Journal of Communications and Networks

  • 제17권4호
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  • Pages.328-338
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  • 2015
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  • 1229-2370(pISSN)
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  • 1976-5541(eISSN)
한국통신학회 (The Korean Institute of Commucations and Information Sciences)
권호 표지

Challenges and Some New Directions in Channel Coding

  • 투고 : 2014.12.15
  • 발행 : 2015.08.31
⟨ 이전 논문 다음 논문 ⟩

초록

Three areas of ongoing research in channel coding are surveyed, and recent developments are presented in each area: Spatially coupled low-density parity-check (LDPC) codes, nonbinary LDPC codes, and polar coding.

키워드

참고문헌

  1. C. E. Shannon, "A mathematical theory of communications," Bell Syst. Tech. J., vol. 27, pp. 379-423, July 1948. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
  2. M. J. E. Golay, "Notes on digital coding," Proc. IRE, vol. 37, p. 657, 1949.
  3. R. W. Hamming, "Error detecing and error correcting codes," Bell Syst. Tech. J., vol. 29, pp. 147-160, 1950. https://doi.org/10.1002/j.1538-7305.1950.tb00463.x
  4. I. Reed and G. Solomon, "Polynomial codes over certain finite fields," J. Soc. Indust. Appl. Math., vol. 8, June 1960.
  5. P. Elias, "Coding for noisy channels," IRE Convention Record, Part 4, pp. 37-46, 1955.
  6. A. J. Viterbi, "Error bounds for convolutional codes and an asymptotically optimum decoding algorithm," IEEE Trans. Inf. Theory, vol. 13, pp. 260-269, Apr. 1960.
  7. G. Forney, "The Viterbi algorithm," Proc. IEEE, vol. 61, pp. 258-278, Mar. 1973.
  8. L. Bahl et al., "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory, pp. 284-287, Mar. 1974.
  9. G. Ungerboeck, "Channel coding with multilevel/phase signals," IEEE Trans. Inf. Theory, pp. 55-67, Jan. 1982.
  10. C. Berrou, A. Glavieux, and P. Thitimajshima, "Near shannon limimt error correcting coding and decoding: Turbo-codes," in Proc. IEEE Int. Conf. Commun. (ICC), (Geneva, Switzerland), vol.2, May 1993, pp. 1064-1070.
  11. R. G. Gallager, Low Density Parity Check Codes. Ph.D. thesis,Massachussetts Institute of Technology (MIT), Cambridge, Mass., 1963.
  12. D. J. C. MacKay, "Good error-correcting codes based on very sparse matrices," IEEE Trans. Inf. Theory, vol. 45, pp. 399-431, Mar. 1999. https://doi.org/10.1109/18.748992
  13. J. L. Massey, "Coding and modulation in digital communications," in Proc. Int. Zurich Seminar on Communications (IZS), (Zurich, Switzerland), 1974.
  14. E. Arikan, "Channel polarization: A method for constructing capacityachieving codes for symmetric binary-input memoryless channels," IEEE Trans. Inf. Theory, vol. 55, pp. 3051-3073, July 2009. https://doi.org/10.1109/TIT.2009.2021379
  15. T. Richardson, M. Shokrollahi, and R. Urbanke, "Design of capacityapproaching irregular low-density parity-check codes," IEEE Trans. Inf. Theory, vol. 47, pp. 619-637, Feb. 2001. https://doi.org/10.1109/18.910578
  16. A. Jimenez Felstrom and K. Zigangirov, "Time-varying periodic convolutional codes with low-density parity-check matrix," IEEE Trans. Inf. Theory, vol. 45, pp. 2181 -2191, Sept. 1999. https://doi.org/10.1109/18.782171
  17. J. Thorpe, "Low-density parity-check (LDPC) codes constructed from protographs," IPN Progress Rep. 42-154, JPL, Aug. 2003.
  18. M. Lentmaier et al., "Approaching capacity with asymptotically regular LDPC codes," in Proc. Inf. Theory and Applications Workshop, (San Diego, CA, USA), Feb. 2009, pp. 173 -177.
