On $2k$ -Variable Symmetric Boolean Functions With Maximum Algebraic Immunity $k$

Given a positive even integer n, it is found that the weight distribution of any n-variable symmetric Boolean function with maximum algebraic immunity (AI) n/2 is determined by the binary expansion of n . Based on the foregoing, all n-variable symmetric Boolean functions with maximum AI are constructed. The amount is (2 wt(n)+1)2
^[log
₂
^n]
.