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A $\{00,01,10,11\}$-valued function on the vertices of the $n$-cube is calleda $t$-resilient $(n,2)$-function if it has the same number of $00$s, $01$s,$10$s and $11$s among the vertices of every subcube of dimension $t$. TheFriedman and Fon-Der-Flaass bounds on the correlation immunity order say thatsuch a function must satisfy $t\le 2n/3-1$; moreover, the $(2n/3-1)$-resilient$(n,2)$-functions correspond to the equitable partitions of the $n$-cube withthe quotient matrix $[[0,r,r,r],[r,0,r,r],[r,r,0,r],[r,r,r,0]]$, $r=n/3$. Wesuggest constructions of such functions and corresponding partitions, showconnections with Latin hypercubes and binary $1$-perfect codes, characterizethe non-full-rank and the reducible functions from the considered class, anddiscuss the possibility to make a complete characterization of the class.

On $(2n/3-1)$-resilient $(n,2)$-functions