Balancedness and correlation immunity of symmetric Boolean functions (Q2384416)

Consider the following Boolean function properties: C1. balancedness; C2. nonaffinity; C3. nondegeneracy; C4. correlation immunity; C5. symmetry and denote by \(A_{n}(i_1,\ldots ,i_t)\) the set of all \(n\)-variable Boolean functions having the properties \(C_{i_{1}},\ldots ,C_{i_{t}}\). By using simple binomial coefficient identities sufficient conditions for symmetric functions to be balanced or correlation immune are deduced. On this way construction of new functions in the sets \(A_{n}(1,2,3,5)\) and \(A_{n}(2,3,4,5)\) is proposed. They are used to improve known lower bounds on the sizes of such sets, e. g., if \(n\geq 14\) and \(n+2\) is a perfect square then \(| A_{n}(2,3,4,5)| \geq 2^{\lfloor n/2 \rfloor +1}+2^{\lceil (n-1)/2 \rceil }-2\). Finally, a method to construct \(n\)-variable, third order correlation immune function for each perfect square \(n\geq 9\) is presented.