Research On Nonbinary Ldpc Codes Based On Permutation Polynomials

Low-Density Parity-Check(LDPC) codes are one special class of linear block codes which have sparse check matrix, which was first introduced by Dr. R. G. Gallager in1962. Due to the interative soft-input soft-output decoding algorithm discovered in recent years, LDPC codes have exhibited outstanding performance approaching the Shannon limit and thus become a hot spot of academic research in the communication area in this decade. Compared with binary LDPC codes, nonbinary LDPC codes over the finite fields GF(q) have been proved out better performance on error correction, which were first proposed by Davey and Mackay in1998. The binary LDPC codes and nonbinary LDPC codes are both analyzed and simulated in this thesis, moreover, a compact construction for nonbinary LDPC codes over GF(q) is proposed.Firstly, the thesis introduces the relative concepts and basic principles of LDPC Codes, including the definition of LDPC Codes and Tanner graph. Various binary encoding algorithms of LDPC Codes, i.e. Gaussian elimination and approximate lower triangular; and soft decision decoding algorithm, i.e. Belief Propagation(BP) decoding algorithm and LLR BP decoding algorithm are investigated in this chapter.Secondly, the thesis introduces the LDPC codes provided by IEEE802.16e and the recursive encoding method included in IEEE802.16e standards. A simulation of various code lengths and rates of LDPC codes of IEEE802.16e is given to show good error correct performance, using the recursive encoding method and BP decoding algorithm. The decoding algorithms of nonbinary LDPC codes over GF(q), i.e. BP decoding algorithm and Log-BP decoding algorithm over GF(q), are introduced. By a simulation of nonbinary LDPC codes provided by Mackay constructed randomly, nonbinary LDPC codes exhibit excellent ability on error correction.Finally, the thesis researches the permutation polynomials(PPs) and the construction for nonbinary LDPC Codes using PPs. The current constructions for LDPC codes are presented. Then the basic concepts and determine conditions of PPs are demonstrated. Based on the construction for binary LDPC codes using PPs proposed by Oscar Y. Takeshita, a compact construction for nonbinary LDPC codes over GF(q) is proposed, which combines quadratic permutation polynomials(QPPs) for nonzero entries and linear permutation polynomials(LPPs) for substitution of nonzero elements over GF(q). Computer simulation results of nonbinary LDPC codes over GF(8) have shown that the proposed compact construction can attain similar error correction performance with other randomly construction for nonbinary LDPC codes.