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Improved quasi-cyclic LDPC codes based on euclidean geometry

Yuanhua Liu, Zhang Mei-ling

Published Jan 1, 2015• 1 citation
Publisher

Abstract

This paper presents an improved Euclidean geometry (EG) method for constructing parity check matrix of quasi-cyclic low-density parity-check (QC-LDPC) codes to decrease the performance degradation of short cycles to the iterative decoding. The new method results in a class of QC-LDPC codes with much fewer short cycles than usual EG-LDPC codes with the same parameters. Simulation results show that the designed QC-LDPC codes with fewer short cycles perform better than the existing Euclidean geometry LDPC codes under iterative decoding.

DOI
10.1049/cp.2015.0209

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This paper presents an improved Euclidean geometry (EG) method for constructing parity check matrix of quasi-cyclic low-density parity-check (QC-LDPC) codes to decrease the performance degradation of short cycles to the iterative decoding. The new method results in a class of QC-LDPC codes with much fewer short cycles than usual EG-LDPC codes with the same parameters. Simulation results show that the designed QC-LDPC codes with fewer short cycles perform better than the existing Euclidean geometry LDPC codes under iterative decoding.
A new method for constructing quasi-cyclic (QC) low-density parity-check (LDPC) codes based on the structural properties of Euclidean geometries is presented. The proposed method changes the incidence vectors of some lines in the original Euclidean geometry method to construct much more parity check matrices and results in a large class of QC-LDPC codes with girth of at least 6. Simulations show that some codes in this class have better performance and lower error floor than the original Euclidean geometry codes.
The existing design of quasi-cyclic low-density parity-check(short for QC-LDPC) codes has not considered the problems of small stopping sets and small cycles together,while their existences will lead to the BER performance of QC-LDPC codes to be much poorer than that of the randomly constructed LDPC codes even decoding failure.To solve the problem,some theorems of the QC-LDPC codes without small stopping sets and small cycles were proposed which provide a theoretical basis for the construction of parity-check matrix by analysis method;meanwhile,good QC-LDPC codes without small stopping sets and small cycles were designed.Since preventing the small stopping sets and small cycles also prevents a low minimum distance,our construction method also leads to QC-LDPC codes with a higher minimum distance.The simulation results show that the QC-LDPC codes can effectively avoid specified stopping sets and small cycles and achieve good coding and decoding property due to the proposed constructions.
Focused on the problem that short cycles degrade the iterative decoding performance of finite length LDPC, a class of improved low density parity check (LDPC) codes constructed based on Gallager's form is designed. On the basis of keeping down the basic form of Gallager's construction, a quasi-cyclic form is used, then the new codes are free of cycles of length 4 and have good minimum distance. Simulation results demonstrated that the improved codes are better than Gallager's codes when block length is small
Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes have practical significance in the field of engineering, thus their construction algorithm is a hot issue in the field of LDPC codes. Based on existing construction methods, particularly the method based on Progressive Edge-Growth (PEG) algorithm, a new construction method of shift matrix is proposed. It can effectively avoid the copy of cycle structures, especially short cycles in base matrix after extension. Besides, it also relieves the time cost caused by random search, and greatly improves the length of cycles when compared with algorithms based on methods such as quadratic congruence and arithmetic sequence. Those algorithms can only eliminate short cycles with length 4, while our method can provide QC-LDPC codes without length 6 at least. Cycle analysis and simulation results show that the newly constructed QC-LDPC codes have good cycle structure and error correction performance.
Short cycles in the Tanner graph of a given LDPC code has negative effect on the performance of iterative decoding. To improve the performance, we presented an approach for constructing Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes whose associated Tanner graph has large girth and a small amount of short cycles. Each column block of the parity check matrix was designed by adding circulant permutation sub-matrices one after another, provided the current parity check matrix has minimal number of short cycles. The condition for the proposed QC-LDPC codes to have girth at least 6 was also analyzed. Regular and irregular QC-LDPC codes with various lengths and rates can be constructed. Simulations show that the proposed QC-LDPC codes have less short cycles and better performance than existing progressive edge growth based QC-LDPC codes.