Non-binary low-density parity-check (NB-LDPC) codes can be directly constructed by using algebraic methods, or indirectly constructed by mapping well-designed binary parity-check matrices to non-binary parity-check matrices. Given the Tanner graph (TG) of a NB-LDPC code, the selection of edge weights in the TG significantly affects the performance of the NB-LDPC code. The authors introduce an edge weight distribution (EWD) parameter for the TG of NB-LDPC codes. By utilising particle swarm optimisation (PSO), the EWD is optimised and it has been demonstrated that the optimal EWD approaches a two-element distribution for large field size and high average variable-node degree. With the optimised EWD, the authors construct a class of field-compatible LDPC (FC-LDPC) codes over GF(q) whose parity-check matrices only include elements 0, 1 and 2, and can be encoded and decoded over different field sizes. The simulations demonstrate that the performance of the proposed FC-LDPC codes improves monotonically with increasing field size, and significantly outperforms that of the corresponding algebraic NB-LDPC codes or NB-LDPC codes generated with uniform distribution of non-zero elements over GF(q).

References

1. 1)
  - 17. Kasai, K., Poulliat, C., Declercq, D., Sakaniwa, K.: ‘Weight distributions of non-binary LDPC codes’, IEICE Trans. Fundam., 2011, E94-A, (4), pp. 1106–1115 (doi: 10.1587/transfun.E94.A.1106).
2. 2)
  - 18. Hu, X.-Y., Eleftheriou, E.: ‘Binary representation of cycle Tanner-graph GF(2b) codes’. Proc. Int. Conf. Communications (ICC), Ruschlikon, Switzerland, July 2004, pp. 528–532.
3. 3)
  - 8. Zhou, B., Kang, J., Tai, Y.-Y., Lin, S., Ding, Z.: ‘High performance nonbinary quasi-cyclic LDPC codes on Euclidean geometries’, IEEE Trans. Commun., 2009, 57, (4), pp. 545–554.
4. 4)
  - 29. Song, S., Zhou, B., Lin, S., Abdel-Ghaffar, K.A.S.: ‘A unified approach to the construction of binary and nonbinary QC-LDPC codes based on finite fields’, IEEE Trans. Commun., 2009, 57, (1), pp. 84–93 (doi: 10.1109/TCOMM.2009.0901.060129).
5. 5)
  - 4. Song, H., Cruz, J.R.: ‘Reduced-complexity decoding of Q-ary LDPC codes for magnetic recording’, IEEE Trans. Magn., 2003, 39, (2), pp. 1081–1087 (doi: 10.1109/TMAG.2003.808600).
6. 6)
  - X.Y. Hu , E. Eleftheriou , D.M. Arnold . Regular and irregular progressive edge-growth Tanner graphs. IEEE Trans. Inf. Theory , 1 , 386 - 398
7. 7)
  - S.-Y. Chung , T.J. Richardson , R.L. Urbanke . Analysis of sum-product decoding of low-density parity-check codes using a Gaussian approximation. IEEE Trans. Inf. Theory , 657 - 670
8. 8)
  - A. Ashikhim , G. Kramer , S. ten Brink . Extrinsic information transfer functions: model and erasure channel properties. IEEE Trans. Inf. Theory , 11 , 2657 - 2673
9. 9)
  - M.C. Davey , D. MacKay . Low-density parity check codes over GF(q). IEEE Commun. Lett. , 6 , 165 - 167
10. 10)
  - 2. Mackay, D.J.C., Davey, M.C.: ‘Evaluation of Gallager codes for short block length and high rate applications’. Proc. IMA Workshop on Codes, Systems and Graphical Models, 1999, pp. 108–112.
11. 11)
  - 20. Kim, K.: ‘Future memory technology: Challenges and opportunities’. Proc. Int. Symp. VLSI Technology, Systems Applications, Hsinchu, April 2008, p. 59.
