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In this paper, we construct high-rate quasi-cyclic low-density parity-check (QC-LDPC) codes with girth-10 based on shortened array codes. First, we derive analytic results on the maximum number of columns for shortened array codes of different girths. A code construction method inspired by the analysis is proposed for column-weight-three codes and is compared with 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 our proposed method to achieve different code rates is comparative or much shorter than those constructed using the greedy construction.