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Part of the book series: The Springer International Series in Engineering and Computer Science ((SECS,volume 276))

Abstract

The concept of a trellis and the proof of the optimality of the Viterbi algorithm grew out of early work at Codex Corporation, here published for the first time. A recently observed flaw in this proof is noted. Trellises for block codes and lattices are of current interest. An absolutely minimal trellis is exhibited for the E8 lattice. This trellis gives a succinct summary of the algebraic, geometrical, and dynamical structure of E8 and its sublattices.

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References

  1. J. L. Massey, Threshold Decoding. Cambridge, MA: MIT Press, 1963.

    Google Scholar 

  2. R. G. Gallager, Low-Density Parity-Check Codes. Cambridge, MA: MIT Press, 1962.

    Google Scholar 

  3. G. D. Forney Jr., “Final report on a coding system design for advanced solar missions,” Contract NAS2–3637, NASA CR73176, NASA Ames Research Center, Moffett Field, CA, Dec. 1967.

    Google Scholar 

  4. A. J. Viterbi, “Error bounds for convolutional codes and an asymptotically optimum decoding algorithm,” IEEE Trans. Inform. Theory, vol. IT-13, pp. 260–269, 1967.

    Article  Google Scholar 

  5. G. J. Minty, “A comment on the shortest-route problem,” Oper. Res., vol. 5, p. 724, 1957.

    Article  Google Scholar 

  6. A. J. Viterbi, “From proof to product,” 1990 IEEE Communication Theory Workshop, Ojai, CA, 1990.

    Google Scholar 

  7. P. Elias, “Error-free coding,” IRE Trans. Inform. Theory, vol. IT-4, pp. 29–37, 1954.

    MathSciNet  Google Scholar 

  8. J. L. Massey and M. K. Sain, “Codes, automata, and continuous systems: Explicit interconnections,” IEEE Trans. Auto. Control, vol. AC-12, pp. 644–650, 1967.

    Article  Google Scholar 

  9. J. L. Massey and M. K. Sain, “Inverses of linear sequential circuits,” IEEE Trans. Computers, vol. C-17, pp. 330–337, 1968.

    Article  Google Scholar 

  10. G. D. Forney Jr., “Convolutional codes I: Algebraic structure,” IEEE Trans. Inform. Theory, vol. IT-16, pp. 720–738, 1970.

    Article  MathSciNet  Google Scholar 

  11. J. K. Wolf, “Efficient maximum likelihood decoding of linear block codes using a trellis,” IEEE Trans. Inform. Theory, vol. IT-24, pp. 76–80, 1978.

    Article  Google Scholar 

  12. J. L. Massey, “Foundation and methods of channel coding,” Proc. Intl. Conf. on Inform. Theory and Systems, Berlin, NTG-Fachberichte, vol. 65, pp. 148–157, 1978.

    Google Scholar 

  13. G. D. Forney Jr., “Coset codes — Part II: Binary lattices and related codes,” IEEE Trans. Inform. Theory, vol. IT-34, pp. 1152–1187, 1988.

    Article  MathSciNet  Google Scholar 

  14. A. D. Kot and C. Leung, “On the construction and dimensionality of linear block code trellises,” Proc. 1993 IEEE Intl. Symp. Inform. Theory, San Antonio, TX, p. 291, 1993.

    Google Scholar 

  15. J. L. Massey, T. Mittelholzer, T. Riedel, and M. Vollenweider, “Ring convolutional codes for phase modulation,” Proc. 1990 IEEE Intl. Symp. Inform. Theory, San Diego, CA, p. 176, 1990.

    Google Scholar 

  16. G. D. Forney Jr. and M. D. Trott, “The dynamics of group codes: State spaces, trellis diagrams and canonical encoders,” IEEE Trans. Inform. Theory, vol. 39, pp. 1491–1513, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  17. H.-A. Loeliger and T. Mittelholzer, “Convolutional codes over groups,” submitted to IEEE Trans. Inform. Theory, 1992.

    Google Scholar 

  18. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 2d ed. New York: Springer-Verlag, 1992.

    Google Scholar 

  19. G. D. Forney Jr., “Density/length profiles and trellis complexity of lattices,” in preparation.

    Google Scholar 

  20. R. C. Davis and H.-A. Loeliger, “A nonalgorithmic maximum likelihood decoder for trellis codes,” IEEE Trans. Inform. Theory,vol. 39, pp. 1450–1453, 1993.

    Article  MATH  Google Scholar 

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© 1994 Springer Science+Business Media New York

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Forney, G.D. (1994). Trellises Old and New. In: Blahut, R.E., Costello, D.J., Maurer, U., Mittelholzer, T. (eds) Communications and Cryptography. The Springer International Series in Engineering and Computer Science, vol 276. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2694-0_13

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  • DOI: https://doi.org/10.1007/978-1-4615-2694-0_13

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-6159-6

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