12. 12)
  - 3. Barnault, L., Declercq, D.: ‘Fast decoding algorithm for LDPC over GF(2q)’. Proc. IEEE Information Theory Workshop, Paris, France, 31 March–4 April 2003, pp. 70–73.
13. 13)
  - D. Vukobratović , V. Šenk . Generalized ACE constrained progressive edge-growth LDPC code design. IEEE Commun. Lett. , 1 , 32 - 34
14. 14)
  - T. Richardson , R. Urbanke . The capacity of low-density parity check codes under message-passing decoding. IEEE Trans. Inf. Theory , 599 - 618
15. 15)
  - 19. Kennedy, J., Eberhart, R.C.: ‘Particle swarm optimization’. Proc. IEEE Int. Conf. Neural Networks (ICNN), Perth, Australia, November/December 1995.
16. 16)
  - D. Declercq , M. Fossorier . Decoding algorithms for nonbinary LDPC codes over GF(q). IEEE Trans. Commun. , 4 , 633 - 643
17. 17)
  - 27. Kliewer, J., Ng, S.X., Hanzo, L.: ‘Efficient computation of EXIT functions for nonbinary iterative decoding’, IEEE Trans. Commun., 2006, 54, (12), pp. 2133–2136 (doi: 10.1109/TCOMM.2006.885050).
18. 18)
  - 26. Byers, G.J., Takawira, F.: ‘EXIT charts for non-binary LDPC codes’. Proc. Int. Conf. Communications (ICC), May 2005, pp. 652–657.
19. 19)
  - 6. Voicila, A., Declercq, D., Verdier, F., et al: ‘Low complexity, low-memory EMS algorithm for non-binary LDPC codes’. Proc. IEEE ICC 2007, Glasgow, Scotland, June 2007, pp. 671–676.
20. 20)
  - 15. Poulliat, C., Fossorier, M., Declercq, D.: ‘Design of regular (2, dc)-LDPC codes over GF(q) using their binary images’, IEEE Trans. Commun., 2008, 56, (10), pp. 1626–1635 (doi: 10.1109/TCOMM.2008.060527).
21. 21)
  - 24. Bennatan, A., Burshtein, D.: ‘Design and analysis of nonbinary LDPC codes for arbitrary discrete-memoryless channels’, IEEE Trans. Inf. Theory, 2006, 52, (2), pp. 549–583 (doi: 10.1109/TIT.2005.862080).
22. 22)
  - 10. Han, G., Guan, Y.L., Kong, L., Cai, K.: ‘Construction of irregular QC-LDPC codes via masking with ACE optimization’, IEEE Commun. Lett., 2014, 18, (2), pp. 348–351 (doi: 10.1109/LCOMM.2014.010214.132463).
23. 23)
  - 14. Fang, Y., Chen, P., Wang, L., Lau, F.C.M.: ‘Design of protograph LDPC codes for partial response channels’, IEEE Trans. Commun., 2012, 60, (10), pp. 2809–2819 (doi: 10.1109/TCOMM.2012.072412.110464).
24. 24)
  - 13. Chang, B.Y., Divsalar, D., Dolecek, L.: ‘Non-binary protograph-based LDPC codes for short block-lengths’. Proc. IEEE Information Theory Workshop (ITW), Lausanne, Switzerland, September 2012, pp. 282–286.
25. 25)
  - V. Rathi , R. Urbanke . Density evolution, thresholds and the stability condition for non-binary LDPC codes. IEE Proc. Commun. , 6 , 1069 - 1074
26. 26)
  - 10. Zeng, L., Lan, L., Tai, Y., Zhou, B., Lin, S., Khaled, A.: ‘Construction of nonbinary cyclic, quasi-cyclic and regular LDPC codes: a finite geometry approach’, IEEE Trans. Commun., 2008, 56, (3), pp. 378–387 (doi: 10.1109/TCOMM.2008.060025).
27. 27)
  - 28. http://www.inference.phy.cam.ac.uk/mackay/codes/data.html.

Towards optimal edge weight distribution and construction of field-compatible low-density parity-check codes over GF(q)

References

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