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Cyclic liftings are proposed to lower the error floor of low-density parity-check (LDPC) codes. The liftings are designed to eliminate dominant trapping sets of the base code by removing the short cycles which are part of the trapping sets. We derive a necessary and sufficient condition for the cyclic permutations assigned to the edges of a cycle ξ of length l(ξ) in the base graph such that the inverse image of ξ in the lifted graph consists of only cycles of length strictly larger than l(ξ). The proposed method is universal in the sense that it can be applied to any LDPC code over any channel and for any iterative decoding algorithm. It also preserves important properties of the base code such as degree distributions, and in some cases, the code rate. The constructed codes are quasi-cyclic and thus attractive from a practical point of view. The proposed method is applied to both structured and random codes over the binary symmetric channel (BSC). The error floor improves consistently by increasing the lifting degree, and the results show significant improvements in the error floor compared to the base code, a random code of the same degree distribution and block length, and a random lifting of the same degree. Similar improvements are also observed when the codes designed for the BSC are applied to the additive white Gaussian noise (AWGN) channel.