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Binary Codes, Graphs, and Trellises

  • Chris Heegard
  • Stephen B. Wicker
Chapter
Part of the The Springer International Series in Engineering and Computer Science book series (SECS, volume 476)

Abstract

In this chapter we focus on the structure and description of binary convolutional codes and their encoders. Both algebraic and graph-based methods are used to develop a generic description of the codes. It is shown that both recursive and feedforward encoders can be used to generate the same convolutional code. The distinctions between various encoders for a given code are discussed, and an emphasis is placed on the properties of recursive, systematic convolutional encoders. It is shown that such descriptive and analytic techniques can also be applied to block codes through the BCJR trellis construction technique.

Keywords

Generator Matrix Binary Code Finite Impulse Response Finite State Machine Turbo Code 
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.

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Bibliography

  1. [BCJR74]
    L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv. Optimal decoding of linear codes for minimizing symbol error rate. IEEE Transactions on Information Theory IT-20:284–287, 1974.MathSciNetMATHCrossRefGoogle Scholar
  2. [Ber80]
    E. R. Berlekamp. The technology of error-correcting codes. Proceedings of the IEEE 68:564–593, 1980.CrossRefGoogle Scholar
  3. [FBEI97]
    G. D. Forney Jr., L. Brown, M. V. Eyuboglu, and J. L. Moran III. The v.34 high-speed modem standard. IEEE Communications Magazine 34(12):28–33, 1997.CrossRefGoogle Scholar
  4. [FJxW96]
    G. D. Forney Jr., R. Johannesson, and Z. X. Wan. Minimal and canonical rational generator matrices for convo-lutional codes. IEEE Transactions on Information Theory IT-42(6):1865–1880, 1996.MathSciNetMATHCrossRefGoogle Scholar
  5. [For70]
    G. D. Forney Jr.. Convolutional codes I: Algebraic structure. IEEE Transactions on Information Theory IT-16(6):720–738, 1970f.MathSciNetMATHCrossRefGoogle Scholar
  6. [GCCC81]
    G. C. Clark Jr. and J. B. Cain. Error-Correction Coding for Digital Communications. New York: Plenum Press, 1981.Google Scholar
  7. [HamSO]
    R. W. Hamming. Error Detecting and Error Correcting Codes. Bell System Technical Journal 29:147–160, 1950.MathSciNetGoogle Scholar
  8. [Ksh96]
    F. R. Kshischang. The trellis structure of maximal fixed-cost codes. IEEE Transactions on Information Theory IT-42(6):1828–1839, 1996.CrossRefGoogle Scholar
  9. [LDJC83]
    S. Lin and D. J. Costello, Jr.. Error Control Coding: Fundamentals and Applications. Englewood Cliffs: Prentice Hall, 1983.Google Scholar
  10. [LM96]
    H.-A. Loeliger and T. Mittleholzer. Convolutional codes over groups. IEEE Transactions on Information Theory IT-42(6):1660–1686, 1996.MATHCrossRefGoogle Scholar
  11. [LV96]
    A. Lafourcade and A. Vardy. Optimal sectionalization of a trellis. IEEE Transactions on Information Theory pages 689–703, 1996.Google Scholar
  12. [Mas78]
    J. L. Massey. Foundations and methods of channel encoding. In Proceedings of the International Conference on Information Theory and Systems Berlin, Germany, 1978.Google Scholar
  13. [McE96]
    R. J. McEliece. On the BCJR trellis for linear block codes. IEEE Transactions on Information Theory IT-42(4):1072–1092, 1996.MathSciNetMATHCrossRefGoogle Scholar
  14. [ML96]
    R. J. McEliece and W. Lin. The trellis complexity of convo-lutional codes. IEEE Transactions on Information Theory IT-42(6):1855–1864, 1996.MathSciNetMATHCrossRefGoogle Scholar
  15. [MS68]
    J. L. Massey and M. K. Sain. Inverses of linear sequential circuits. IEEE Transactions on Communications COM-17(4):330–337, 1968.MATHCrossRefGoogle Scholar
  16. [MW86]
    H. H. Ma and J. K. Wolf. On tail biting convolutional codes. IEEE Transactions on Communications COM-34(2):104–110, 1986.MATHCrossRefGoogle Scholar
  17. [Pir88]
    P. Piret. Convolutional Codes: An Algebraic Approach Cambridge: MIT Press, 1988.MATHGoogle Scholar
  18. [Sha48]
    C. E. Shannon. A mathematical theory of communication. Bell System Technical Journal 27:379–423, 623–656, October 1948.MathSciNetMATHGoogle Scholar
  19. [VK96]
    A. Vardy and F. R. Kshischang. Proof of a conjecture of McEliece regarding the expansion index of the minimal trellis. IEEE Transactions on Information Theory IT-42(6):2027–2034, 1996.MATHCrossRefGoogle Scholar
  20. [Wei8 7]
    L.-F. Wei. Trellis-coded modulation with multidimensional constellations. IEEE Transactions on Information Theory IT-33:483–501, 1987.CrossRefGoogle Scholar
  21. [Wic95]
    S. B. Wicker. Error Control Systems for Digital Communications and Storage. Englewood Cliffs: Prentice Hall, 1995.Google Scholar
  22. [Wol78]
    J. K. Wolf. Efficient maximum likelihood decoding of linear block codes. IEEE Transactions on Information Theory IT-24(l):76–80, 1978.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Chris Heegard
    • 1
    • 2
  • Stephen B. Wicker
    • 2
  1. 1.Alantro Communications, Inc.USA
  2. 2.Cornell UniversityUSA

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