On the q-Bentness of Boolean Functions

For each non-constant q in the set of n-variable Boolean functions, the qtransform 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 ±2n/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.