Balanced Boolean functions with few-valued Walsh spectra parameterized by <inline-formula><tex-math id="M1">$ P(x^2+x) $</tex-math></inline-formula>

Boolean functions with few-valued spectra have wide applications in cryptography, coding theory, sequence designs, etc. In this paper, we further study the parametric construction approach to obtain balanced Boolean functions using $ 2 $-to-$ 1 $ mappings of the form $ P(x^2+x) $, where $ P $ denotes carefully selected permutation polynomials. The key contributions of this work are twofold: (1) We establish a new family of four-valued spectrum Boolean functions. This family includes Boolean functions with good cryptographic properties, e.g., the same nonlinearity as semi-bent functions, the maximal algebraic degree, and the optimal algebraic immunity for dimensions $ n \leq 14 $. (2) We derive seven distinct classes of plateaued functions, including four infinite families of semi-bent functions and a class of near-bent functions.