Perfect colorings of the 12-cube that attain the bound on correlation immunity

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. This is a translation into English of the original paper by D. G. Fon-Der-Flaass, “Perfect colorings of the 12-cube that attain the bound on correlation immunity”, published in Russian in Siberian Electronic Mathematical Reports [0]. Let Hn be the hypercube of dimension n. Its vertices are the binary vectors of length n (we will identify such a vector with the set of its nonzero coordinates); two vertices are adjacent if their vectors differ in exactly one coordinate. A coloring of the vertices into black and white colors is called a perfect coloring with parameters (a, b, c, d) if every black vertex has a black and b white neighbors and every white vertex has c black and d white neighbors. (For a general definition of perfect coloring and main properties, see [1], [3].) In [2], it is proved that for every perfect 2-coloring of Hn with b 6= c, it holds