Pseudocodeword-free criterion for codes with cycle-free Tanner graph

  • Wittawat Kositwattanarerk


Iterative decoding and linear programming decoding are guaranteed to converge to the maximum-likelihood codeword when the underlying Tanner graph is cycle-free. Therefore, cycles are usually seen as the culprit of low-density parity-check codes. In this paper, we argue in the context of graph cover pseudocodeword that, for a code that permits a cycle-free Tanner graph, cycles have no effect on error performance as long as they are a part of redundant rows. Specifically, we characterize all parity-check matrices that are pseudocodeword-free for such class of codes.


Iterative decoding Linear programming decoding Low-density parity-check (LDPC) code Tanner graphs Pseudocodewords 

Mathematics Subject Classification




The author wishes to thank Gretchen L. Matthews for her support while the author is at Clemson University, Patanee Udomkavanich for her advice, and the anonymous reviewers for their helpful suggestions and comments. This work is supported by the Thailand Research Fund under Grant TRG5880116 and the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.


  1. 1.
    Axvig N., Dreher D., Morrison K., Psota E., Perez L.C., Walker J.L.: Analysis of connections between pseudocodewords. IEEE Trans. Inform. Theory 55(9), 4099–4107 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barahona F., Grötschel M.: On the cycle polytope of a binary matroid. J. Comb. Theory Ser. B 40, 40–62 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Etzion T., Trachtenberg A., Vardy A.: Which codes have cycle-free Tanner graphs? IEEE Trans. Inform. Theory 45(6), 2173–2181 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Feldman J., Wainwright M.J., Karger D.R.: Using linear programming to decode binary linear codes. IEEE Trans. Inform. Theory 51(3), 954–972 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Guichard D.: An Introduction to Combinatorics and Graph Theory.
  6. 6.
    Kashyap N.: A decomposition theory for binary linear codes. IEEE Trans. Inform. Theory 54(7), 3035–3058 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kelley C., Sridhara D.: Pseudocodewords of Tanner graphs. IEEE Trans. Inform. Theory 53(11), 4013–4038 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Koetter R., Li W.-C.W., Vontobel P.O., Walker J.: Characterizations of pseudo-codewords of (low-density) parity-check codes. Adv. Math. 213(1), 205–229 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kositwattanarerk W., Matthews G.L.: Lifting the fundamental cone and enumerating the pseudocodewords of a parity-check code. IEEE Trans. Inform. Theory 57(2), 898–909 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kou Y., Lin S., Fossorier M.P.C.: Low-density parity-check codes based on finite geometries: a rediscovery and new results. IEEE Trans. Inform. Theory 47(7), 2711–2736 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kschischang F.R., Frey B.J., Loeliger H.-A.: Factor graphs and the sum-product algorithm. IEEE Trans. Inform. Theory 47(2), 498–519 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lechner G.: The effect of cycles on binary message-passing decoding of LDPC codes. In: Proc. Comm. Theory Workshop, Australia, IEEE (2010).Google Scholar
  13. 13.
    MacKay D.J.C., Neal R.M.: Near Shannon limit performance of low density parity check codes. Electron. Lett. 32, 1645–1646 (1996).CrossRefGoogle Scholar
  14. 14.
    Richardson T., Shokrollahi A., Urbanke R.: Design of capacity-approaching irregular low-density parity-check codes. IEEE Trans. Inform. Theory 47(2), 619–637 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Seymour P.D.: Decomposition of regular matroids. J. Comb. Theory Ser. B 52, 305–359 (1980).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Tian T., Jones C.R., Villasenor J.D., Wesel R.D.: Selective avoidance of cycles in irregular LDPC code construction. IEEE Trans. Commun. 52, 1242–1247 (2004).CrossRefGoogle Scholar
  17. 17.
    Wiberg N.: Codes and decoding on general graphs. Ph.D. thesis, Linköping University, Linköping (1996).Google Scholar
  18. 18.
    Xia S.-T., Fu F.-W.: Minimum pseudoweight and minimum pseudocodewords of LDPC codes. IEEE Trans. Inform. Theory 54, 480–485 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Xu J., Chen L., Djurdjevic I., Lin S., Abdel-Ghaffar K.: Construction of regular and irregular LDPC codes: geometry decomposition and masking. IEEE Trans. Inform. Theory 53, 121–134 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zumbragel J., Skachek V., Flanagan M.F.: On the pseudocodeword redundancy of binary linear codes. IEEE Trans. Inform. Theory 58, 4848–4861 (2012).MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceMahidol UniversityBangkokThailand
  2. 2.Centre of Excellence in MathematicsThe Commission on Higher EducationBangkokThailand

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