# On unbalanced Boolean functions with best correlation immunity

## TL;DR

This paper investigates the structure of unbalanced Boolean functions with maximum possible correlation immunity, establishing divisibility conditions, classifying certain equitable partitions, and characterizing related orthogonal arrays.

## Contribution

It proves divisibility constraints for correlation-immunity bounds, classifies specific equitable partitions of the 12-cube, and characterizes related orthogonal arrays, advancing understanding of Boolean function structures.

## Key findings

- CB is divisible by 3 for functions attaining the correlation-immunity bound
- Exactly 2 equivalence classes of equitable partitions for a specific quotient matrix in 12-cube
- Characterization of orthogonal arrays OA(1024,12,2,7) and OA(512,11,2,6)

## Abstract

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$(512,11,2,6)$.