Abstarct: As quasi cyclic low density parity check (QC-LDPC) codes are implementable on hardwares, also because of their capability of high speed encoding/decoding, they are of most desired codes in coding theory and telecommunications. Among the variety of methods in constructing QC-LDPC codes, those constructed baxsed on protographs have great importance. In fact, due to small size of their baxse graphs, one would be able to do further investigations on them to design codes with better parameters and performances. Parameters of QC-LDPC codes that hugely impact on the performance curevs are their high girths as well as their low lengths. Indeed, given a fixed girth, the smaller length of a QC-LDPC code, the less complexity of its encoding/decoding is. The other drastic parameter that effects the performance of a QC-LDPC code is existence of small elementary trapping sets (ETSs) in Tanner graph of the code. Simulations reveal that in Gaussian channels and under an iterative decoding algorithm, by avoiding these tapping sets, we encounter with a better performance of the code in the sense of error floor region. Recognizing the most harmful ETS’s and preventing them from occurance in the Tanner graph of QC-LDPC codes are two interesting open problems in the field of coding theory. In the first part of this thesis, we investigate regular QC-LDPC codes whose baxse graph are simple. QC-LDPC codes constructed from these simple graphs are single-edge QC-LDPC codes. In order to construct them with a specific girth, we consider the necessary and sufficient conditions for the their exponent matrices. Our goal is to obtain all non-isomorphic QC-LDPC codes with a certain degree distribution, girths 6 to 12 and minimum lifting degree. To reach our goal, we introduce difference matrices which have a remarkable contribution to reduce the size of the search space to obtain the exponent matrices. Finally, all non-isomorphic QC-LDPC codes (with the same girth, degree distribution and minimum distance) are provided in Tables. In the second part of this thesis, we investigate multiple-edge QC-LDPC codes with girth 6 whose baxse graph are multiple. We obtain the necessary and sufficient conditions for exponent matrices of codes which simplify considering 4-cycles in an exponent matrix. Similar to single-edge QC-LDPC codes, we introduce difference matrices. As a result of this study, an analytical lower bound on the lifting degree is derived which is applicable to both regular and irregular multipleedge QC-LDPC codes with girth 6. We also provide QC-LDPC codes whose lifting degrees meet our proposed lower bound and in comparison with single-edge QC-LDPC codes, the presented codes with minimum lifting degrees have shorter lengths. In addition, we propose a method to reduce the size of the search space to obtain exponent matrices. In the third part of this thesis, we consider elementary trapping sets in the Tanner graph of variable-regular LDPC codes. Using substantial concepts in the graph theory, we study the parameters of these sets which are the size of ETSs and the number of odd degree check check nodes to improve the results existing in the literature.