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Recently, a class of low-density parity-check codes based on affine permutation matrices, called APM-LDPC codes, have been considered which have some advantages than quasi-cyclic (QC) LDPC codes in terms of minimum-distance, cycle distribution, and error-rate performance. Moreover, some explicit constructions for exponent matrices of conventional APM-LDPC codes with girth at least 6 have been investigated. In this paper, a class of 4-cycle free APM-LDPC codes is constructed by a new explicit method such that the constructed codes have better cycle distributions rather than the recently proposed APM codes with girth 6. As simulation results show, the constructed codes outperform PEG and random-like LDPC codes with the same rates and lengths.