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This paper presents a hybrid ARQ (HARQ) scheme based on shortened low-density parity-check (LDPC) codes. Although conventional incremental redundancy HARQ (IR-HARQ) schemes based on punctured LDPC codes can achieve high throughput, puncturing slows the convergence of iterative decoding, and moreover, LDPC codes originally designed to be high-rate generally have lower decoding complexity for each iteration than low-rate codes. This makes computational complexity at high rates a problem for conventional IR-HARQ schemes. The proposed HARQ scheme uses a high-rate mother code: for a retransmission request, information bits are divided into smaller sub-blocks, and additional parity bits are generated for one of the sub-blocks to be used as a shortened form of the mother code. In order to achieve high throughput with the proposed HARQ scheme, it is necessary for the shortened codes of all the sub-blocks to perform well. A multi-edge type code-design is employed to construct irregular LDPC codes that meet this requirement. Simulation results show that the proposed HARQ scheme can achieve throughput comparable to that of a conventional IR-HAQR scheme over a wide range of signal-to-noise ratios and with lower computational complexity.