimization (TS‐PSO) algorithm, in which both the threshold and girth distribution of QC‐LDPC codes are considered. The proposed scheme is composed of two stages. In the first stage, we construct a binary base matrix of QC‐LDPC code with the best threshold. The matrix is constructed by combining a binary PSO algorithm and the protograph extrinsic information transfer (PEXIT) method. In the second stage, we search an exponent matrix of the QC‐LDPC code with the best girth distribution. This exponent matrix is based on the base matrix obtained in the first stage. Consequently, the parity‐check matrix of the QC‐LDPC code with the best threshold and best girth distribution are constructed. Furthermore, bit error rate performances are compared for the QC‐LDPC codes constructed by proposed scheme, the QC‐LDPC code in 802.16e standard, and the QC‐LDPC code in Tam's study. Simulation results show that the QC‐LDPC codes proposed in this study are superior to both the 802.16e code and the Tam code on the additive white Gaussian noise (AWGN) and Rayleigh channels. Moreover, proposed scheme is easily implemented, and is flexible and effective for constructing QC‐LDPC codes with low encoding complexity. Copyright © 2012 John Wiley &amp; Sons, Ltd.

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 method of constructing quasi-cyclic low-density parity-check (QC-LDPC) codes of large length by combining QC-LDPC codes of small length as their component codes, via the Chinese Remainder Theorem. The girth of the QC-LDPC codes obtained by the proposed method is always larger than or equal to that of each component code. By applying the method to array codes, we present a family of high-rate regular QC-LDPC codes with no 4-cycles. Simulation results show that they have almost the same performance as random regular LDPC codes.

Array based quasi-cyclic LDPC codes and their tight lower bounds on the lifting degree

Quasi-cyclic LDPC (QC-LDPC) codes have been accepted as the standard codes of 5G enhanced mobile broadband data channel. These standard codes are designed to support multiple lifting sizes and possess rate-compatible property, which can help adapt various information lengths and code rates well. In this paper, we propose an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">algebra-assisted</i> method for constructing QC-LDPC codes with such properties. We will first review the encoding mechanism and requirements of 5G LDPC codes, and present <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">cycle analysis</i> for such emerging codes. We then propose a metric, referred to as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">weighted average number of cycles</i> (WANC), from the perspective of cycle structure for constructing the QC-LDPC codes that can support <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">multiple lifting sizes</i> . Based on the WANC metric and algebraic methods, we develop a simple and practical algorithm to construct this kind of QC-LDPC codes. We finally apply the proposed algorithm to construct the exponent matrices for cases of 5G LDPC codes and the standard LDPC codes of consultative committee for space data systems, respectively. Simulation results show that the proposed WANC metric and designed algorithm are feasible and effective, and thus can be utilized to design other similar QC-LDPC codes.

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

Recently, Low Density Parity-Check (LDPC) codes based on Affine Permutation Matrices (APM) drew lots of attention. Compared with the Quasi-Cyclic LDPC (QC-LDPC) codes, these kinds of codes have some advantages. APM-LDPC codes obtain lower cycle-distributions, minimum hamming distance and greater girth. This paper explains the importance of cyclic distribution by comparing APM-LDPC codes with QC-LDPC codes. Then a particular form of APM-LDPC codes is proposed and researched. The new codes can low down the cycle-distribution to larger extent. In the following research, an effective method, which constructs the new codes with fixed girth, is proposed. Simulations show that the construction method is reasonable and effective. The transmission performances are better than the traditional methods, as well. Finally, the implementation and verification are carried out on FPGA.

In this paper, we exploit construction of Reed-Solomon(RS) based Quasi-Cyclic Low-Density Parity-Check(QC-LDPC) codes using protograph combining two existing QC-LDPC codes construction and experiment the performance of the codes. Two constructions have benefits: One is the construction RS based QC-LDPC codes whose girth is at least 8 and another is protograph based QC-LDPC codes to increase the upper bounds of minimum Hamming distance. We construct the protographs with three strategies and two of them have coding gain in sense of bit error rate compare to existing Reed Solomon based QC-LDPC codes.

In this paper, we study no 4-cycle, high-rate LDPC codes based on finite geometries for use in data storage devices and prove that these codes cannot be classified as quasi-cyclic (QC) codes but should be considered as broader generalized quasi-cyclic (GQC) codes. Because of the GQC structure of such codes, they can be systematically encoded using Groebner bases and their encoder can be implemented using simple feedback-shift registers. In order to demonstrate the efficiency of the encoder, we show that the hardware complexity of the serial-in serial-out encoder architecture of these codes is of linear order O(n). To encode a binary codeword of length n, less than 2n adders and 3n memory elements are required. Furthermore, we evaluated the error performances of these codes with sum product algorithm (SPA) decoding over additive white Gaussian noise (AWGN) channels. At a bit error rate (BER) of 10^-5, they perform 1-dB away from the Shannon limit after 10 decoding iterations.