Low Density Parity Check Codes

  • Giovanni CancellieriEmail author
Part of the Signals and Communication Technology book series (SCT)


The Tanner graph is described. A girth in the Tanner graph is equivalent to a short cycle of 1- symbols in the parity check matrix. The condition called row-column constraint is introduced, in order to allow practical decoding procedures. They are based on the sum-product algorithm, which is briefly outlined. Regular and irregular LDPC codes are introduced. The firstly proposed LDPC codes by Gallager and MacKay-Neal are reviewed. The following main families of more recent LDPC codes are briefly described: codes based on protographs, codes constructed employing repeat and accumulation devices, codes derived from the decomposition of finite geometries, codes obtained starting from MDS codes. Furthermore codes obtained from superimposition or by circulant expansion are analysed. Masking and row or column splitting can be employed for reducing the 1-symbol density and breaking short cycles. The effects of such procedures on the code rate are stressed. For instance, circulant expansion and masking do not vary the code rate, whereas row (column) splitting increases (reduces) it. Irregular LDPC codes can be designed by means of proper rules progressively adding edges to the Tanner graph. A statistical treatment, called density evolution, is presented, in order to obtain asymptotic best performance at different code rates, taking into account the intrinsic nature of the sum-product decoding algorithm. A first approach to LDPC convolutional codes is presented, starting from good known LDPC block codes. The use of unwrapping is suggested, in particular with suitable QC codes, or a derivation from properly modified H-extension is reviewed.


Code Rate LDPC Code Turbo Code Convolutional Code Variable Node 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Abbasfar A, Divsalar D, Yao K (2007) Accumulate-repeat-accumulate codes. IEEE Trans Commun 55:692–702CrossRefGoogle Scholar
  2. Ashikhmin A, Kramer G, Ten Brink S (2004) Extrinsic information transfer functions: model and erasure channel properties. IEEE Trans Inf Theory 50:2657–2673CrossRefzbMATHGoogle Scholar
  3. Baldi M, Chiaraluce F (2008) A simple scheme for belief propagation decoding of BCH and RS codes in multimedia transmissions. Int J Dig Multim Broadcas 8:ID957846Google Scholar
  4. Baldi M, Bianchi M, Cancellieri G et al (2012) On the generator matrix of array codes. In: Proceedings of Softcom 2012, Split (Croatia)Google Scholar
  5. Baldi M, Cancellieri G, Chiaraluce F (2014) Array convolutional low-density parity-check codes. IEEE Commun Lett 18:336–339CrossRefGoogle Scholar
  6. Chen J, Tanner RM, Jones C, Li Y (2005) Improved min-sum decoding algorithms for irregular LDPC codes. In: Proceedings of international symposium on information theory, 2005, Adelaide (Australia), pp 449–453Google Scholar
  7. Costello DJ, Pusane AE, Jones CR, Divsalar D (2007) A comparison of ARA- and protograph-based LDPC block and convolutional codes. In: Information theory and applications workshop, San Diego (CA)Google Scholar
  8. Divsalar D, Jin H, McEliece R (1998) Coding theorems for turbo-like codes. In: Proceedings of 36th annual allerton conference on communication, Monticello (IL), pp 201–210Google Scholar
  9. Divsalar D, Jones C, Dolinar S, Thorpe J (2005) Protograph based LDPC codes with minimum distance linearly growing with block size. In: Proceedings of IEEE global telecommunications conference, St. Louis (MI), pp 1152–1156Google Scholar
