Constructions And Performance Analysis Of Quasi-cyclic LDPC Codes

LDPC (Low Density Parity Check) code is a kind of linear block codes based onsparse check matrix. The structure, performance and complexity of a LDPC code aredetermined by its parity check matrix. Thus the key of the construction of LDPC codesis the construction of their parity check matrices.According to the method of construction, LDPC codes can be divided intorandom codes and structure codes. By random methods we can obtain good LDPCcodes which have good performance. Code parameters of the random codes can beselected flexibly. But the code structures are not clear; encoding complexity is high;large storage spaces are needed to store the parity check matrices. These defectsinfluence the applications of the random LDPC codes. The structure codes areconstructed based on algebra, geometry and combinatorial design, and so on. Theyhave good structures, simplify the encoding and decoding process, and overcome thedefects of the random LDPC codes.Quasi-Cyclic LDPC code is a kind of structure LDPC codes. QC-LDPC codeswith short and medium length have high error correcting capability. Theirperformances are close to the random LDPC codes which have the best performances.With respect to the random LDPC codes, the QC-LDPC codes have regular structure,good hardware implementation and great flexibility in code length and rate,furthermore remain performances close to the Shannon limit.In this paper, two approaches to construct QC-LDPC codes based oncombinatorial sequences and finite fields are proposed respectively. Firstly, based onthe sufficient and necessary condition that the parity check matrix has no cycles oflength four, using the special properties of combinatorial sequences, this paperproposes six methods to construct the parity check matrices of the binary QC-LDPC codes. Two classes of binary and q-ary quasi-cyclic LDPC codes are constructedbased on primitive elements and additive subgroups of finite fields. The QC-LDPCcodes constructed have a girth of at least6. Cyclic shift coefficients of base matricescould be expressed in simple analytic expressions, so the sparse check matrices needlittle storage spaces. Simulation results demonstrate that the codes constructed performwell.