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Low-density parity-check (LDPC) codes are attractive since their performance is known to approach the Shannon limits for suitably large block lengths. However, for moderate block lengths, error floors still jeopardize the performance even of well-designed LDPC codes. Previous work has shown that the error floor of a broad class of LDPC codes is due to certain graphical structures called absorbing sets. Separable, circulant-based (SCB) codes represent a general family of high-performance, hardware-friendly LDPC codes built out of circulants. A recently proposed technique applies row selection and column elimination methods to SCB codes to dramatically decrease error floors by avoiding certain small dominant absorbing sets in a principled way. This paper focuses on improving the greedy column elimination method to achieve greater flexibility in code rate while provably avoiding small dominant absorbing sets. Flexibility and low implementation complexity are therefore possible without sacrificing SCB code performance.