Constructions Of Quasi-Cyclic Low-Density Parity-Check Codes Based On Integer Sequences

As a Shannon-limit approaching coding technique, low-density parity-check (LDPC) codes perform well in a wide range of channels. Hence the construction of high performance LDPC codes has been a focus of research in the field. Quasi-cyclic LDPC (QC-LDPC) codes, an important subclass of LDPC codes, are getting more and more attention due to their easy implementation and other graceful features. Three construction methods of QC-LDPC codes based on integer sequences are proposed in this thesis.Firstly, an approach to the construction a class of (3, k) regular QC-LDPC codes is proposed, which is based on combinatorial objects termed difference sequences (DS). The elements in shift matrix are selected from difference sequences generated from quadratic polynomials. Then a class of QC-LDPC codes without4-cycles can be obtained by replacing each element in shift matrix with a circulant permutation matrix (CPM). Girth is a key factor to affect the performance of LDPC codes, and a relatively large girth can help to improve the bit error rate (BER) performance. So an efficient algorithm for searching good difference sequences which can achieve girth eight is devised. After the comparison between codes with girth8and codes with girth6, we draw a conclusion that the larger the girth is, the better the error performance of the code is.Secondly, an algebraic method for constructing QC-LDPC codes based on Hoey sequence is proposed. The shift matrix consists of the elements from Hoey sequences, and then the parity-check matrix can be obtained by replacing each element in shift matrix with a relevant CPM. Two classes of regular QC-LDPC codes with colomun weight two and three are constructed, called Class-Ⅰ codes and Class-Ⅱ codes, respectively. The shift matrices of Class-I codes consist of one row of Hoey sequences and one row of zeros, and the shift matrices of Class-Ⅱ codes.are composed of two rows of Hoey sequences and one row of zeros. With the properties of Hoey sequence, it is easy to show that the girths of these two classes of QC-LDPC codes are eight and six, respectively.Finally, a construction method based on perfect cyclic difference sets is introduced. The parity-check matrix of the codes is composed of weight-2circulant matrices (W-2CMs). The corresponding shift matrix is composed of order pairs formed by the elements of perfect cyclic difference sets. We can acquire the parity-check matrix by substituting each order pair in shift matrix with a relevant W-2CM. Given that W-2CM will increase the probability of the occurrence of short cycles, a necessary and sufficient condition for the code to have girth at least six is presented. Since the shift numbers in shift matrices are selected from perfect cyclic difference sets, if different intervals are set, a variety of codes are obtained. In view of this situation, the error performances of codes with various intervals are shown.Simulation results show that these four classes of codes slightly outperform the counterpart PEG codes and have better performance than the corresponding MacKay codes and array codes in additive white Gaussion noise channel. Furthermore, no error floor can be observed for all the four classes of codes until the BER down to10"7even10-8, and they have fast decoding convergence rates.