New characterizations of generalized Boolean functions

This paper focuses on providing the characteristics of generalized Boolean functions from a new perspective. We first generalize the classical Fourier transform and correlation spectrum into what we will call the \(\rho\) $\rho$ -Walsh–Hadamard transform ( \(\rho\) $\rho$ -WHT) and the \(\rho\) $\rho$ -correlation spectrum, respectively. Then a direct relationship between the \(\rho\) $\rho$ -correlation spectrum and the \(\rho\) $\rho$ -WHT is presented. We investigate the characteristics and properties of generalized Boolean functions based on the \(\rho\) $\rho$ -WHT and the \(\rho\) $\rho$ -correlation spectrum, as well as the sufficient (or also necessary) conditions and subspace decomposition of \(\rho\) $\rho$ -bent functions. We also derive the \(\rho\) $\rho$ -autocorrelation for a class of generalized Boolean functions on \((n+2)\) $(n+2)$ -variables. Secondly, we present a characterization of a class of generalized Boolean functions with \(\rho\) $\rho$ -WHT in terms of the classical Boolean functions. Finally, we demonstrate that \(\rho\) $\rho$ -bent functions can be obtained from a class of composite construction if and only if \(\rho =1\) $\rho =1$ . Some examples of non-affine \(\rho\) $\rho$ -bent functions are also provided.