Abstract
The performance of a linear error-detecting code in a symmetric memoryless channel is characterized by its probability of undetected error, which is a function of the channel symbol error probability, involving basic parameters of a code and its weight distribution. However, the code weight distribution is known for relatively few codes since its computation is an NP-hard problem. It should therefore be useful to have criteria for properness and goodness in error detection that do not involve the code weight distribution. In this work we give two such criteria. We show that a binary linear code C of length n and its dual code C ⊥ of minimum code distance d ⊥ are proper for error detection whenever d ⊥ ≥ ⌊n/2⌋ + 1, and that C is proper in the interval [(n + 1 − 2d ⊥)/(n − d ⊥); 1/2] whenever ⌈n/3⌉ + 1 ≤ d ⊥ ≤ ⌊n/2⌋. We also provide examples, mostly of Griesmer codes and their duals, that satisfy the above conditions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Klove, T. and Korzhik, V.I., Error Detecting Codes: General Theory and Their Application in Feedback Communication Systems, Boston: Kluwer, 1995.
Leung-Yan-Cheong, S.K., Barnes, E.R., and Friedman, D.U., On Some Properties of the Undetected Error Probability of Linear Codes, IEEE Trans. Inform. Theory, 1979, vol. 25, no.1, pp. 110–112.
Massey, J., Coding Techniques for Digital Data Networks, in Proc. Int. Conf. on Information Theory and Systems, Berlin, 1978, NTG-Fachberichte, vol. 65, 1978, pp. 307–315.
Dodunekova, R., Dodunekov, S.M., and Nikolova, E., A Survey on Proper Codes, Discr. Appl. Math., special issue (to appear).
Dodunekova, R., Dodunekov, S.M., and Nikolova, E., On the Error-Detecting Performance of Some Classes of Block Codes, Probl. Peredachi Inf., 2004, vol. 40, no.4, pp. 68–78 [Probl. Inf. Trans. (Engl. Transl.), 2004, vol. 40, no. 4, pp. 356–364].
Berlekamp, E.R., McEliece, R.J., and van Tilborg, H.C.A., On the Inherent Intractability of Certain Coding Problems, IEEE Trans. Inform. Theory, 1978, vol. 24, no.3, pp. 384–386.
Dodunekova, R., Extended Binomial Moments of a Linear Code and the Undetected Error Probability, Probl. Peredachi Inf., 2003, vol. 39, no.3, pp. 28–39 [Probl. Inf. Trans. (Engl. Transl.), 2003, vol. 39, no. 3, pp. 255–265].
Pless, V., Introduction to the Theory of Error-Correcting Codes, New York: Wiley, 1998.
Dodunekov, S.M. and Simonis, J., Codes and Projective Multisets, Electronic J. Combin., 1998, vol. 5, no.1, Research Paper R37.
Faldum, A. and Willems, W., A Characterization of MMD Codes, IEEE Trans. Inform. Theory, 1998, vol. 44, no.4, pp. 1555–1558.
Dodunekova, R. and Dodunekov, S.M., The MMD Codes Are Proper for Error Detection, IEEE Trans. Inform. Theory, 2002, vol. 48, no.12, pp. 3109–3111.
Dodunekova, R., The Duals of MMD Codes Are Proper for Error Detection, IEEE Trans. Inform. Theory, 2003, vol. 49, no.8, pp. 2034–2038.
Dodunekova, R. and Dodunekov, S.M., Sufficient Conditions for Good and Proper Error Detecting Codes, IEEE Trans. Inform. Theory, 1997, vol. 43, no.6, pp. 2023–2026.
MacWilliams, F.J. and Sloane, N.J.A., The Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977. Translated under the title Teoriya kodov, ispravlyayushchikh oshibki, Moscow: Svyaz', 1979.
Manev, N.L., On the Uniqueness of Certain Codes Meeting the Griesmer Bound, Pliska Stud. Math. Bulgar., 1986, vol. 8, pp. 47–53.
Dodunekova, R. and Dodunekov, S.M., Linear Block Codes for Error Detection, Preprint of Chalmers Univ. of Technology and Goteborg Univ., 1996, no. 1996-07.
Dodunekova, R. and Dodunekov, S.M., Sufficient Conditions for Good and Proper Error Detecting Codes via Their Duals, Math. Balkanica (N.S.), 1997, vol. 11, no.3–4, pp. 375–381.
Helleseth, T., Further Classiffication of Codes Meeting the Griesmer Bound, IEEE Trans. Inform. Theory, 1984, vol. 30, no.2, pp. 395–403.
Manev, N.L., A Characterization up to Isomorphism of Some Classes of Codes Meeting the Griesmer Bound, C. R. Acad. Bulgare Sci., 1984, vol. 37, no.4, pp. 481–483.
van Tilborg, H.C.A., On the Uniqueness Resp. Nonexistence of Certain Codes Meeting the Griesmer Bound, Inform. Control, 1980, vol. 44, no.1, pp. 16–35.
Dodunekov, S.M. and Manev, N.L., Characterization of Two Classes of Codes Meeting the Griesmer Bound, Probl. Peredachi Inf., 1983, vol. 19, no.4, pp. 3–10 [Probl. Inf. Trans. (Engl. Transl.), 1983, vol. 19, no. 4, pp. 253–259].
Helleseth, T. and van Tilborg, H.C.A., A New Class of Codes Meeting the Griesmer Bound, IEEE Trans. Inform. Theory, 1981, vol. 27, no.5, pp. 548–555.
Delsarte, P. and Goethals, J.-M., Irreducible Binary Cyclic Codes of Even Dimension, in Proc. 2nd Chapel Hill Conf. on Combinatorial Mathematics and Its Applications, Univ. of North Carolina at Chapel Hill, 1970, pp. 100–113.
Dodunekova, R., Rabaste, O., and Paez, J.L.V., Error Detection with a Class of Irreducible Binary Cyclic Codes and Their Dual Codes, IEEE Trans. Inform. Theory, 2005, vol. 51, no.3, pp. 1206–1209.
Dodunekova, R., On the Error-Detecting Performance of the Delsarte-Goethals Irreducible Binary Cyclic Codes and Their Duals, Preprint of Dept. Math. Sciences, Chalmers Univ. of Technology and Goteborg Univ., 2004, no. 2004-17.
Author information
Authors and Affiliations
Additional information
__________
Translated from Problemy Peredachi Informatsii, No. 3, 2005, pp. 3–16.
Original Russian Text Copyright © 2005 by Dodunekova, Nikolova.
Supported by the Swedish Research Council under grant 621-2003-5325.
Partially supported by the Bulgarian NSF under contracts MM901/99 and MM1405/2004.
Rights and permissions
About this article
Cite this article
Dodunekova, R., Nikolova, E. Sufficient Conditions for Monotonicity of the Undetected Error Probability for Large Channel Error Probabilities. Probl Inf Transm 41, 187–198 (2005). https://doi.org/10.1007/s11122-005-0023-5
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11122-005-0023-5