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).Comment: v3: final; title changed; revised; OA(512,11,2,6) discusse