Circulant Permutation Matrix Research Articles

Constructing Grith Eight GC-LDPC Codes Based on the GCD FLRM Matrix with a New Lower Bound.

By connecting multiple short, local low-density parity-check (LDPC) codes with a global parity check, the globally coupled (GC) LDPC code can attain high performances with low complexities. The typical design of a local code is a quasi-cyclic (QC) LDPC for which the code length is proportional to the size of circulant permutation matrix (CPM). The greatest common divisor (GCD)-based full-length row multiplier (FLRM) matrix is constrained by a lower bound of CPM size to avoid six length cycles. In this paper, we find a new lower bound for the CPM size and propose an algorithm to determine the minimum CPM size and the corresponding FLRM matrix. Based on the new lower bound, two methods are proposed to construct the GC-QC-LDPC code of grith 8 based on the GCD based FLRM matrix. With the proposed algorithm, the CPM size can be 45% less than that given by sufficient condition of girth 8. Compared with the conventional GC-LDPC construction, the codes constructed with the proposed method have improved performance and are more flexible in code length and code rate design.

On the search of smallest QC-LDPC code with girth six and eight

In this paper, a new and simple method for the construction of Girth-6 (J,L) Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes is proposed. The method is further extended to the search of Girth-8 QC-LDPC codes with base matrices of order 3 × L and 4 × L. The construction is based on three different forms of exponent matrices and the parameters α, p, and q which satisfy the necessary algebraic conditions for a QC-LDPC code having girth 6 and 8. The proposed (J,L) QC-LDPC codes with girth at least six have optimal circulant permutation matrix (CPM) size for the cases where q = α + 1. Moreover, most of the girth-8 QC-LDPC codes searched by the proposed method have smaller CPM size than the existing codes of the same girth. In several cases, the method gives more than one exponent matrices for a code, as most of the existing methods cannot do so. Besides this, the proposed method not only search the QC-LDPC codes with smaller CPM size but also takes much less time than the existing search based methods to search code.

Construction of girth-8 (3,L)-QC-LDPC codes of smallest CPM size using column multipliers

In this paper, a new method for the construction of the exponent matrix of quasi-cyclic low-density parity-check (QC-LDPC) codes is proposed. The entries of the exponent matrix are based on the column multipliers. To find the column multipliers, a parameter $$ {\varvec{S}_{\varvec{\alpha}}} $$ is defined which gives the value of column multiplier of the $$ {\varvec{\alpha}} $$th column. The proposed method reduced the complexity related to the formation of the exponent matrix and results in (3,L)-QC-LDPC codes with girth at least eight, for $$ {\varvec{L} > 3} $$. Also, a lower bound on the size of the circulant permutation matrix (CPM) for a QC-LDPC code is derived, and the codes constructed by this method are optimal to the given bound. Further, most of the codes constructed using this method are of smaller CPM size. Specifically, for $$ {\varvec{L} > 25} $$, our constructed QC-LDPC codes have the shortest CPM size compared to the existing ones in the literature.

Design of Quantum LDPC Codes From Quadratic Residue Sets

We design classes of quantum low-density parity-check (LDPC) codes, called quasi-cyclic stabilizer (QCS) codes, from conventional QC-LDPC codes. The proposed QCS codes belong to the family of non-Calderbank–Shor–Steane stabilizer codes. The QC-LDPC codes are self-orthogonal with respect to the symplectic inner product (SIP) and are constructed from submatrices of nonorthogonal Latin squares via array dispersion. The Latin squares are constructed using quadratic (non)-residue sets of prime modulus $p$ , where $p = 4n\pm 1$ . For $p = 4n-1$ , two constructions, namely, Type-I-A and Type-I-B QCS codes, are proposed based on matrix superposition and matrix concatenation, respectively. For $p = 4n+1$ , Type-II QCS codes are proposed based on permutations of a base matrix. We show that the parity-check matrix for Type-I-B and Type-II QCS codes is self-orthogonal with respect to the SIP for all orders of circulant permutation matrix. This resulting in ensembles of QCS codes characterized by a single base matrix. We show that the minimum distance of Type-II QCS codes can be lower bounded by the minimum distance of the QC-LDPC codes. Simulation results show that the proposed QCS codes outperform some codes in the literature with a noteworthy low-error floor, below 10−7, over quantum depolarizing channels.

