  A $\{00,01,10,11\}$-valued function on the vertices of the $n$-cube is called
a $t$-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 $t$. The
Friedman and Fon-Der-Flaass bounds on the correlation immunity order say that
such 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 with
the quotient matrix $[[0,r,r,r],[r,0,r,r],[r,r,0,r],[r,r,r,0]]$, $r=n/3$. We
suggest constructions of such functions and corresponding partitions, show
connections with Latin hypercubes and binary $1$-perfect codes, characterize
the non-full-rank and the reducible functions from the considered class, and
discuss the possibility to make a complete characterization of the class.
