Cyclic Low-density Parity-check Research Articles

Huffrith Algorithm with Quasi Cyclic Low Density Parity Check in NOMA Systems

Huffrith Algorithm with Quasi Cyclic Low Density Parity Check in NOMA Systems

A Joint Encryption-Encoding Scheme using QC-LDPC Codes based on Finite Geometry

Joint encryption-encoding schemes have been released to fulfill both reliability and security desires in a single step. Using Low Density Parity Check (LDPC) codes in joint encryption-encoding schemes, as an alternative to classical linear codes, would shorten the key size as well as improving error correction capability. In this article, we present a joint encryption-encoding scheme using Quasi Cyclic-Low Density Parity Check (QC-LDPC) codes based on finite geometry. We observed that our proposed scheme not only outperforms its predecessors in key size and transmission rate, but also remains secure against all known cryptanalyses of code-based secret key cryptosystems. We subsequently show that our scheme benefits from low computational complexity. In our proposed joint encryption-encoding scheme, by taking the advantage of QC-LDPC codes based on finite geometries, the key size decreases to 1/5 of that of the so far best similar system. In addition, using our proposed scheme a wide range of desirable transmission rates are achievable. This variety of codes makes our cryptosystem suitable for a number of different communication and cryptographic standards.

FER Performance Analysis and Optimization of Diagonal Structure Based QC-LDPC Codes with Girth 12 Using LU Decomposition

The manuscript presents the construction of a diagonally structured Parity check matrix using lower and upper (LU) decomposition for regular Quasi Cyclic-Low Density Parity Check Codes (QC-LDPC) with girth 12. This diagonally structured parity check matrix reduces the sparsity characteristics of QC-LDPC codes. The regular Quasi Cyclic-LDPC code has been constructed using circulant shifting and randomization method in which uniform distribution of non-zero elements, XOR operation, and pairing of row and column has been done. A Simplified Log Domain Sum-Product Algorithm (SLDSPA) is used for the decoding of the constructed code. To reduce the CPU run time of the constructed code, a column-wise loop optimization technique is applied. The frame error rate and CPU run time is calculated for different code lengths such as code1 (816, 408), code2 (1008, 504), code3 (1032, 516), code4 (1944, 972), code5 (672, 336) and code6 (504, 252). The construction of regular QC-LDPC codes on half code rate is obtained for the weight of each row is 6 and each column is given by 3. The results showing that the constructed code is better performing in terms of Low FER (10−1 to 10−7), Low SNR (0 to 3.5 dB) and Less CPU run time in milliseconds.

Cyclic Low Density Parity Check Codes With the Optimum Burst Error Correcting Capability

The paper presents a new scheme of cyclic codes suitable for the correction of burst errors. This is accomplished by the proper definition of their parity check polynomials in which the difference between orders of every two consecutive elements of the utilized polynomial is unique and in order of power of 2. In the proposed polynomials, the number of applied elements is much lower than their orders (or codes' lengths). This leads to represent codes as a class of low-density parity check (LDPC) codes, while they do not have any 4 cycle in their Tanner graphs. Considering the properties of the circulant matrix and structure of the defined polynomials, it is proven that codes have the optimum burst error-correcting capability. This is evident for short and long length codes. Moreover, it is shown that constructed codes can be combined with Fire codes and demonstrate cyclic codes that are applicable for the simultaneous correction of random and burst errors.

Half Rate Quasi Cyclic Low Density Parity Check Codes Based on Combinatorial Designs

This paper presents new half rate Quasi Cyclic Low Density Parity Check (QC- LDPC) codes formed on the basis of combinatorial designs. In these codes, circulant matrices of the parity check matrix are formed on the basis of subsets in which the difference between any two elements of a subset is unique with all differences obtained from the same or different subsets. This structure of circulant matrices guarantees non-existence of cycle-4 in the Tanner graph of QC-LDPC codes. First, an irregular code with girth 6 constituted by two rows of circulant matrices is proposed. Then, more criteria will be considered on the structure of subsets with the mentioned feature aiming to represent a new scheme of regular QC-LPDC codes with girth at least 8. From simulations, it is confirmed that codes have similar to or better performance than other well-known half rate codes, while require lower complexity in their design.

Joint LDPC and Physical-Layer Network Coding for Asynchronous Bi-Directional Relaying

For practical bi-directional relaying, symbols transmitted by two sources cannot arrive at the relay with perfect symbol and frame alignments and the asynchronous multiple-access channel (MAC) should be seriously considered. In this paper, we consider the Low-Density Parity-Check (LDPC)-coded BPSK signalling over the general asynchronous MAC with both frame and symbol misalignments. For the symbol-asynchronous MAC, we present a formal log-domain generalized sum-product-algorithm (Log-G-SPA) for efficient decoding. When the frame-asynchronism is encountered at the relay, we propose an original approach by employing the cyclic LDPC codes and the simple cyclic-redundancy-check (CRC) coding technique. Simulation results demonstrate the effectiveness of the proposed approach.

