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.
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.
This paper proposes a coset-based Quasi-Cyclic Low Density Parity Check(QC LDPC) construction method and similar SFT-LDPC code construction method in order to guarantee the girth under high bitrate.It gives two necessary conditions of girth at least 6 when the size of cyclic permutation matrix is a composite number.This construction method with various lengths,rates and girth at least 6 can be flexiblely constructed.Simulation results show that the code constructed by this method is comparable to random ones in error correction performance over additive white Gaussian noise channel.
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 + 3 L /4 + 3/4) + 1 and P ≥ ( L − 1)( L 2 + 3 L /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.
In this paper, we proposed a novel method for constructing quasi-cyclic low-density parity-check (QC LDPC) codes based on cyclic subgroup over GF(q) . LDPC codes constructed by this method are quasi-cyclic (QC) and they perform very well over the additive white Gaussian noise (AWGN) with sum product algorithm (SPA). The main advantage is that QC LDPC codes with a variety of block lengths and rates can be easily constructed with no cycles of length four or less. Simulation results show that the proposed QC LDPC codes perform slight better than the random regular LDPC codes for short to moderate block lengths. Since the codes are QC, they can be encoded using simple shift registers with linear complexity.
This study considers the construction of four‐cycle free quasi‐cyclic low‐density parity‐check (QC‐LDPC) codes. These codes are based on the cyclotomic cosets of q modulo n where q is a prime power, n is a prime and ( n , q ) = 1. If s is the order of q modulo n , then n − 1 = sl for some integer l . Then there are l distinct cyclotomic cosets , 1 ≤ i ≤ l . Concatenation of the l circulant matrices formed using these sets gives an s × sl matrix M co ( n , q ). In addition, the Kronecker product of the transpose of ∪ i with itself modulo n gives an sl × sl matrix M kr ( n , q ). Replacing the entries of these matrices with their associated dispersed binary n × n matrices provides binary QC‐LDPC codes with girth at least six. Furthermore, for any prime power q ′ such that q ′ = n + 1 or q ′ ≥ 2 n , these two matrices can be used to construct q ′‐ary QC‐LDPC codes with girth at least six. These constructions produce large classes of four‐cycle free binary and non‐binary QC‐LDPC codes. Among the structured LDPC codes, the introduced technique, from code‐construction‐complexity perspective, which is an engineering factor, is at least among the best ones if not the best. Performance results are presented which show that the codes obtained perform well over additive white Gaussian noise channel with the iterative sum‐product decoding algorithm.
We propose a new algorithm based on PEG algorithm to construct high-girth quasi-cyclic low-density parity check (QC-LDPC) codes. Before our improved algorithm in introduction, we depict the general algorithm based on PEG algorithm to construct QC - LDPC code, if necessary to eliminate the short cycles, that would bring in cycles-search. When code length of basic matrix is not too long, the algorithm is feasible, but with the code length growth, cycles-search will bring large of calculation and time-consuming sharply. Therefore we put forward an improved method by using the specific relation of basic matrix rows to replace traditional cycles-search, effectively eliminates short cycles, and largely reduces the time complexity at the same time. Through the simulation analysis, the performance can also be comparable with random constructed LDPC codes.
In this paper, we propose an effective and efficient searching method for constructing quasi-cyclic low-density parity-check (QC-LDPC) codes with a desired girth g. We begin with an arbitrary QC-LDPC code with girth-4 and we evaluate only the number of cycles with length 4. When all the cycles with length 4 are removed by adjusting the elements of the QC-LDPC code, we form a QC-LDPC code with girth-6. Subsequently, we consider only the numbers of cycles with length 4 and length 6. In general, knowing that the current QC-LDPC code has a girth of g', we only consider the numbers of cycles with length up to g' even though g' may be smaller than the desired girth g. By using an adaptive cost function, which is defined as the number of cycles of length g', in the optimization/searching process, we are able to reduce the computational effort tremendously compared with Wang's searching algorithm [1]. Consequently, our proposed method can generate QC-LDPC codes with the desired girth much more efficiently.