Abstract
We prove that the (10,40,4) code found by Best is unique. We then employ this fact to show that A(10,3)= A(11,4)≤78 and A(11,3)=A(12,4)≤156.
The paper is a part of [9] submitted to IEEE Transactions on Information Theory.
Research supported by a grant from the Israeli Ministry of Science and Technology and the Guastallo Fellowship.
Research supported in part by the Rothschild Fellowship.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
M.R. Best, “Binary codes with a minimum distance of four,” IEEE Trans. Inform. Theory, vol. IT-26, pp. 738–742, 1980.
M.R. Best, A.E. Brouwer, F.J. MacWilliams, A.M. Odlyzko, and N.J.A. Sloane, “Bounds for binary codes of length less than 25,” IEEE Trans. Inform. Theory, vol. IT-24, pp. 81–93, 1978.
J.-M. Goethals, “The extended Nadler code is unique,” IEEE Trans. Inform. Theory, vol. IT-23, pp. 132–135, 1977.
J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, New York: Springer-Veralg, 1988.
J.H. Conway and N.J.A. Sloane, “Quaternary constructions for the binary codes of Julin, Best, and others,” preprint.
P. Delsarte and J.-M. Goethals, “Unrestricted codes with the Golay parameters are unique,” Discr. Math., vol. 12, pp. 211–224, 1975.
D. Julin, “Two improved block codes,” JEEE Trans. Inform. Theory, vol. IT-11, p. 459, 1965.
D.E. Knuth, The Art of Computer Programming: Sorting and Searching, Reading: Addison-Wesley, 1973.
S. Litsyn and A. Vardy, “The uniqueness of the Best code,” submitted to JEEE Trans. Inform. Theory.
F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, New York: North-Holland, 1977.
V. Pless, “On the uniqueness of the Golay codes,” J. Comb. Theory, vol. 5, pp. 215–228, 1968.
S. Snover, “The uniqueness of the Nordstrom-Robinson and the Golay binary codes,” Ph.D. Thesis, Michigan State University, 1973.
N. Wax, “On upper bounds for error-detecting and error-correcting codes of finite length,” IEEE Trans. Inform. Theory, vol. IT-5, pp. 168–174, 1959.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1994 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Litsyn, S., Vardy, A. (1994). Two new upper bounds for codes of distance 3. In: Cohen, G., Litsyn, S., Lobstein, A., Zémor, G. (eds) Algebraic Coding. Algebraic Coding 1993. Lecture Notes in Computer Science, vol 781. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57843-9_26
Download citation
DOI: https://doi.org/10.1007/3-540-57843-9_26
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-57843-7
Online ISBN: 978-3-540-48357-1
eBook Packages: Springer Book Archive