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IEEE International Symposium on Information Theory (ISIT)

Extension of quasi-cyclic LDPC codes by lifting

Seho Myung, Kyeongcheol Yang

Published Jan 1, 2005• 16 citations
Publisher

Abstract

In this paper, we analyze some cycle properties of quasi-cyclic low-density parity-check (QC-LDPC) codes and show that the girth of a QC-LDPC code is upper bounded by a certain number introduced by the structure of its mother matrix. We also propose a simple method to extend QC-LDPC codes of large length by lifting QC-LDPC codes of smaller length. In particular, it is possible to generate them from a single exponent matrix by a proper modulo-operation. Simulation results show that the more we apply the lifting to QC-LDPC codes, the more memory efficiency becomes better, but it may induce a little performance degradation

DOI
10.1109/isit.2005.1523759

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In this paper, we analyze some cycle properties of quasi-cyclic low-density parity-check (QC-LDPC) codes and show that the girth of a QC-LDPC code is upper bounded by a certain number introduced by the structure of its mother matrix. We also propose a simple method to extend QC-LDPC codes of large length by lifting QC-LDPC codes of smaller length. In particular, it is possible to generate them from a single exponent matrix by a proper modulo-operation. Simulation results show that the more we apply the lifting to QC-LDPC codes, the more memory efficiency becomes better, but it may induce a little performance degradation
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In this paper, we present an improved method for obtaining quasi-cyclic low-density parity-check (QC-LDPC) codes of larger length by combining some component QC-LDPC codes of smaller length.Large girth can be obtained by carefully designing one large girth component code of smaller length.The other component codes are properly designed to obtain a lot of QC-LDPC codes with much few short cycles.Simulations show that compared with CRT based QC-LDPC codes and PEG based QC-LDPC codes, the proposed QC-LDPC codes have much less short cycles and better performance.
In this letter, we present an efficient method for constructing a family of quasi-cyclic low-density parity-check (QC-LDPC) codes specified only by one single exponent matrix that can support multiple lifting sizes. Our method adopts the small-lifting-size-first design strategy and the resulting exponent matrix of larger lifting size has non-decreasing girth. Numerical results show that all constituent codes can perform as well as (outperform) the CCSDS (5G) LDPC codes over the AWGN channel with the iterative decoding algorithms. This fact demonstrates the feasibility and validity of our proposed method and makes it competitive in some applications, e.g., enhancing short and medium-length LDPC codes to support longer length.
In this paper, we define two “difference matrices” which correspond to an exponent matrix. We present necessary and sufficient conditions for these difference matrices to have quasi-cyclic low-density parity-check codes (QC-LDPC) codes with a certain girth. We achieve all non-isomorphic QC-LDPC codes with the shortest length, girth 6, the column weight, m = 4, and the row weight, 5 ≤ n ≤ 11. Moreover, a method to obtain an exponent matrix with girth 10 is presented which reduces the complexity of the search algorithm. If the first row and the first column of the exponent matrix are all-zero, then by applying our method we do not need to test Fossorier's lemma to avoid 6-cycles and 8-cycles. For girth 10, we also provide a lower bound on the lifting degree which is tighter than the existing bound. For girth 12, a new lower bound on the lifting degree is achieved.
Short cycles in the Tanner graph of a low-density parity-check (LDPC) code may cause a severe performance degradation. In this paper, we investigate the cycle properties of quasi-cyclic LDPC (QC-LDPC) codes. We first analyze a necessary and sufficient condition for a cycle of a given length to exist, by using the sequence representation of a parity-check matrix for a QC-LDPC code. We then derive bounds which are necessary conditions for a QC-LDPC code to have a given girth in terms of its parameters. Our necessary conditions are applicable to any regular or irregular QC-LDPC codes as well as they improve the existing bounds for many classes of regular QC-LDPC codes.
Multitype quasi-cyclic (QC) low-density parity-check (LDPC) codes are a class of protograph LDPC codes lifted cyclically from protographs with multiple edges, represented by two weight and slope matrices. For a given weight-matrix, an approach is proposed to find the maximum-achievable girthgmax⁡g_{\max }gmax​of the corresponding multitype QC-LDPC codes by some inevitable chains having less complexity than the existing methods. This advantage leads to some new patterns of the weight matrices such that the corresponding codes have some improvements in terms of the maximum-achievable girths or the minimum-distance upper-bounds. In continue, for a given weight-matrix with maximum-achievable girthgmax⁡g_{\max }gmax​, some slope-matrices are constructed by a depth-first search algorithm for which the corresponding multitype QC-LDPC codes with even girthggg,g≤gmax⁡g\le g_{\max }g≤gmax​, have smaller lengths, higher rates, or larger minimum-distances than the state-of-the-art achievements. Simulation results show that the constructed codes have some error-rate advantages than PEG, random-like, CCSDS, and 802.11n/ac IEEE standard LDPC codes.