In this paper

we study a method for constructing a full-rank Quasi-Cyclic Low Density Parity Check（QC-LDPC） code

which can reduce encoding complexity and improve the performance of Bite Error Rate（BER） while reducing the number of short cycles.Furthermore

we present an improved Belief Propagation（BP） decoding algorithm

in which by selecting the variables node logarithmic likelihood ratio decision threshold T=5.3 and the number of parity node edges q=3

we can improve the decoding performance to a certain extent under the conditions of weak turbulences with different intensity.Our studies show the construction method for LDPC code and the improved decoding algorithm presented in this paper have broad prospects in atmospheric laser communications.