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Lowering the error floor of LDPC codes is extremely attractive to some systems, such as deep space communication systems and storage systems, which desire very low error rates. The error floor phenomenon of LDPC codes, which is associated with their message passing decoding algorithms, is mainly caused by some unfavorable combinatorial characteristics (or error-prone patterns) of LDPC codes. In this paper, a mathematical analysis method which enables the identification of some significant features of error-prone patterns is presented with the help of the beliefs passed in their decoders. Based on the analysis, an improved decoder is proposed which can effectively deal with the traversable trapping sets and achieve significantly improved error floor performance compared with current decoders. More importantly, this proposed decoder does not require the information of all the possible trapping sets of an individual LDPC code when correcting the error bits in its trapping sets.