Construction of QC-LDPC Codes Based on Pre-Masking and Local Optimal Searching

In this letter, we consider the construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes by jointly optimizing the girth ${g}$ and the number of ${g}$ -cycles. We first propose a sub-array detection method to accelerate the detection of cycles, and by detecting the non-repeated automatic enumerative cycles in the frame of sub-arrays, cycle detection time can be greatly reduced. Furthermore, a local optimal searching algorithm combined with pre-masking is presented, by orderly searching the local optimal ${p}_{j,l}$ for each circulant permutation matrix (CPM), a class of QC-LDPC codes with possibly enlarged ${g}$ , less number of ${g}$ -cycles, and arbitrary attainable degree distribution could be constructed. Simulation results show that the codes constructed by the proposed method perform well in error floor region.

High Rate APPS and Bi-APPS LDPC Codes Design with Low Error Floor and Large Girth

In this paper, two new methods to construct low-density parity-check (LDPC) codes with low error floor and large girth are proposed. The first one is APPS-LDPC codes based on Arithmetic Progression theory and cycle classification, whose girth is at least eight. Based on the designed APPS-LDPC codes, we further construct Bi-diagonal APPS-LDPC codes with column degree 4, whose circulant permutation matrix is combined by two shifted identity matrix. The designed APPS-LDPC code has 0.25 and 0.2 dB coding gain compared to partition-and-shift (PS)-LDPC code and progressive-edge-growth (PEG)-LDPC code. And the Bi-APPS-LDPC code has similar performance to T2 LDPC code in CCSDS standard, but its effective structure is more suitable for high throughput decoder implementation on FPGA. Both codes have less construction complexity than PS-LDPC code and PEG-LDPC code.

An Approach to Evaluating the Number of Closed Paths in an All-One Base Matrix

Given an all-one base matrix of size $M \times N$ , a closed path of different lengths can be formed by starting at an arbitrary element and moving horizontally and vertically alternatively before terminating at the same “starting” element. When the closed-path length is small, say 4 or 6, the total number of combinations can be evaluated easily. When the length increases, the computation becomes non-trivial. In this paper, a novel method is proposed to evaluate the number of closed paths of different lengths in an all-one base matrix. Theoretical results up to closed paths of length 10 have been derived and are verified by the exhaustive search method. Based on the theoretical work, results for closed paths of length larger than 10 can be further derived. Note that each of such closed paths may give rise to one or more cycles in a low-density parity-check (LDPC) code when the LDPC code is constructed by replacing each “1” in the base matrix with a circulant permutation matrix or a random permutation matrix. Since LPDC codes with short cycles are known to give unsatisfactory error correction capability, the results in this paper can be used to estimate the amount of effort required to evaluate the number of potential cycles of an LDPC code or to optimize the code.

An explicit method for construction of CTBC codes with girth 6

A cylinder-type block-circulant (CTBC) code with circulant permutation matrix (CPM) size p is presented by a binary base matrix with a cylindrical structure and a slope-vector on Zp. The slope-vector is called proper if the corresponding CTBC code is 4-cycle free. In this paper, an algorithm is presented that for a given base matrix with a cylindrical structure, explicitly generates a proper slope-vector. This proper slope-vector provides a lower-bound on the CPM-sizes of the corresponding CTBC codes required to achieve girth at least 6. Interestingly, the lower bound, minimum distance and 6-cycle multiplicity of the constructed CTBC codes are better than some masked and unmasked known explicitly constructed 4-cycle free QC LDPC codes with the same regularity. Simulation results show that the constructed CTBC codes outperform some 4-cycle free QC LDPC codes in binary and nonbinary cases.

Recursive construction of $(J,L)$ QC LDPC codes with girth 6

‎In this paper‎, ‎a recursive algorithm is presented to generate some exponent matrices which correspond to Tanner graphs with girth at least 6‎. ‎For a $J times L$ exponent matrix $E$‎, ‎the lower bound $Q(E)$ is obtained explicitly such that $(J,L)$ QC LDPC codes with girth at least 6 exist for any circulant permutation matrix (CPM) size $m geq Q(E)$‎. ‎The results show that the exponent matrices constructed with our recursive algorithm have smaller lower-bound than the ones proposed recently with girth 6‎.

