  For each non-constant Boolean function $q$, Klapper introduced the notion of
$q$-transforms of 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.
  In this work we discuss the existence of $q$-nearly bent functions, a new
family of Boolean functions characterized by the $q$-transform. Let $q$ be a
non-affine Boolean function. We prove that any balanced Boolean functions
(linear or non-linear) are $q$-nearly bent if $q$ has weight one, which gives a
positive answer to an open question (whether there exist non-affine $q$-nearly
bent functions) proposed by Klapper. We also prove a necessary condition for
checking when a function isn't $q$-nearly bent.
