On the $q$-Bentness of Boolean Functions

For each non-constant $q$ in the set of $n$-variable Boolean functions, the {\em $q$-transform} of a Boolean function $f$ is related to the Hamming distances from $f$ to the functions obtainable from $q$ by nonsingular linear change of basis. Klapper conjectured that no Boolean function exists with its $q$-transform coefficients equal to $\pm 2^{n/2}$ (such function is called $q$-bent). In our early work, we only gave partial results to confirm this conjecture for small $n$. Here we prove thoroughly that the conjecture is true by investigating the nonexistence of the partial difference sets in Abelian groups with special parameters. We also introduce a new family of functions called almost $q$-bent functions, which are close to $q$-bentness.