Column-Weight Three QC LDPC Codes with Girth 20

In this paper, some well-structured block designs are presented whose incidence matrices can be considered as the mother matrices of some regular and irregular column-weight three quasi cyclic low-density parity-check codes of girth 20. The class of LDPC codes derived from our construction possesses much more flexibility in terms of code length and rate. Moreover, the length of the regular constructed codes with rate at most 0.5 is at least 4.9 times smaller than the length of the voltage graph LDPC codes with the same rate and girth. Simulation results show that the constructed codes outperform progressive edge growth, repeat accumulate and random LDPC codes.

Cyclic and Quasi-Cyclic LDPC Codes on Constrained Parity-Check Matrices and Their Trapping Sets

This paper is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly low-density parity-check (LDPC) codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths and rates. Some fundamental structural properties of these descendant codes are developed, including the characterization of the roots of the generator polynomial of a cyclic descendant code. The second part of the paper shows that cyclic and quasi-cyclic descendant LDPC codes can be derived from cyclic finite-geometry LDPC codes using the results developed in the first part of the paper. This enlarges the repertoire of cyclic LDPC codes. The third part of the paper analyzes the trapping set structure of regular LDPC codes whose parity-check matrices satisfy a certain constraint on their rows and columns. Several classes of finite-geometry and finite-field cyclic and quasi-cyclic LDPC codes with large minimum distances are shown to have no harmful trapping sets of size smaller than their minimum distances. Consequently, their error-floor performances are dominated by their minimum distances.

Counting Short Cycles of Quasi Cyclic Protograph LDPC Codes

An efficient method for counting short cycles in the Tanner graphs of quasi cyclic (QC) protograph low-density parity-check (LDPC) codes is presented. The method is based on the relationship between the number of short cycles in the graph and the eigenvalues of the directed edge matrix of the graph. We demonstrate that for a QC protograph LDPC code, the complexity of computing such eigenvalues can be reduced significantly by representing the directed edge matrix as a block circulant matrix. Numerical results are presented to show the lower complexity of the proposed method compared to the existing algorithms for counting short cycles. These results also reveal that QC LDPC codes on average have a superior short cycle and girth distribution compared to similar randomly constructed codes.

Nonbinary Cyclic LDPC Codes Derived from Idempotents and Modular Golomb Rulers

Based jointly on idempotents and modular Golomb rulers, we construct a class of nonbinary cyclic low-density parity-check (LDPC) codes. The defining parity-check matrix is a sparse circulant, on which we put two constraints: 1) the characteristic polynomial is an idempotent, 2) the nonzero elements of the first row are located on a modular Golomb ruler. We show that the second constraint forms a necessary and sufficient condition for the Tanner graph to have no cycles of length 4. The minimum distance of the code is proved equal to the column weight of the parity-check matrix plus one. A search algorithm is presented, with which we obtain some high rate codes with large minimum distances. The issue of code equivalence is also discussed. Simulation results show that the obtained codes perform well under iterative decoding.

Memory Access Optimized Implementation of Cyclic and Quasi-Cyclic LDPC Codes on a GPGPU

Software based decoding of low-density parity-check (LDPC) codes frequently takes very long time, thus the general purpose graphics processing units (GPGPUs) that support massively parallel process...

Burst Decoding of Cyclic Codes Based on Circulant Parity-Check Matrices

An error-burst correcting algorithm is developed based on a circulant parity-check matrix of a cyclic code. The proposed algorithm is more efficient than error trapping if the code rate is less than about 2/3. It is shown that for any (n, k) cyclic code, there is an n × n circulant parity-check matrix such that the algorithm, applied to this matrix, corrects error bursts of lengths up to the error-burst correction limit of the cyclic code. This same matrix can be used to efficiently correct erasure bursts of lengths up to n - k. The error-burst correction capabilities of a class of cyclic low-density parity-check (LDPC) codes constructed from finite geometries are also considered.

Lowering Error Floor of Irregular LDPC Codes by CRC and OSD Algorithm

Irregular Low-Density Parity-Check (LDPC) codes generally achieve better performance than regular LDPC codes at low E b /N 0 values. They have, however, higher error floors than regular LDPC codes. With respect to the construction of the irregular LDPC code, it can achieve the trade-off between the performance degradation of low E b /N 0 region and lowering error floor. It is known that a decoding algorithm can achieve very good performance if it combines the Ordered Statistic Decoding (OSD) algorithm and the Log Likelihood Ratio-Belief Propagation (LLR-BP) decoding algorithm. Unfortunately, all the codewords obtained by the OSD algorithm satisfy the parity check equation of the LDPC code. While we can not use the parity check equation of the LDPC code to stop the decoding process, the wrong codeword that satisfies the parity check equation raises the error floor. Once a codeword that satisfies the parity check equation is generated by the LLR-BP decoding algorithm, we regard that codeword as the final estimate and halt decoding; the OSD algorithm is not performed. In this paper, we propose a new encoding/decoding scheme to lower the error floor created by irregular LDPC codes. The proposed encoding scheme encodes information bits by Cyclic Redundancy Check (CRC) and LDPC code. The proposed decoding scheme, which consists of the LLR-BP decoding, CRC check, and OSD decoding, detects errors in the codewords obtained by the LLR-BP decoding algorithm and the OSD decoding algorithm using the parity check equations of LDPC codes and CRC. Computer simulations show that the proposed encoding/decoding scheme can lower the error floor of irregular LDPC codes.