  19. M. Lentmaier et al., "Iterative decoding threshold analysis for LDPC convolutional codes," IEEE Trans. Inf. Theory, vol. 56, pp. 5274 -5289, Oct. 2010. https://doi.org/10.1109/TIT.2010.2059490
  20. S. Kudekar, T. Richardson, and R. Urbanke, "Threshold saturation via spatial coupling: Why convolutional LDPC ensembles perform so well over the BEC," IEEE Trans. Inf. Theory, vol. 57, pp. 803-834, Feb. 2011. https://doi.org/10.1109/TIT.2010.2095072
  21. S. Kudekar, T. Richardson, and R. Urbanke, "Spatially coupled ensembles universally achieve capacity under belief propagation," in Proc. IEEE Int. Symp. Inform. Theory (ISIT), (Cambridge, MA, USA), July 2012, pp. 453-457.
  22. S. Kumar et al., "A proof of threshold saturation for spatially-coupled LDPC codes on BMS channels," in Proc. Allerton Conf. on Communications, Control, and Computing, Oct. 2012, pp. 176-184.
  23. C. Measson and R. Urbanke, "Maximum a posteriori decoding and turbo codes for general memoryless channels," in Proc. IEEE Int. Symp. Inform. Theory (ISIT), (Adelaide, Australia), Sept. 2005, pp. 1241-1245.
  24. D.Mitchell, M. Lentmaier, and D. Costello, "New families of LDPC block codes formed by terminating irregular protograph-based LDPC convolutional codes," in Proc. IEEE Int. Symp. Inform. Theory (ISIT), (Austin, Texas, USA), June 2010, pp. 824-828.
  25. W. Nitzold, G. Fettweis, and M. Lentmaier, "Spatially-coupled nearlyregular LDPC code ensembles for rate-flexible code design," in Proc. IEEE Int. Conf. Commun. (ICC), (Sydney, Australia), June 2014, pp. 2027-2032.
  26. A. R. Iyengar et al., "Windowed decoding of protograph-based LDPC convolutional codes over erasure channels," IEEE Trans. Inf. Theory, vol. 58, pp. 2303-2320, Apr. 2012. https://doi.org/10.1109/TIT.2011.2177439
  27. N. Ul Hassan, M. Lentmaier, and G. Fettweis, "Comparison of LDPC block and LDPC convolutional codes based on their decoding latency," in Proc. Int. Symp. on Turbo Codes & Iterative Inf. Proc., (Gothenburg, Sweden), Aug. 2012.
  28. T. Hehn and J. Huber, "LDPC codes and convolutional codes with equal structural delay: a comparison," IEEE Trans. Commun., vol. 57, pp. 1683- 1692, June 2009. https://doi.org/10.1109/TCOMM.2009.06.080014
  29. S. Maiya et al., "Coding with a latency constraint: The benefits of sequential decoding," in Proc. Allerton Conf. on Communications, Control, and Computing, Oct. 2010, pp. 201-207.
  30. E. Sharon, N. Presman, and S. Litsyn, "Convergence analysis of generalized serial message-passing schedules," IEEE J. Sel. Areas Commun., vol. 27, pp. 1013-1024, Aug. 2009. https://doi.org/10.1109/JSAC.2009.090819
  31. N. Ul Hassan et al., "Reduced complexity window decoding schedules for coupled LDPC codes," in Proc. IEEE Inform. Theory Workshop (ITW), (Lausanne, Switzerland), Sept. 2012, pp. 20-24.
  32. N. Ul Hassan et al., "Non-uniform windowed decoding schedules for spatially coupled codes," in Proc. IEEE Glob. Comm. Conf. (GLOBECOM), (Atlanta, GA, USA), Dec. 2013, pp. 1862-1867.
  33. L. Ozarow, S. Shamai (Shitz), and A. Wyner, "Information theoretic considerations for cellular mobile radio," IEEE Trans. Veh. Technol., vol. 43, no. 2, pp. 359-378, 1994. https://doi.org/10.1109/25.293655
  34. E. Biglieri, G. Caire, and G. Taricco, "Coding for the fading channel: a survey," Signal Processing (EURASIP), vol. 80, no. 7, pp. 1135-1148, 2000. https://doi.org/10.1016/S0165-1684(00)00027-X
  35. J. Boutros et al., "Low-density parity-check codes for nonergodic blockfading channels," IEEE Trans. Inf. Theory, vol. 56, no. 9, pp. 4286-4300, 2010. https://doi.org/10.1109/TIT.2010.2053890
  36. N. Ul Hassan et al., "Improving code diversity on block-fading channels by spatial coupling," in Proc. IEEE Int. Symp. Inform. Theory (ISIT), (Honolulu, Hawaii, USA), pp. 2311-2315, June 2014.
