||With superior performance and manageable decoding complexity, Low-density Parity-check Code (LDPC) codes have been intensively studied and play a significant role in practical communication systems. Generally speaking, for practical communication systems, there are three important issues for the LDPC codes' application. Firstly, most real time communication systems require low decoding latency and power consumption, which calls for low complexity decoding scheme without sacrificing performance. Secondly, since the throughput performance of incremental redundancy (IR) hybrid automatic-repeat-request (HARQ) mechanism depends on the performance of rate-compatible (RC) codes, especially high rate codes, efficiently constructing RC LDPC codes is of great importance to practical communication systems. Thirdly, in some application, it is hard to avoid short cycles, especially cycles of length 4 in the LDPC code graph, which degrade the sum product decoding performance of the LDPC code. Therefore, low complexity decoding strategy, the design of good rate-compatible codes, and girth 4 problem are still challenging for the LDPC code study. In this thesis, we first consider the efficient design of RC LDPC codes by puncturing and splitting a LDPC code matrix. Specifically, we propose a novel criterion to rank candidate puncturing and splitting patterns according to their estimated performance so that good RC LDPC codes can be obtained. Efficient cost functions are devised to compare the expected performance of candidate patterns and to sort out good ones. The proposed scheme has a big advantage in reducing decoding complexity without sacrificing performance. With comparable (even better) decoding performance, the proposed RC LDPC codes are always excelled at lower decoding latency and nearly 25% hardware complexity saving can be obtained. We also consider the universal girth 4 problem in a LDPC code graph. A novel mismatched decoding is presented to overcome the performance degradation resulting from cycles of length 4. Specifically, a mismatched decoding graph is derived by introducing invisible nodes in appropriate positions of the original girth 4 graph, so that the message dependency within the cycles can be meliorated and the sum product decoding performance can be improved. An efficient placing algorithm is provided to find an appropriate mismatched graph so that the mismatched decoding can be optimized in terms of error performance and decoding convergence. Nearly 1dB gain is obtained in the target example by applying the proposed placing algorithm. In addition, some important extensions of mismatched decoding will be presented as well: 1) efficient ranking algorithms are introduced for the puncturing problem, with an aim to provide good rate compatible codes based on effective mismatched decoding; 2) a low complexity decoder is proposed for gaining lower decoding latency and low complexity by selecting the decoding matrix. Numerical simulations demonstrate the superiority of the proposed algorithms and verify the effectiveness of the ranking criterion and the mismatched decoding algorithm.