Type‐II quasi‐cyclic low‐density parity‐check codes from Sidon sequences

From Sidon sequences, two explicit methods for constructing (2J, 2L)-regular type-II quasi-cyclic (QC) low-density parity-check (LDPC) codes are proposed without four cycles. The advantage of the new methods is two-fold: (i) compared with the existing lower bound of circulant permutation matrix (CPM) sizes above which type-II QC-LDPC codes exist without four cycles, the novel constructions enable much tighter lower bounds; and (ii) the codes constructed by the first method perform almost as well as type-II QC-LDPC codes from perfect cyclic difference set, while possessing a much more flexible CPM sizes.

A Novel Construction Approach of Irregular LDPC Codes Based on QC Structure and Zigzag Pattern

This paper presents a novel construction method of irregular Low-density parity-check (LDPC)codes based on Quasi-cyclic (QC) structure and zigzag pattern. By using the proposed method, a class of irregularand highly structured LDPC codes can be designed withthe advantages of low storage requirement and linear timeencoding complexity. The constructed codes are called Irregular repeat-accumulate like (IRA-like) codes since theirparity-check matrices are similar with those of IRA codes,which all contain a sparse zigzag pattern submatrix. Theleft part of the parity-check matrix of IRA-like codes is akind of circulant permutation matrix. A best-effort analyzing method for optimizing the cycle structure of IRAlike codes is presented. We further details the proper constraints for avoiding short cycles and low-weight codewords. Simulation results show that the proposed IRA-likecodes have low encoding complexity, good iterative decoding performance and flexible choice of code parameters.

Construction of girth‐eight QC‐LDPC codes from arithmetic progression sequence with large column weight

For any row weight L, a novel class of (8, L) quasi-cyclic low-density parity-check (QC-LDPC) codes is deterministically constructed with girth eight. Compared with a similar construction, the proposed (7, L) and (8, L) QC-LDPC codes exist for any circulant permutation matrix of size P ≥ (L − 1)(L 2 + 3L/4 + 3/4) + 1 and P ≥ (L − 1)(L 2 + 3L/4 + 7/4) +1, which remarkably improve the existing bounds of (L − 1)(L 2 + L) +1 and (L − 1)(L 2 + L + 1) +1, respectively. Moreover, for any column weight J, any row weight L and any common difference, a construction for (J, L) QC-LDPC codes with girth eight is also proposed. This construction has high flexibility with respect to the design of code rate and code length. Simulation results show that the novel codes with moderate lengths perform well in the additive white Gaussian noise channel.

Deterministic Girth-Eight QC-LDPC Codes with Large Column Weight

For any row weight L, several novel classes of (J = 5, L) and (J = 6, L) quasi-cyclic LDPC codes are deterministically constructed with girth eight. From these results, it is proved that (5, L) QC-LDPC codes with girth eight exist for any circulant permutation matrix (CPM) size P ≥ (2L + 3)(L - 1) + 1, and that girth-eight (6, L) QC-LDPC codes exist for any P ≥ 2(L+5)(L-1)+1. The two novel bounds remarkably improve the existing bounds of L^2(L-1) + 1 and (L^2+1)(L-1)+1, respectively. Moreover, for any column weight J and any row weight L, a construction for (J,L) QC-LDPC codes with girth eight is also proposed. This is the first deterministic and systematic construction which can generate girth-eight QC-LDPC codes with J ≥ 7.

Construction of Irregular QC-LDPC Codes in Near-Earth Communications

High-speed, high-reliability requirements and strict power limitation of near-earth communications yield that the channel coding must have high coding gain and lower error floor. In this paper, an irregular quasi-cyclic low-density parity- check (QC-LDPC) code for specific use in near-earth communications is proposed. In order to eliminate the impact of short-cycles and further improve the performance of codes, a base matrix with girth at least 8 is employed. Then dual- diagonal structure is adopted in parity check matrix to reduce encoding complexity. Besides, circulant permutation matrices and zero matrices with different sizes are used to fill the nonzero positions and zero positions of the base matrix so as to remove the complexity of the selection of cyclic shift coefficients. The adoption of circulant permutation matrices with different sizes ensures that the parity check matrix is full rank, which makes the code rate controllable. Finally, simulation results are given to demonstrate the performance of the constructed codes by comparing with several existing codes. Index Terms—Low-density parity-check codes, parity check matrix, circulant permutation matrix, near-earth communications

