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Low-density parity-check (LDPC) codes are an important choice of the error correction codes in broadband powerline communication (PLC) systems. In this paper, a "column extension" design method for rate-compatible quasi-cyclic (QC) LDPC codes is proposed, which can provide efficient multi-rate QC-LDPC schemes for PLC systems. Based on the optimization of degree distributions among multi-rate LDPC codes, excellent performance can be achieved at each code rate. Benefiting from the nested matrix structure, the designed QC-LDPC codes facilitate the implementation of multi-rate encoder/decoder and offer the advantage of low-complexity. A comparison with the QC-LDPC codes in ITU-T G.9960 specification is performed, where the effectiveness of the proposed method and the superior performance of our designed codes are validated via extrinsic information transfer (EXIT) chart analysis and bit error rate (BER) simulations.