  37. M. C. Davey and D. J. C. MacKay, "Low density parity check codes over GF(q)," in Proc. IEEE Inform. Theory Workshop (ITW), (Killarney, Ireland), June 1998, pp. 70-71.
  38. D. Declercq and M. P. Fossorier, "Decoding algorithms for nonbinary LDPC codes over GF(q)," IEEE Trans. Commun., vol. 55, pp. 633-643, Apr. 2007. https://doi.org/10.1109/TCOMM.2007.894088
  39. A. Voicila et al., "Low-complexity decoding for non-binary LDPC codes in high order fields," IEEE Trans. Commun., pp. 1365-1375, May 2010.
  40. E. Boutillon and L. Conde-Canencia, "Bubble check: a simplified algorithm for elementary check node processing in extended min-sum nonbinary LDPC decoders," IET Electronic Lett., vol. 46, pp. 633-634, Apr. 2010. https://doi.org/10.1049/el.2010.0566
  41. G. Montorsi, "Analog digital belief propagation," IEEE Commun. Lett., vol. 16, no. 7, pp. 1106-1109, 2012. https://doi.org/10.1109/LCOMM.2012.020712.112133
  42. J. Sayir, "Non-binary LDPC decoding using truncated messages in the walsh-hadamard domain," in Proc. Int. Symp. Inf. Theory and Its App. (ISITA), (Melbourne, Australia), Oct. 2014.
  43. A. Bennatan and D. Burshtein, "On the application of LDPC codes to arbitrary discrete memoryless channels," IEEE Trans. Inf. Theory, vol. 50, pp. 417-438, Mar. 2004. https://doi.org/10.1109/TIT.2004.824917
  44. U. Erez and G. Miller, "The ML decoding performance of LDPC ensembles over $Z_q$," IEEE Trans. Inf. Theory, vol. 51, no. 5, pp. 1871-1879, 2005. https://doi.org/10.1109/TIT.2005.846431
  45. H.Wymeersch, H. Steendam, and M.Moeneclaey, "Log-domain decoding of LDPC codes over GF(q)," in Proc. IEEE Int. Conf. Commun. (ICC), vol. 2, (Paris, France), vol. 2, 2004, pp. 772-776.
  46. A. Voicila et al., "Architecture of a low-complexity non-binary LDPC decoder for high order fields," in Communications and Information Technologies, 2007. ISCIT '07. International Symposium on, 2007, pp. 1201-1206.
  47. V. Savin, "Min-Max decoding for non binary LDPC codes," in Proc. IEEE Int. Symp. Inform. Theory (ISIT), (Toronto, Canada), 2008, pp. 960-964.
  48. C. Spagnol, E. Popovici, and W. Marnane, "Hardware Implementation of GF(q) LDPC Decoders," IEEE Trans. Circuits Syst. I, vol. 56, no. 12, pp. 2609-2620, 2009. https://doi.org/10.1109/TCSI.2009.2016621
  49. J. Lin et al., "An Efficient VLSI architecture for nonbinary LDPC decoders," IEEE Trans. Circuits Syst. II, vol. 57, no. 1, pp. 51-55, 2010. https://doi.org/10.1109/TCSII.2009.2036542
  50. C. Zhang and K. Parhi, "A network-efficient nonbinary QC-LDPC decoder architecture," IEEE Trans. Circuits Syst. I, vol. 59, no. 6, pp. 1359-1371, 2012. https://doi.org/10.1109/TCSI.2011.2177001
  51. X. Chen, S. Lin, and V. Akella, "Efficient configurable decoder architecture for nonbinary quasi-cyclic LDPC codes," IEEE Trans. Circuits Syst. I, vol. 59, no. 1, pp. 188-197, 2012. https://doi.org/10.1109/TCSI.2011.2161416
  52. X. Chen and C.-L. Wang, "High-throughput efficient non-binary LDPC decoder based on the simplified min-sum algorithm," IEEE Trans. Circuits Syst. I, vol. 59, no. 11, pp. 2784-2794, 2012. https://doi.org/10.1109/TCSI.2012.2190668
  53. Y.-L. Ueng et al., "An efficient layered decoding architecture for nonbinary QC-LDPC codes," IEEE Trans. Circuits Syst. I, vol. 59, no. 2, pp. 385-398, 2012. https://doi.org/10.1109/TCSI.2011.2163889
  54. Y.-L. Ueng et al., "A high-throughput trellis-based layered decoding architecture for non-binary LDPC codes using Max-Log-QSPA," IEEE Trans. Signal Process., vol. 61, no. 11, pp. 2940-2951, 2013. https://doi.org/10.1109/TSP.2013.2256905
  55. T. P. Minka, "Expectation propagation for approximate bayesian inference," in Proc. AUAI, Morgan Kaufmann Publishers Inc., 2001, pp. 362-369.
