In this paper, we aim at constructing high-rate quasi-cyclic low-density parity-check (QC-LDPC) codes with girth-10 based on shortened array codes. Our first contribution is the derivation of analytic results on the maximum number of columns for shortened array codes of different girths. Then, inspired by the analysis, we propose a code construction method for column-weight-three codes. We further compare the minimum length and the error performance of the column-weight-three codes constructed by the proposed algorithm and those found by the conventional greedy construction algorithm. We show that the proposed method is more effective than the conventional greedy algorithm in the sense that the minimum length of the codes constructed using the proposed method to achieve different code rates is comparative or much shorter than those constructed using the greedy construction.