  10. Djurdjevic I, Xu J, Abdel-Ghaffar K, Lin S (2003) A class of low-density parity-check codes constructed based on Reed-Solomon codes with two information symbols. IEEE Commun Lett 7:317–319CrossRefGoogle Scholar
  11. Esmaeli M, Tadayon MH, Gulliver TA (2011) More on the stopping and minimum distances of array codes. IEEE Trans Commun 59:750–757CrossRefGoogle Scholar
  12. Fossorier MPC (2004) Quasi-cyclic low-density parity check codes from circulant permutation matrices. IEEE Trans Inf Theory 50:1788–1793CrossRefzbMATHMathSciNetGoogle Scholar
  13. Gallager RG (1962) Low-density parity-check codes. IRE Trans Inf Theory 8:21–28CrossRefzbMATHMathSciNetGoogle Scholar
  14. Halford TR, Grant AJ, Chugg KM (2006) Which codes have 4-cycle-free Tanner graphs? IEEE Trans Inf Theory 52:4219–4223CrossRefzbMATHMathSciNetGoogle Scholar
  15. Han Y, Ryan WE (2009) Low-floor decoders for LDPC codes. IEEE Trans Commun 57:1663–1773CrossRefGoogle Scholar
  16. Hu XY, Eleftheriou E, Arnold DM (2001) Progressive edge-growth Tanner graphs. In: Proceedings on 2001 IEEE global telecommunications conference, San Antonio (TX), pp 995–1001Google Scholar
  17. Hu XY, Eleftheriou E, Arnold DM (2005) Regular and irregular progressive edge-growth Tanner graphs. IEEE Trans Inf Theory 51:386–398CrossRefzbMATHMathSciNetGoogle Scholar
  18. Jacobsen N, Soni R (2007) Design of rate compatible irregular LDPC codes based on edge growth and parity splitting. In: Proceedings IEEE 66th vehicular technology conference, Baltimore (MD), pp 1052–1056Google Scholar
  19. Jimenez-Felstrom AJ, Zigangirov KS (1999) Time-varying periodic convolutional codes with low density parity-check matrix. IEEE Trans Inf Theory 45:2181–2191CrossRefMathSciNetGoogle Scholar
  20. Jin H, Khandekar A, McEliece R (2000) Irregular repeat-accumulate codes. In: Proceedings of 2nd international conference on turbo codes, Brest (France), pp 1–8Google Scholar
  21. Johnson SJ, Weller SR (2004) Codes for iterative decoding from partial geometries. IEEE Trans Commun 52:236–243CrossRefGoogle Scholar
  22. Kamiya N (2007) High-rate quasi-cyclic low-density parity-check codes derived from finite affine planes. IEEE Trans Inf Theory 53:1444–1459CrossRefMathSciNetGoogle Scholar
  23. Kou Y, Lin S, Fossorier MPC (2001) Low-density parity-check codes based on finite geometries: a rediscovery and new results. IEEE Trans Inf Theory 47:2711–2736CrossRefzbMATHMathSciNetGoogle Scholar
  24. Kschischang F, Frey B, Loeliger HA (2001) Factor graphs and the sum-product algorithm. IEEE Trans Inf Theory 47:485–519CrossRefMathSciNetGoogle Scholar
  25. Lin S, Costello DJ (2004) Error control coding. Pearson/Prentice-Hall, Upper Saddle RiverGoogle Scholar
  26. MacKay DJC (1999) Good error-correcting codes based on very sparse matrices. IEEE Trans Inf Theory 45:399–431CrossRefzbMATHMathSciNetGoogle Scholar
  27. MacKay DJC, Davey M (1999) Evaluation of Gallager codes for short block length and high rate applications. In: Proceedings of IMA Workshop codes systems and graphical models, Minneapolis (MN), pp 113–130Google Scholar
  28. MacKay DJC, Neal RM (1995) Good codes based on very sparse matrices. In: Boyd C (ed) Proceedings of cryptography and coding 5th IMA conference. Lecture notes computer science. Springer, Berlin, pp 110–111Google Scholar
  29. MacKay DJC, Neal RM (1997) Near Shannon limit performance of low density parity check codes. Electron Lett 33:457–458CrossRefGoogle Scholar
