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