Abstract
We have recently found tail-biting trellis representations of the binary Golay code ℂ24 that have considerably fewer states than the best conventional trellis. In particular, we have exhibited a 16-state, twelve-section tail-biting trellis for ℂ24, whereas a conventional trellis must have at least 256 states at its midpoint. This trellis and the corresponding set of generators for ℂ24 have an exceptionally regular structure. In this paper, we classify all the possible sets of generators of this type, up to automorphisms in the Mathieu group M 24. Our analysis is based on the Miracle Octad Generator representation of ℂ24 and M 24.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A.R. Calderbank, G.D. Forney, Jr., and A. Vardy, Minimal tail-biting trellises: the Golay code and more, submitted to the IEEE Trans. Inform. Theory, August 1997.
J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, New York: Springer-Verlag, 1993.
G.D. Forney, Jr., Coset codes — Part II: Binary lattices and related codes, IEEE Trans. Inform. Theory, vol.34, pp. 1152–1187, 1988.
—, The forward-backward algorithm, in Proc. 34th Allerton Conference on Comm., Control, and Computing, Monticello, IL., pp. 432–446, October 1996.
R. Kötter and A. Vardy, Construction of minimal tail-biting trellises, in Proc. IEEE Int. Workshop on Inform. Theory, pp. 73–75, Killarney, Ireland, June 1998.
A. Vardy —, The theory of tail-biting trellises, in preparation, 1998.
B.D. Kudryashov and T.G. Zakharova, Block codes from convolutional codes, Problemy Peredachi Informatsii, vol. 25, pp. 98–102, 1989.
S. Lin and D.J. Costello, Jr., Error Control Coding: Fundamentals and Applications, Englewood Cliffs, NJ: Prentice-Hall, 1983.
J.H. Ma and J.K. Wolf, On tail-biting convolutional codes, IEEE Trans. Commun., vol.34, pp. 104–111, 1986.
D.J. Muder, Minimal trellises for block codes, IEEE Trans. Inform. Theory, vol.34, pp. 1049–1053, 1988.
G. Solomon and H.C.A. van Tilborg, A connection between block and convolutional codes, SIAM J. Applied Math., vol. 37, pp. 358–369, 1979.
N. Wiberg, H.-A. Loeliger, and R. Kötter, Codes and iterative decoding on general graphs, Euro. Trans. Telecommun., vol. 6, pp. 513–526, 1995.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media New York
About this chapter
Cite this chapter
Calderbank, A.R., Forney, G.D., Vardy, A. (1998). Classification of Certain Tail-Biting Generators for the Binary Golay Code. In: Vardy, A. (eds) Codes, Curves, and Signals. The Springer International Series in Engineering and Computer Science, vol 485. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5121-8_11
Download citation
DOI: https://doi.org/10.1007/978-1-4615-5121-8_11
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7330-8
Online ISBN: 978-1-4615-5121-8
eBook Packages: Springer Book Archive