  It is known that the order of correlation immunity of a nonconstant
unbalanced Boolean function in $n$ variables cannot exceed $2n/3-1$; moreover,
it is $2n/3-1$ if and only if the function corresponds to an equitable
$2$-partition of the $n$-cube with an eigenvalue $-n/3$ of the quotient matrix.
The known series of such functions have proportion $1:3$, $3:5$, or $7:9$ of
the number of ones and zeros. We prove that if a nonconstant unbalanced Boolean
function attains the correlation-immunity bound and has ratio $C:B$ of the
number of ones and zeros, then $CB$ is divisible by $3$. In particular, this
proves the nonexistence of equitable partitions for an infinite series of
putative quotient matrices. We also establish that there are exactly $2$
equivalence classes of the equitable partitions of the $12$-cube with quotient
matrix $[[3,9],[7,5]]$ and $16$ classes, with $[[0,12],[4,8]]$. These
parameters correspond to the Boolean functions in $12$ variables with
correlation immunity $7$ and proportion $7:9$ and $1:3$, respectively (the case
$3:5$ remains unsolved). This also implies the characterization of the
orthogonal arrays OA$(1024,12,2,7)$ and OA$(512,11,2,6)$.
