Abstract
Let K be a permutation group acting on binary vectors of length n and F K be a code of length 2n consisting of all binary functions with nontrivial inertia group in K. We obtain upper and lower bounds on the covering radii of F K , where K are certain subgroups of the affine permutation group GA n . We also obtain estimates for distances between F K and almost all functions in n variables as n → ∞. We prove the existence of functions with the trivial inertia group in GA n for all n ≥ 7. An upper bound for the asymmetry of a k-uniform hypergraph is obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
MacWilliams, F.J. and Sloane, N.J.A., The Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977. Translated under the title Teoriya kodov, ispravlyayuschikh oshibki, Moscow: Svyaz, 1979.
Kuznetsov, Yu.V. and Shkarin, S.A., Reed-Muller Codes (Survey), Mat. Vopr. Kibern., Moscow: Nauka, 1996, vol. 6, pp. 5–50.
Chechiota, S.I., On the Limit Distribution of the Distance between a Random Vector and Some Binary Codes, Probl. Peredachi Inf., 1995, vol. 31, no. 1, pp. 90–98 [Probl. Inf. Trans. (Engl. Transl.), 1995, vol. 31, no. 1, pp. 78–85].
Erdõs, P. and Spencer, J., Probabilistic Methods in Combinatorics, Budapest: Acad. Kiadó, 1974. Translated under the title Veroyatnostnye metody v kombinatorike, Moscow: Mir, 1976.
Ambrosimov, A.S. and Sharov, N.N., Some Asymptotic Decompositions of the Number of Functions with Nontrivial Inertia Group, Probl. Kibern., 1979, vol. 36, pp. 65–84.
Petrov, V.V., Summy nezavisimykh sluchaynykh velichin, Moscow: Nauka, 1972. Translated under the title Sums of Independent Random Variables, Berlin: Springer, 1975.
Shannon, C.E., The Synthesis of Two-Terminal Switching Circuits, Bell Syst. Tech. J., 1949, vol. 98, no. 1, pp. 59–98. Translated in Raboty po teorii informatsii i kibernetike (Works on Information Theory and Cybernetics), Moscow: Inostr. Lit., 1963.
Strazdin', I.E., Inertia Groups of Boolean Functions in Four Variables, Avtomat. Vych. Tekhn., 1968, no. 5, pp. 18–22.
Harary, F., Graph Theory, Reading, Massachusetts: Addison-Wesley, 1969. Translated under the title Teoriya grafov, Moscow: Mir, 1973.
Strazdin', I.E., Affine Classification of Boolean Functions in Five Variables, Avtomat. Vych. Tekhn., 1975, no. 1, pp. 1–9.
Kloss, B.M. and Nechiporuk, E.I., On Classification of Functions of Many-valued Logic, Probl. Kibern., 1963, vol. 9, pp. 27–36.
Rights and permissions
About this article
Cite this article
Denisov, O.V. Binary Codes Formed by Functions with Nontrivial Inertia Groups. Problems of Information Transmission 37, 339–352 (2001). https://doi.org/10.1023/A:1013875401571
Issue Date:
DOI: https://doi.org/10.1023/A:1013875401571