  56. G. Montorsi, "Analog digital belief propagation: from theory to practice," in Proc. IEEE Int. Conf. Commun. (ICC), 2012, pp. 2591-2595.
  57. M. Awais Aslam et al., "VLSI implementation of a non-binary decoder based on the analog digital belief propagation," IEEE Trans. Signal Process., pp. 3965-3975, July 2014.
  58. E. Arikan and E. Telatar, "On the rate of channel polarization," in Proc. IEEE Int. Symp. Inform. Theory (ISIT), (Seoul, South Korea), June 28-July 3, 2009, pp. 1493-1495.
  59. S. H. Hassani et al., "Rate-dependent analysis of the asymptotic behavior of channel polarization," IEEE Trans. Inf. Theory, vol. 59, no. 4, pp. 2267-2276, 2013. https://doi.org/10.1109/TIT.2012.2228295
  60. R. Mori and T. Tanaka, "Performance of polar codes with the construction using density evolution," IEEE Commun. Lett., vol. 13, pp. 519-521, July 2009. https://doi.org/10.1109/LCOMM.2009.090428
  61. I. Tal and A. Vardy, "How to construct polar codes," IEEE Trans. Inf. Theory, vol. 59, pp. 6562-6582, Oct. 2013. https://doi.org/10.1109/TIT.2013.2272694
  62. R. Pedarsani et al., "On the construction of polar codes," in Proc. IEEE Int. Symp. Inform. Theory (ISIT), (St. Petersburg, Russia), 2011, pp. 11-15.
  63. N. Hussami, S. B. Korada, and R. Urbanke, "Performance of polar codes for channel and source coding," in Proc. IEEE Int. Symp. Inform. Theory (ISIT), (Seoul, South Korea), 2009, pp. 1488-1492.
  64. S. B. Korada and R. Urbanke, "Polar codes for slepian-wolf, wyner-ziv, and gelfand-pinsker," in Proc. IEEE Inform. Theory Workshop (ITW), (Cairo, Egypt), 2010.
  65. E. Sasoglu, E. Telatar, and E. Yeh, "Polar codes for the two-user binaryinput multiple-access channel," in Proc. IEEE Inform. Theory Workshop (ITW), (Cairo, Egypt), 2010, pp. 1-5.
  66. S. B. Korada et al., "An empirical scaling law for polar codes," in Proc. IEEE Int. Symp. Inform. Theory (ISIT), (Austin, Texas, USA), 2010, pp. 884-888.
  67. E. Abbe and I. Telatar, "Mac polar codes and matroids," in Proc. Inf. Theory and Applications Workshop, (San Diego, CA, USA), 2010, pp. 1-8.
  68. E. Sasoglu, I. Telatar, and E. Arikan, "Polarization for arbitrary discrete memoryless channels," in Proc. IEEE Inform. Theory Workshop (ITW), (Taormina, Italy), 2009, pp. 144-148.
  69. M. Karzand and I. Telatar, "Polar codes for q-ary source coding," in Proc. IEEE Int. Symp. Inform. Theory (ISIT), (Austin, TX, USA), 2010, pp. 909-912.
  70. W. Park and A. Barg, "Polar codes for q-ary channels," IEEE Trans. Inf. Theory, vol. 59, no. 2, pp. 955-969, 2013. https://doi.org/10.1109/TIT.2012.2219035
  71. A. G. Sahebi and S. S. Pradhan, "Multilevel polarization of polar codes over arbitrary discrete memoryless channels," in Proc. Allerton Conf. on Communications, Control, and Computing, 2011, pp. 1718-1725.
