Skip to main content


High Girth Column-Weight-Two LDPC Codes Based on Distance Graphs


LDPC codes of column weight of two are constructed from minimal distance graphs or cages. Distance graphs are used to represent LDPC code matrices such that graph vertices that represent rows and edges are columns. The conversion of a distance graph into matrix form produces an adjacency matrix with column weight of two and girth double that of the graph. The number of 1's in each row (row weight) is equal to the degree of the corresponding vertex. By constructing graphs with different vertex degrees, we can vary the rate of corresponding LDPC code matrices. Cage graphs are used as examples of distance graphs to design codes with different girths and rates. Performance of obtained codes depends on girth and structure of the corresponding distance graphs.



  1. 1.

    Chung S-Y, Forney GD Jr., Richardson TJ, Urbanke R: On the design of low-density parity-check codes within 0.0045 dB of the Shannon limit. IEEE Communications Letters 2001,5(2):58-60. 10.1109/4234.905935

  2. 2.

    Gallager RG: Low-density parity-check codes. IRE Transactions on Information Theory 1962,8(1):21-28. 10.1109/TIT.1962.1057683

  3. 3.

    Song H, Liu J, Vijaya Kumar BVK: Low complexity LDPC codes for partial response channels. Proceedings of IEEE Global Telecommunications Conference (GLOBECOM '02), November 2002, Taipei, Taiwan 2: 1294-1299.

  4. 4.

    Song H, Liu J, Vijaya Kumar BVK: Large girth cycle codes for partial response channels. IEEE Transactions on Magnetics 2004,40(4, part 2):3084-3086. 10.1109/TMAG.2004.829197

  5. 5.

    Moura JMF, Lu J, Zhang H: Structured low-density parity-check codes. IEEE Signal Processing Magazine 2004,21(1):42-55. 10.1109/MSP.2004.1267048

  6. 6.

    Biggs N: Cubic graphs with large girth. Processdings of the 3rd International Conference on Combinatorial Mathematics, June 1989, New York, NY, USA 56-62.

  7. 7.

    Exoo G: A simple method for constructing small cubic graphs of girths 14, 15, and 16. Electronic Journal of Combinatorics 1996,3(1):1-3.

  8. 8.

    Wong P: Cages—a survey. Journal of Graph Theory 1982, 6: 1-22. 10.1002/jgt.3190060103

  9. 9.

    Meringer M: Fast generation of regular graphs and construction of cages. Journal of Graph Theory 1999,30(2):137-146. 10.1002/(SICI)1097-0118(199902)30:2<137::AID-JGT7>3.0.CO;2-G

  10. 10.

    Meringer M: Genreg-download manual.

  11. 11.

    Royle G: Cages of higher valency.

  12. 12.

    Biggs N: Constructions for cubic graphs with large girth. Electronic Journal of Combinatorics 1998.,5(1)

  13. 13.

    Weisstein E: Cage graph. From MathWorld-A Wolfram Web Resource,

  14. 14.

    Zhong H, Zhang T: Design of VLSI implementation-oriented LDPC codes. Proceedings of 58th IEEE Vehicular Technology Conference (VTC '03), October 2003, Orlando, Fla, USA 1: 670-673.

Download references

Author information

Correspondence to Gabofetswe Malema.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Malema, G., Liebelt, M. High Girth Column-Weight-Two LDPC Codes Based on Distance Graphs. J Wireless Com Network 2007, 048158 (2007) doi:10.1155/2007/48158

Download citation


  • Cage
  • Information System
  • Minimal Distance
  • Matrix Form
  • System Application