Study On LDPC Codes:Analysis,Design,and Construction

With the development of iterative decoding,it was rediscovered in the late 1990's that low-density parity-check(LDPC)codes are a class of capacity-approaching channel codes.At present,binary LDPC code has been selected as the coding scheme for eMBB(Enhanced Mobile Broadband)data channel.Research shows that belief propagation(BP)decoding on cycle-free Tanner graph(or factor graph)is a maximum a posteriori(MAP)decoding(optimal),but BP decoding on Tanner graph with cycles is suboptimal.However,for finite block lengths,the Tanner graph of an LDPC code inevitably contains short cycles.Moreover,for short to medium block lengths,nonbinary LDPC codes outperform binary LDPC codes.Hence,this dissertation studies the analysis,design,construction,decoding,and etc.of binary and nonbinary LDPC codes.The main contributions are summarized as follows.To analyze the girth of Tanner(J,L)quasi-cyclic(QC)LDPC codes and optimize these codes,the method for computing girth and the code design scheme based on masking are proposed,respectively.By checking the existence of the 35th root and 33rd root for the equations over the prime fields,the girths of Tanner(5,7)QC LDPC codes and Tanner(3,11)QC LDPC codes are derived,respectively.For any two positive integers,J and L,an algorithm for computing the girth of QC LDPC codes is proposed,and then the girths of Tanner(J,L)QC LDPC codes are determined for a given block length.Moreover,the masking technique is investigated.Based on Tanner(J,L)QC LDPC codes with the known girth,binary and nonbinary QC LDPC codes are constructed.Numerical simulation results show that the constructed QC LDPC codes have good iterative decoding performance.To construct the structural LDPC codes,the construction method based on the combinatorial structures in combinatorial design theory is proposed.The relation between group divisible design(GDD)and partial geometry(PaG)is first analyzed,and then a class of regular LDPC codes is constructed employing the resolvable property of GDD.Combined with superposi-tion and masking,much more structural LDPC codes are obtained.Finally,nonbinary cyclic LDPC codes and QC LDPC cycle codes over high-order finite fields are constructed based on cyclic difference family(CDF),respectively,and their girths and minimum distances are also analyzed.To construct LDPC codes which can be fast encoded,the construction methods based on computer are presented.The submatrix constraint for the parity-check matrices of general LDPC codes is first studied,and for QC LDPC codes,the submatrix constraints for their exponent matrices are also proposed.Next,the general form of the parity-check matrices of array-based QC LDPC codes is presented.Combined with masking,a class of QC LDPC codes with large girth and few short cycles is constructed.Under the framework of superpo-sition,the construction of QC LDPC codes is changed into the design of three parameters.The exhaustive search for the combination of these three parameters can be performed by means of computer,and then QC LDPC codes with optimal" cycle distributions are ob-tained.At last,for LDPC codes with bidiagonal structure of the parity-check matrices,the cycle structure of their Tanner graphs is first analyzed,and then the method for constructing parity-check matrices with bidiagonal structure is proposed.To construct QC LDPC codes,a method for constructing QC LDPC codes based on iso-morphism theory is presented.Based on graph isomorphism theory,the isomorphism of LDPC codes(and their parity-check matrices)is first presented,and then an algorithm for counting and enumerating cycles of QC LDPC codes is also proposed.According to the isomorphism theory of QC LDPC codes and the proposed algorithm for counting cycles,the construction of binary and nonbinary QC LDPC codes is presented.Combined with mask-ing technique,QC LDPC codes with much more optimal cycle distributions are obtained.Furthermore,given degree distribution and code length,QC LDPC codes constructed from the other methods can be divided into several isomorphic classes and then which one is the best can be determined based on their cycle distributions.Finally,with the cycle elimination method,the nonzero elements of the parity-check matrices of nonbinary LDPC codes are optimized.Finally,targeting on two types of application scenarios,two classes of nonbinary LDPC codes are constructed.First,nonbinary LDPC codes with flexible code rates and field order-s are constructed for low-latency and high-reliability applications in 5G communications.Second,for uplink communication in space communications,a class of nonbinary LDPC codes with fast iterative convergence is proposed.To improve the decoding performance of nonbinary LDPC codes,an algorithm based on the maximum likelihood(or approximate maximum likelihood)decoding of binary LDPC codes for nonbinary LDPC codes with short block lengths is proposed.Numerical simulation results show that the proposed algorithm achieves about 0.3 dB coding gain over the Q-ary sum-product algorithm.