Abstract
We refine a lower bound on the spectrum of a binary code. We give a simple derivation of the known bound on the undetected error probability of a binary code.
Similar content being viewed by others
References
McEliece, R.J., Rodemich, E.R., Rumsey, H., Jr., and Welch, L.R., New Upper Bounds on the Rate of a Code via the Delsarte-MacWilliams Inequalities, IEEE Trans. Inform. Theory, 1977, vol. 23, no. 2, pp. 157–166.
Litsyn, S., New Upper Bounds on Error Exponents, IEEE Trans. Inform. Theory, 1999, vol. 45, no. 2, pp. 385–398.
Burnashev, M.V., Code Spectrum and the Reliability Function: Binary Symmetric Channel, Probl. Peredachi Inf., 2006, vol. 42, no. 4, pp. 3–22 [Probl. Inf. Transm. (Engl. Transl.), 2006, vol. 42, no. 4, pp. 263–281].
Levenshtein, V.I., On the Straight-Line Bound for the Undetected Error Exponent, Probl. Peredachi Inf., 1989, vol. 25, no. 1, pp. 33–37 [Probl. Inf. Transm. (Engl. Transl.), 1989, vol. 25, no. 1, pp. 24–27].
Burnashev, M.V., Sharpening of an Upper Bound for the Reliability Function of a Binary Symmetric Channel, Probl. Peredachi Inf., 2005, vol. 41, no. 4, pp. 3–22 [Probl. Inf. Transm. (Engl. Transl.), 2005, vol. 41, no. 4, pp. 301–318].
Burnashev, M.V., On the BSC Reliability Function: Expanding the Region Where It Is Known Exactly, Probl. Peredachi Inf., 2015, vol. 51, no. 4, pp. 3–22 [Probl. Inf. Transm. (Engl. Transl.), 2015, vol. 51, no. 4, pp. 307–325].
Burnashev, M.V., Amari, S., and Han, T.S., On Some Problems of Hypothesis Testing with Information Constraints, Teor. Veroyatnost. i Primenen., 2000, vol. 45, no. 4, pp. 625–638 [Theory Probab. Appl. (Engl. Transl.), 2002, vol. 45, no. 4, pp. 558–568].
Shimokawa, H., Han, T.S., and Amari, S., Error Bounds of Hypothesis Testing with Data Compression, in Proc. 1994 IEEE Int. Sympos. on Information Theory (ISIT’94), Trondheim, Norway, June 27–July 1, 1994, p. 114.
Ore, Ø., Theory of Graphs, Providence, RI: Amer. Math. Soc., 1962. Translated under the title Teoriya grafov, Moscow: Nauka, 1980.
Levenshtein, V.I., Bounds on the Probability of Undetected Error, Probl. Peredachi Inf., 1977, vol. 13, no. 1, pp. 3–18 [Probl. Inf. Transm. (Engl. Transl.), 1977, vol. 13, no. 1, pp. 1–12].
Levenshtein, V.I., Bounds for Packings of Metric Spaces and Some of Their Applications, Probl. Kibern., 1983, vol. 40, pp. 43–110.
Acknowledgement
The author is grateful to a reviewer for constructive critical remarks, which improved the paper.
Funding
The research was supported in part by the Russian Foundation for Basic Research, project no. 19-01-00364.
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2019, published in Problemy Peredachi Informatsii, 2019, Vol. 55, No. 4, pp. 76–85.
Rights and permissions
About this article
Cite this article
Burnashev, M.V. On Lower Bounds on the Spectrum of a Binary Code. Probl Inf Transm 55, 366–375 (2019). https://doi.org/10.1134/S0032946019040033
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0032946019040033