# Recursive construction of $(J,L)$ QC LDPC codes with girth 6

Document Type: Research Paper

Authors

1 Shahrekord University

2 University of Shahrekord,

Abstract

‎In this paper‎, ‎a recursive algorithm is presented to generate some exponent matrices which correspond to Tanner graphs with girth at least 6‎. ‎For a $J \times L$ exponent matrix $E$‎, ‎the lower bound $Q(E)$ is obtained explicitly such that $(J,L)$ QC LDPC codes with girth at least 6 exist for any circulant permutation matrix (CPM) size $m \geq Q(E)$‎. ‎The results show that the exponent matrices constructed with our recursive algorithm have smaller lower-bound than the ones proposed recently with girth 6‎.

Keywords

Main Subjects

### References

[1] R. G. Gallager, Low density parity check codes, IRE Trans., 8 (1962) 21–28.

[2] R. M. Tanner, A recursive approach to low complexity codes, IEEE Trans. Inform. Theory, 27 (1981) 533–547.

[3] R. J. McEliece, D. J. C. Mackay and J. F. Cheng, Turbo decoding as instance of pearl$^{,}$s “belief propagation” algorithm, IEEE J. Sel. Areas Commun, 16 (1998) 140–152.

[4] J. L. Cheng, U. N. Peled, I. Perepelitsa and V. Pless, Explicit construction of families of LDPC codes with girth at least six, in Proc. 40th Annu. Allerton Conf. Communication, control and computing, Monticello, IL, (2002) 1024–1031.

[5] M. E. $O^{, ‎}$Sullivan, Algebraic construction of sparse matrices with large girth, IEEE Trans. Inform. Theoy, 8 (2006) 1788–1793.

[6] K. Lally, Explicit construction of Type-II QC-LDPC codes with girth at least 6, IEEE Int. Symp. on Inf. Theory, (2007) 2371–2375.

[7] B. Ammar, B. Honary, Y. Kou, J. Xu and S. Lin, Construction of low-density parity-check codes based on balanced incomplete block designs, IEEE Trans. Inform. Theoy, 50 (2004) 1257–1268.

[8] L. Lan, L. Zeng, Y. Y. Tai, L. Chen, S. Lin and K. A. Ghaffar, Construction of QC-LDPC codes for AWGN and
binary erasure channels: A finite field approach, IEEE Trans. Inform. Theoy, 53 (2007) 2429–2457.

[9] C. M. Huang, J. F. Huang and C. C. Yang, Construction of QC-LDPC codes from quadratic congruences, IEEE Comm. Lett., 12 (2008) 313–315.

[10] M. Karimi and A. H. Banihashemi, On the girth of quasi cyclic protograph LDPC codes, IEEE Trans. Inform.
Theoy, 59 (2013) 4542–4552.

[11] G. Zhang, R. Sun and X. Wang, Construction of girth-eight QC-LDPC codes from greatest common divisor, IEEE Comm. Lett., 17 (2013) 369–372.