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Construction of Low-density Parity-check Codes Based on Cyclic Difference Sets

Guo Zhi-rong

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

This paper proposes a method to construct Quasi-Cyclic Low-Density Parity-Check(QC-LDPC) codes based on Cyclic Difference Sets(CDS) of combinatory. The constructed regular QC-LDPC codes are free of 4-cycles and with low complexity shift register encoding. Simulations results show that the constructed LDPC codes perform well under sum-product decoding algorithm.

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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.

Journal of Beijing Jiaotong University•Jan 1, 2010•0 citations
This paper proposes a method to construct Quasi-Cyclic Low-Density Parity-Check(QC-LDPC) codes based on Cyclic Difference Sets(CDS) of combinatory. The constructed regular QC-LDPC codes are free of 4-cycles and with low complexity shift register encoding. Simulations results show that the constructed LDPC codes perform well under sum-product decoding algorithm.
Difference sets are used to give a novel method for constructing structured regular quasi-cyclic low-density parity-check codes. Let Fqbe a field with q elements and assume that D={d1,..., dk} is a (v,k,1)-difference set for Zvwith d12k. Depending on v=q-1, v≥ 2dkor v≥ 2dk-1, three code construction methods are given that produce regular 4-cycle free codes, that is codes with girth at least 6. Simulation results show that the constructed codes perform well over the AWGN channel with an iterative sum-product message-passing decoding algorithm.
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.
In order to solve the problems of high coding complexity and inflexible code length selection in quasi-cyclic low-density Parity-check (QC-LDPC) codes, a deterministic structure is proposed based on the perfect cyclic difference sets. The number of shifts of base matrix can be obtained by a simple addition and subtraction. Because of the special structure of the base matrix and the perfect cyclic difference sets, it saves the storage space and reduces the complexity of the hardware implementation. The girth is at least six and the code-length and code-rate can be selected flexibly. Simulation results demonstrate that the Net Coding Gain of the irregular PCDS-QC-LDPC(2680, 1340) code based on perfect cyclic difference sets has respectively improved 0.13dB and 0.32dB than the code based on PEG-QC-LDPC(2680,1340) and array dispersion and masking AD-MASK-QC-LDPC(2680,1340) with the rate of 0.5 and the bit error rate of 10 -6 by using the sum-product algorithm iterative decoding in the additive white Gaussian channel.
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.