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IEEE East-West Design & Test Symposium (EWDTS)

Generalization of Floor Lifting for QC-LDPC Codes: Theoretical Properties and Applications

Ilya Vorobyev, Nikita Polyanskii, German Svistunov, Егоров Сергей Иванович, Vasiliy S. Usatyuk

Published Sep 1, 2018• 3 citations
Publisher

Abstract

The given paper presents several probabilistic statements for the methods related to constructing a QC-LDPC code of shorter length by length adaption from a given QC-LDPC code of maximal length. Several probabilistic statements concerning a theoretical improvement of the generalization of floor lifting method with respect to the number of small cycles are proved. Simulation results show that the generalization of floor lifting method under base graph 2 (BG2) allow to achieves from 0.1 to 0.3 dB coding gain.

DOI
10.1109/ewdts.2018.8524862

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The given paper presents several probabilistic statements for the methods related to constructing a QC-LDPC code of shorter length by length adaption from a given QC-LDPC code of maximal length. Several probabilistic statements concerning a theoretical improvement of the generalization of floor lifting method with respect to the number of small cycles are proved. Simulation results show that the generalization of floor lifting method under base graph 2 (BG2) allow to achieves from 0.1 to 0.3 dB coding gain.
The given paper presents a method for constructing a QC-LDPC code of shorter length by length adaption from a given QC-LDPC code of maximal length. The proposed method can be considered as a generalization of floor lifting. Making some offline calculation it is possible to construct a sequence of QC-LDPC codes with different circulant sizes generated from a single exponent matrix of QC-LDPC code which has the largest length. The only parameter stored in memory is a constant needed for scaling in the lifting procedure. Simulation results show that the proposed lifting method achieves around a 0.1 dB coding gain on short lengths over previous lifting approaches.
In the given paper we present a novel approach for constructing a QC-LDPC code of smaller length by lifting a given QC-LDPC code. The proposed method can be considered as a generalization of floor lifting. Also we prove several probabilistic statements concerning a theoretical improvement of the method with respect to the number of small cycles. Making some offline calculation of scale parameter it is possible to construct a sequence of QC-LDPC codes with different circulant sizes generated from a single exponent matrix using only floor and scale operations. The only parameter we store in memory is a constant needed for scaling.
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 lengthl(ξ) in the base graph such that the inverse image of ξ in the lifted graph consists of only cycles of length strictly larger thanl(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.
In this paper, we present two simple methods to construct a quasi-cyclic LDPC (QC-LDPC) code of larger length by lifting a given QC-LDPC code. We show that its girth is always larger than or equal to that of the given QC-LDPC code. Applying the methods recursively, it is possible to construct a sequence of QC-LDPC codes generated from a single exponent matrix by proper floor or modulo operations. Simulation results show that they have no serious performance degradation, as compared with randomly constructed LDPC codes
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 form the trapping sets. We derive a necessary and sufficient condition for the cyclic permutations assigned to the edges of a cycle c of length ℓ(c) in the base graph such that the inverse image of c in the lifted graph consists of only cycles of length strictly larger than ℓ(c). 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. 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.
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 form the trapping sets. We derive a necessary and sufficient condition for the cyclic permutations assigned to the edges of a cycle ccc of length ℓ(c)\ell(c)ℓ(c) in the base graph such that the inverse image of ccc in the lifted graph consists of only cycles of length strictly larger than ℓ(c)\ell(c)ℓ(c). 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, encoder and decoder structure, and in some cases, the code rate. 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.