zbMATH — the first resource for mathematics

On the search of smallest QC-LDPC code with girth six and eight. (English) Zbl 1453.94038
Summary: In this paper, a new and simple method for the construction of girth-6 \((J,L)\) quasi-cyclic low-density parity-check (QC-LDPC) codes is proposed. The method is further extended to the search of Girth-8 QC-LDPC codes with base matrices of order \(3 \times L\) and \(4 \times L\). The construction is based on three different forms of exponent matrices and the parameters \(\alpha \), p, and \(q\) which satisfy the necessary algebraic conditions for a QC-LDPC code having girth 6 and 8. The proposed \((J,L)\) QC-LDPC codes with girth at least six have optimal circulant permutation matrix (CPM) size for the cases where \(q = \alpha + 1\). Moreover, most of the girth-8 QC-LDPC codes searched by the proposed method have smaller CPM size than the existing codes of the same girth. In several cases, the method gives more than one exponent matrices for a code, as most of the existing methods cannot do so. Besides this, the proposed method not only search the QC-LDPC codes with smaller CPM size but also takes much less time than the existing search based methods to search code.
94A24 Coding theorems (Shannon theory)
Full Text: DOI
[1] Gallager, RG, Low-density parity-check codes, IRE Transactions on Information Theory., 8, 1, 21-28 (1962) · Zbl 0107.11802
[2] Mackay, DJC; Neal, RM, Near shannon limit performance of low density parity check codes, Electron. Lett., 32, 18, 1645-1645 (1996)
[3] Berrou C., Glavieus A., Thitimajshima P.: Near shannon limit error-correcting coding and decoding: turbo-codes 1. IEEE International Conference on Communications. In: Proceeding of IEEE International Conference on Communications. (1993). 10.1109/icc.1993.397441
[4] Myung, S.; Yang, K., A combining method of quasi-cyclic LDPC codes by the chinese remainder theorem, IEEE Commun. Lett., 9, 9, 823-825 (2005)
[5] Tanner, RM, A recursive approach to low complexity codes, IEEE Trans. Inf. Theory, 27, 5, 533-547 (1981) · Zbl 0474.94029
[6] Bajpai, A.; Srirutchataboon, G.; Kovintavewat, P., Wuttisittikulkij L: A new construction method for large girth quasi-cyclic LDPC codes with optimized lower bound using chinese remainder theorem, Wirel. Pers. Commun., 91, 1, 369-381 (2016)
[7] Gholami, M.; Gholami, Z., An explicit method to generate some QC-LDPC codes with girth 8, Iranian Journal of Science and Technology, Transactions A: Science., 40, 2, 145-149 (2016) · Zbl 1342.94114
[8] Karimi, M.; Banihashemi, AH, On the girth of quasi-cyclic protograph LDPC codes, IEEE Trans. Inf. Theory, 59, 7, 4542-4552 (2013) · Zbl 1364.94611
[9] Mellinger, KE, LDPC codes from triangle-free line sets, Des. Codes Crypt., 32, 1-3, 341-350 (2004) · Zbl 1052.51008
[10] Sakzad, A.; Sadeghi, M.; Panario, D., Codes with girth 8 tanner graph representation, Des. Codes Crypt., 57, 1, 71-81 (2010) · Zbl 1202.94235
[11] Tasdighi, A.; Banihashemi, AH; Sadeghi, MR, Efficient search of girth-optimal QC-LDPC codes, IEEE Trans. Inf. Theory, 62, 4, 1552-1564 (2016) · Zbl 1359.94739
[12] Tasdighi, A.; Banihashemi, AH; Sadeghi, MR, Symmetrical constructions for regular girth-8 QC-LDPC codes, IEEE Trans. Commun., 65, 1, 14-22 (2017)
[13] Wang, Y.; Yedidia, JS; Draper, SC, Construction of high-girth QC-LDPC codes, Proceeding of 5th International Symposium on Turbo Codes and Related Topics (2007)
[14] Zhang, G.; Sun, R.; Wang, X., Several explicit constructions for (3, L) QC-LDPC codes with girth at least eight, IEEE Commun. Lett., 17, 9, 1822-1825 (2013)
[15] Zhang, J.; Zhang, G., Deterministic girth-eight QC-LDPC codes with large column weight, IEEE Communications Letters., 18, 4, 656-659 (2014)
[16] Zhang, L.; Li, B.; Cheng, L., Constructions of QC LDPC codes based on integer sequences, SCIENCE CHINA Inf. Sci., 57, 6, 1-14 (2014) · Zbl 1357.94096
[17] Fossorier, MPC, Quasi-cyclic low-density parity-check codes from circulant permutation matrices, IEEE Trans. Inf. Theory, 50, 8, 1788-1793 (2004) · Zbl 1300.94123
[18] Li, L.; Li, H.; Li, J.; Jiang, H., Construction of type-II QC-LDPC codes with fast encoding based on perfect cyclic difference sets, Optoelectron. Lett., 13, 5, 358-362 (2017)
[19] O’sullivan, ME, Algebraic construction of sparse matrices with large girth, IEEE Trans. Inf. Theory, 52, 2, 718-727 (2006) · Zbl 1317.05113
[20] Vandendriessche, P., Some low-density parity-check codes derived from finite geometries, Des. Codes Crypt., 54, 3, 287-297 (2010) · Zbl 1185.05033
[21] Yuan, J.; Liang, M.; Wang, Y.; Lin, J.; Pang, Y., A novel construction method of QC-LDPC codes based on CRT for optical communications, Optoelectron. Lett., 12, 3, 208-211 (2016)
[22] Zhang, G.; Sun, R.; Wang, X., Construction of girth-eight QC-LDPC codes from greatest common divisor, IEEE Commun. Lett., 17, 2, 369-372 (2013)
[23] Bocharova, IE; Hug, F.; Johannesson, R., Searching for voltage graph-based LDPC tailbiting codes with large girth, IEEE Trans. Inf. Theory, 58, 4, 2265-2279 (2012) · Zbl 1365.94537
[24] Kim, KJ; Chung, JH; Yang, K., Bounds on the size of parity-check matrices for quasi-cyclic low-density parity-check codes, IEEE Trans. Inf. Theory, 59, 11, 7288-7298 (2013) · Zbl 1364.94614
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.