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Progressive edge-growth (PEG) algorithm is known to construct low-density parity-check (LDPC) codes at finite code lengths with large girths. A simple variation of the PEG algorithm, linear-encoding PEG (LPEG) algorithm, can be applied to generate linear encodable LDPC codes. This paper presents a modified LPEG algorithm to construct fast encodable LDPC codes with a certain girth constraint. The presented fast encodable LDPC codes facilitate the parallel encoding processes for parity bit generations. Therefore, their encoding time is much shorter than that of LPEG codes.