Abstract
We introduce notions of local and interweight spectra of an arbitrary coloring of a Boolean cube, which generalize the notion of a weight spectrum. The main objects of our research are colorings that are called perfect. We establish an interrelation of local spectra of such a coloring in two orthogonal faces of a Boolean cube and study properties of the interweight spectrum. Based on this, we prove a new metric property of perfect colorings, namely, their strong distance invariance. As a consequence, we obtain an analogous property of an arbitrary completely regular code, which, together with his neighborhoods, forms a perfect coloring.
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Delsarte, P., An Algebraic Approach to the Association Schemes of Coding Theory, Philips Res. Rep. Suppl., 1973, no. 10. Translated under the title Algebraicheskii podkhod k skhemam otnoshenii teorii kodirovaniya, Moscow: Mir, 1976.
Vasil’eva, A.Yu., Local Spectra of Perfect Binary Codes, Diskretn. Anal. Issled. Oper., Ser. 1, 1999, vol. 6, no. 1, pp. 3–11 [Discrete Appl. Math. (Engl. Transl.), 2004, vol. 135, no. 1–3, pp. 301–307].
Vasil’eva, A.Yu., Strong Distance Invariance of Perfect Binary Codes,Diskretn. Anal. Issled. Oper., Ser. 1, 2002, vol. 9, no. 4, pp. 33–40.
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Original Russian Text © A.Yu. Vasil’eva, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 2, pp. 84–90.
Supported in part by the Russian Foundation for Basic Research, project no. 07-01-00248-a.
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Vasil’eva, A.Y. Local and interweight spectra of completely regular codes and of perfect colorings. Probl Inf Transm 45, 151–157 (2009). https://doi.org/10.1134/S0032946009020069
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DOI: https://doi.org/10.1134/S0032946009020069