Abstract
We establish the properness of some classes of binary block codes with symmetric distance distribution, including Kerdock codes and codes that satisfy the Grey-Rankin bound, as well as the properness of Preparata codes, thus augmenting the list of very few known proper nonlinear codes.
Similar content being viewed by others
REFERENCES
Kløve, 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. Inf. Theory, 1979, vol. 25, no. 1, pp. 110–112.
Dodunekova, R., Dodunekov, S.M., and Nikolova, E., A Survey on Proper Codes, in Proc. Workshop on General Theory of Information Transfer and Combinatorics, Bielefeld, Germany, 2002, to appear.
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.
Preparata, F.P., A Class of Optimum Nonlinear Double-Error-Correcting Codes, Inf. Control, 1968, vol. 13, no. 4, pp. 378–400.
Kerdock, A.M., A Class of Low-Rate Nonlinear Binary Codes, Inf. Control, 1972, vol. 20, no. 2, pp. 182–187.
Semakov, N.V., Zinoviev, V.A., and Zaitsev, G.V., Uniformly Packed Codes, Probl. Peredachi Inf., 1971, vol. 7, no. 1, pp. 38–50 [Probl. Inf. Trans. (Engl. Transl.), 1971, vol. 7, no. 1, pp. 30–39].
Semakov, N.V., Zinoviev, V.A., and Zaitsev, G.V., On the Duality of Preparata and Kerdock Codes, in Proc. 5th All-Union Conf. on Coding Theory, Moscow-Gorky, 1972, Part 2, pp. 55–58.
Hammons, A.R., Jr., Kumar, P.V., Calderbank, A.R., Sloane, N.J.A., and Solé, P., The ℤ4-Linearity of Kerdock, Preparata, Goethals, and Related Codes, IEEE Trans. Inf. Theory, 1994, vol. 40, no. 2, pp. 301–319.
Grey, L.D., Some Bounds for Error-Correcting Codes, IEEE Trans. Inf. Theory, 1962, vol. 8, no. 3, pp. 200–202.
Rankin, R.A., The Closest Packing of Spherical Caps in n Dimensions, Proc. Glasgow Math. Assoc., 1955, vol. 2, pp. 139–144.
Rankin, R.A., On the Minimal Points of Positive Definite Quadratic Forms, Mathematika, 1956, vol. 3, pp. 15–24.
McGuire, G., Quasi-Symmetric Designs and Codes Meeting the Grey-Rankin Bound, J. Combin. Theory, Ser. A, 1997, vol. 78, pp. 280–291.
Brouwer, A.E., Some New Two-Weight Codes and Strongly Regular Graphs, Discrete Appl. Math., 1985, vol. 10, no. 4, pp. 455–461.
Jungnickel, D. and Tonchev, V.D., Exponential Number of Quasi-Symetric SDP Designs and Codes Meeting the Grey-Rankin Bound, Des. Codes Cryptogr., 1991, vol. 1, no. 3, pp. 247–253.
Author information
Authors and Affiliations
Additional information
Translated from Problemy Peredachi Informatsii, No. 4, 2004, pp. 68–78.
Original Russian Text Copyright © 2004 by Dodunekova, Dodunekov, Nikolova.
Rights and permissions
About this article
Cite this article
Dodunekova, R., Dodunekov, S.M. & Nikolova, E. On the error-detecting performance of some classes of block codes. Probl Inf Transm 40, 356–364 (2004). https://doi.org/10.1007/s11122-005-0004-8
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11122-005-0004-8