  30. Milenkovic O, Kashyap N, Leyba D (2008) Shortened array codes of large girth. IEEE Trans Inf Theory 52:3707–3722CrossRefMathSciNetGoogle Scholar
  31. Mitchell DGM, Pusane AE, Zigangirov KS, Costello DJ (2008) Asymptotically good LDPC convolutional codes based on protographs. Proc ISIT, Toronto (Canada), pp 1030–1034Google Scholar
  32. Mittelholzer T (2002) Efficient encoding and minimum distance bound for Reed-Solomon-type array codes. In: Proceedings of ISIT 2002, Lousanne (Switzerland), p 282Google Scholar
  33. Oenning TR, Moon J (2001) A low-density generator matrix interpretation of parallel concatenated single bit parity check codes. IEEE Trans Magn Rec 37:737–741CrossRefGoogle Scholar
  34. Papoulis A (1965) Probability, random variables, and stochastic processes. McGraw Hill, New YorkzbMATHGoogle Scholar
  35. Pearl J (1988) Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kauffman, San Mateo (CA)Google Scholar
  36. Richardson T (2003) Error floors of LDPC codes. In: Proceedings of 41th annual, allerton conference on communications, Monticello (IL), pp 1426–1435Google Scholar
  37. Richardson T, Urbanke R (2001) The capacity of low-density parity-check codes under message-passing decoding. IEEE Trans Inf Theory 47:599–618CrossRefzbMATHMathSciNetGoogle Scholar
  38. Richardson T, Urbanke R (2008) Modern coding theory. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  39. Richardson J, Shokrollahi A, Urbanke R (2001) Design of capacity-approaching irregular low-density parity-check codes. IEEE Trans Inf Theory 47:2001MathSciNetGoogle Scholar
  40. Richter G (2006) Finding small stopping sets in the Tanner graphs of LDPC codes. In: Proceedings of 4th international symposium turbo codes, Munich (Germany)Google Scholar
  41. Rudolph LD (1967) A class of majority logic decodable codes. IEEE Trans Inf Theory 13:305–307CrossRefzbMATHGoogle Scholar
  42. Ryan WE, Lin S (2009) Channel codes: classical and modern. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  43. Tanner RM (1981) A recursive approach to low complexity codes. IEEE Trans Inf Theory 27:533–547CrossRefzbMATHMathSciNetGoogle Scholar
  44. Tanner RM, Sridhara D, Sridharan A et al (2004) LDPC block and convolutional codes based on circulant matrices. IEEE Trans Inf Theory 50:2966–2984CrossRefzbMATHMathSciNetGoogle Scholar
  45. Ten Brink S (2001) Convergence behaviour of iteratively decoded parallel concatenated codes. IEEE Trans Commun 49:1727–1737CrossRefzbMATHGoogle Scholar
  46. Ten Brink S, Kramer G (2003) Design of repeat-accumulate codes for iterative detection and decoding. IEEE Trans Sig Proc 51:2764–2772CrossRefGoogle Scholar
  47. Thorpe J (2003) Low-density parity-check (LDPC) codes constructed from protographs. JPL INP Prog Rep 42(154):42–154Google Scholar
  48. Tian T, Jones G, Villasenor JD, Wesel RD (2004) Selective avoidance of cycles in irregular LDPC code construction. IEEE Trans Commun 52:1242–1247CrossRefGoogle Scholar
  49. Wang CL, Fossorier MPC (2009) On asymptotic ensemble weight enumerators of LDPC-like codes. IEEE J Sel Areas Commun 27:899–907CrossRefGoogle Scholar
  50. Xu J, Chen L, Zeng LQ et al (2005) Construction of low-density parity-check codes by superimposition. IEEE Trans Commun 53:243–251CrossRefGoogle Scholar
  51. Zhang Y, Ryan WE (2009) Toward low LDPC-code floor: a case study. IEEE Trans Commun 57:1566–1573CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Information EngineeringPolytechnic University of MarcheAnconaItaly

Personalised recommendations