Geodesic
Perfect colorings of the $12$-cube that attain the bound on correlation immunity
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 4 (2007), pp. 292-295

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We construct perfect $2$-colorings of the $12$-hypercube that attain our recent bound on the dimension of arbitrary correlation immune functions. We prove that such colorings with parameters $(x,12-x,4+x,8-x)$ exist if $x=0,2,3$ and do not exist if $x=1$. 
@article{SEMR_2007_4_a17,
     author = {D. G. Fon-Der-Flaass},
     title = {Perfect colorings of the $12$-cube that attain the bound on correlation immunity},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {292--295},
     publisher = {mathdoc},
     volume = {4},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2007_4_a17/}
}
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D. G. Fon-Der-Flaass. Perfect colorings of the $12$-cube that attain the bound on correlation immunity. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 4 (2007), pp. 292-295. http://geodesic.mathdoc.fr/item/SEMR_2007_4_a17/