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This paper focuses on methods for a systematic modification of the parity check matrix of regular LDPC codes for improved performance in the low BER region (i.e., the error floor). A judicious elimination of dominant absorbing sets strictly improves the absorbing set spectrum and thereby improves the code performance. This absorbing set elimination is accomplished without compromising code properties and parameters such as the girth, node degree, and the structure of the parity check matrix. For a representative class of practical codes we substantiate theoretical analysis with experimental results obtained in the low BER region. Our results demonstrate at least an order of magnitude improvement of the error floor relative to the original code designs. Given that the conventional code parameters remain intact, the new code can easily be implemented on the existing software or hardware platforms employing high-throughput, compact architectures. As such, the proposed approach provides a step towards the improved code design that is compatible with practical implementation constraints.