  72. S. B. Korada, E. Sasoglu, and R. Urbanke, "Polar codes: Characterization of exponent, bounds, and constructions," IEEE Trans. Inf. Theory, vol. 56, no. 12, pp. 6253-6264, 2010. https://doi.org/10.1109/TIT.2010.2080990
  73. R. Mori and T. Tanaka, "Channel polarization on q-ary discrete memoryless channels by arbitrary kernels," in Proc. IEEE Int. Symp. Inform. Theory (ISIT), (Austin, TX, USA), 2010, pp. 894-898.
  74. R. Mori and T. Tanaka, "Source and channel polarization over finite fields and reed-solomon matrix," arXiv preprint arXiv:1211.5264, 2012.
  75. N. Presman, O. Shapira, and S. Litsyn, "Binary polar code kernels from code decompositions," in Proc. IEEE Int. Symp. Inform. Theory (ISIT), (St. Petersburg, Russia), 2011, pp. 179-183.
  76. N. Presman, O. Shapira, and S. Litsyn, "Polar codes with mixed kernels," in Proc. IEEE Int. Symp. Inform. Theory (ISIT), (St. Petersburg, Russia), pp. 6-10, IEEE, 2011.
  77. S. H. Hassani, S. B. Korada, and R. Urbanke, "The compound capacity of polar codes," arXiv preprint arXiv:0907.3291, 2009.
  78. E. Sasoglu, Polar coding theorems for discrete systems. Ph.D. thesis, EPFL, 2011.
  79. E. Sasoglu and L. Wang, "Universal polarization," arXiv preprint arXiv:1307.7495, 2013.
  80. S. H. Hassani and R. Urbanke, "Universal polar codes," arXiv preprint arXiv:1307.7223, 2013.
  81. D. J. Costello and G. D. Forney Jr, "Channel coding: The road to channel capacity," Proc. IEEE, vol. 95, no. 6, pp. 1150-1177, 2007. https://doi.org/10.1109/JPROC.2007.895188
  82. M. Plotkin, "Binary codes with specified minimum distance," IRE Trans. Inform. Theory, vol. 6, pp. 445-450, Sept. 1960. https://doi.org/10.1109/TIT.1960.1057584
  83. M. Bakshi, S. Jaggi, and M. Effros, "Concatenated polar codes," in Proc. IEEE Int. Symp. Inform. Theory (ISIT), (Austin, TX, USA), 2010, pp. 918-922.
  84. E. Arikan and G. Markarian, "Two-Dimensional Polar Coding," in Proc. Int. Symp. on Commun. Theory and App. (ISCTA), (Ambleside, U.K.), July 2009.
  85. H. Mahdavifar et al., "Performance limits and practical decoding of interleaved reed-solomon polar concatenated codes," IEEE Trans. Commun., vol. 62, no. 5, pp. 1406-1417, 2014. https://doi.org/10.1109/TCOMM.2014.050714.130602
  86. P. Trifonov and P. Semenov, "Generalized concatenated codes based on polar codes," in Proc. Int. Symp. on Wireless Comm. Systems (ISWCS), (Aachen, Germany), 2011, pp. 442-446.
  87. E. Arikan, "A performance comparison of polar codes and Reed-Muller codes," IEEE Commun. Lett., vol. 12, pp. 447-449, June 2008. https://doi.org/10.1109/LCOMM.2008.080017
  88. E. Arikan et al., "Performance of short polar codes under ML decoding," in Proc. ICT MobileSummit, (Santander, Spain), June 2009.
  89. Y. S. Park et al., "A 4.68gb/s belief propagation polar decoder with bitsplitting register file," in VLSI Circuits Digest of Technical Papers, 2014 Symposium on, June 2014, pp. 1-2.
  90. I. Tal and A. Vardy, "List decoding of polar codes," in Proc. IEEE Int. Symp. Inform. Theory (ISIT), (St. Petersburg, Russia), 2011, pp. 1-5.
  91. I. Dumer and K. Shabunov, "Soft-decision decoding of reed-muller codes: recursive lists," IEEE Trans. Inf. Theory, vol. 52, no. 3, pp. 1260-1266, 2006. https://doi.org/10.1109/TIT.2005.864443
  92. E. Arikan, "A survey of reed-muller codes from polar coding perspective," in Proc. IEEE Inform. Theory Workshop (ITW), (Cairo, Egypt), Jan 2010, pp. 1-5.
  93. The Coded Modulation Library [Online]. Available: http://www.iterativesolutions.com/Matlab.htm