Research On Construction And Theoretical Analysis Of QC-LDPC Codes

QC-LDPC code, which is a type of the structured LDPC code, is widely applied in several communication systems. QC-LDPC code can be efficiently encoded using simple shift registers with linear complexity, which is mainly owing to the unique structure of the parity-check matrix. This dissertation studies the construction and theoretical analysis ofQC-LDPC code, and stresses several tasks below. At first, a novel construction algorithm of QC-LDPC is proposed which is based on the traditional BIBD. Adopting the multiplicative group over finite fields, the algorithm replaces the location vector of the BIBD's element by the exponent in the multiplicative group, which can avoid a mass of exponential and module calculations and thus simplify the calculation of the location vector. It is strictly proved that the new constructed QC-LDPC codes have girth at least 6 and experimental results show that the performance of these QC-LDPC codes is very close to the random LDPC and converge very fast. Moreover, these QC-LDPC codes have low error-floor and can outperform the traditional BIBD QC-LDPC codes.Secondly, good QC-LDPC codes for correcting erasure-bursts are constructed by the decomposition of the above parity-check matrixes. The ability of correcting erasure-bursts is obtained according to the structure characteristic of the parity-check matrix. In addition, the influence of the code rate caused by decomposition is analyzed, which indicates that the value of t is larger, the influence is smaller and the rate is larger. Simulation results show that codes constructed by decomposition perform well over both AWGN and binary erasure channels and approach the Shannon Limit.Thirdly, the calculation of the minimum Hamming weight and the number of minimum weight codewords of IA-LDPC codes are addressed. The condition, which is equivalent to the cancel-out condition holding for any row of the support matrix, is introduced to prove the judge theorem for the codeword. Based on the above discussion, the minimum Hamming weight and the number of minimum weight codewords of IA-LDPC codes with j = 2,3 or j = 4, q= 5 are deduced in detail, and the same results of IA-LDPC codes with j = 5 or j = 4, q≥7are provided by computer searching. Moreover, the characteristic of the IA-LDPC codeword is analyzed, and the composing structure of the minimum weight codeword set A0 is described, then the relationship between the sizes of the set A0 and A1 is presented, where A1 represents the subset of A0 whose element has the all 0s column in its support matrix. At last, a new lower bound on the error probability is proposed, which is based onDawson-Sankoff inequality. The bound are obtained by applying a new lower bound on the probability of a union of events, derived by improving on Dawson-Sankoff inequality. The improved Dawson-Sankoff inequality is strictly proved. For the AWGN and BSC channel, the expressions of the lower bound on the error probability are presented, which are based on the traditional Dawson-Sankoff inequality and the disadvantages of these lower bounds are pointed out and analyzed. Furthermore, the judge rule of the redundant error events is brought forward and the improved lower bound is achieved. Experiment results show that the new lower bound has tighter performance than the existed lower bounds and is very close to union upper bound.