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Cardinality spectra of components of correlation immune functions, bent functions, perfect colorings, and codes. (English. Russian original) Zbl 1276.06008
Probl. Inf. Transm. 48, No. 1, 47-55 (2012); translation from Probl. Peredachi Inf. 48, No. 1, 54-63 (2012).
Summary: We study cardinalities of components of perfect codes and colorings, correlation-immune functions, and bent functions (sets of ones of these functions). Based on results of Kasami and Tokura, we show that for any of these combinatorial objects the component cardinality in the interval from $$2^k$$ to $$2^{k+1}$$ can only take values of the form $$2^{k+1}-2^p$$, where $$p\in\{0,\dots ,k\}$$ and $$2^k$$ is the minimum component cardinality for a combinatorial object with the same parameters. For bent functions, we prove existence of components of any cardinality in this spectrum. For perfect colorings with certain parameters and for correlation-immune functions, we find components of some of the above-given cardinalities.

MSC:
 06E30 Boolean functions 05C15 Coloring of graphs and hypergraphs 94A60 Cryptography 94B25 Combinatorial codes
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References:
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