Construction of quasi-cyclic low-density parity-check codes for simplifying shuffle networks in layered decoder

Offset Shuffle Networks (OSNs) interleave a-posterior probability messages in the Block Row-Layered Decoder (BRLD) of Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes. However, OSNs usually consume a significant amount of computational resources and limit the clock frequency, particularly when the size of the Circulant Permutation Matrix (CPM) is large. To simplify the architecture of the OSN, we propose a Simplified Offset Shuffle Network Block Progressive Edge-Growth (SOSN-BPEG) algorithm to construct a class of QC-LDPC codes. The SOSN-BPEG algorithm constrains the shift values of CPMs and the difference of the shift values in the same column by progressively appending check nodes. Simulation results indicate that the error performance of the SOSN-BPEG codes is the same as that of the codes in WiMAX and DVB-S2. The SOSN-BPEG codes can reduce the complexity of the OSNs by up to 54.3%, and can improve the maximum frequency by up to 21.7% for various code lengths and rates.

Several Explicit Constructions for (3,L) QC-LDPC Codes with Girth at Least Eight

Some explicit constructions for exponent matrices which correspond to Tanner graphs with girth at least eight are proposed. From these results, for any row weight of L>3, (3,L) quasi-cyclic LDPC codes with girth at least eight are constructed for any circulant permutation matrix (CPM) size P>L(L+mod(L,2))/2.

Deterministic construction of girth‐eight (3, L ) QC‐LDPC codes from quadratic function

For any row weight L and any circulant permutation matrix size P > (L − 1)2 + b 0(L − 1) (b 0 = L − 2 + mod(L,2)), a class of (3,L) quasi-cyclic (QC) low-density parity-check (LDPC) codes is explicitly constructed with girth eight, based on the quadratic function f(x) = x 2 + bx and its derivative, where b is either of the two integers {b 0, − b 0 − 2(L − 2)}. Compared with a similar construction from the so-called difference sequence, the proposed construction is not only much more simple in concept, but also completely deterministic in the sense that no computer search or computing is required. Simulation results show that the new codes significantly outperform the girth-eight codes constructed by the earliest-sequence-based method.

Fiber-based optical parametric amplif ier for 40-Gb/s NRZ-DPSK signal transmission system employing QC-LDPC codes

In this letter, we investigate quasi-cyclic low-density parity-check (QC-LDPC) codes in a 40-Gb/s nonreturn-to-zero differential phase-shift keying (NRZ-DPSK) signal transmission system based on a fiber-based optical parametric amplifier (FOPA). A constructed algorithm of QC-LDPC codes according to the optimizing set of shift values on the circulant permutation matrix (CPM) of the basis matrix is proposed. Simulation results prove that the coding gain in the encoded system can be realized at 10.2 dB under QC-LDPC codes with a code rate of 5/6 when the bit error rate (BER) is 10-9. In addition, the error-floor level originating from the uncoded system is suppressed.

A Design of ALT LDPC Codes Using Circulant Permutation Matrices

In this paper, we propose a simple H parity check matrix from the CPM(circulant permutation matrix), which can easily avoid the cycle-4, and approach to flexible code rates and lengths. As a result, the operations of the submatrices will become the multiplications between several CPMs, the calculations of the LDPC(low density parity check) encoding could be simplest. Also we consider the fast encoding problem for LDPC codes. The proposed constructions could lead to fast encoding based on the simplest matrices operations for both regular and irregular LDPC codes.

Low complexity construction for quasi-cyclic low-density parity-check codes by Progressive-Block Growth

In this paper, a novel method for constructing quasi-cyclic LDPC codes with low complexity is proposed. By choosing each circulant permutation matrix sequentially, the proposed method ensures that the current circulant permutation matrix forms no cycles of length smaller than g with the existent circulant permutation matrices. The construction complexity of the proposed algorithm is much lower than that of the random construction. Simulation results show that the proposed QC-LDPC codes can outperform both the random regular LDPC codes with short to moderate block lengths, and the proposed construction method LDPC codes based on the finite geometries in terms of bit-error-rate (BER).