Cardinality spectra of components of correlation immune functions, bent functions, perfect colorings, and codes

We study cardinalities of components of perfect codes and colorings, correlation immune functions, and bent function (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 a {0, ..., 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.