  The study of non-linearity (linearity) of Boolean function was initiated by
Rothaus in 1976. The classical non-linearity of a Boolean function is the
minimum Hamming distance of its truth table to that of affine functions. In
this note we introduce new "multidimensional" non-linearity parameters
$(N_f,H_f)$ for conventional and vectorial Boolean functions $f$ with $m$
coordinates in $n$ variables. The classical non-linearity may be treated as a
1-dimensional parameter in the new definition. $r$-dimensional parameters for
$r\geq 2$ are relevant to possible multidimensional extensions of the Fast
Correlation Attack in stream ciphers and Linear Cryptanalysis in block ciphers.
Besides we introduce a notion of optimal vectorial Boolean functions relevant
to the new parameters. For $r=1$ and even $n\geq 2m$ optimal Boolean functions
are exactly perfect nonlinear functions (generalizations of Rothaus' bent
functions) defined by Nyberg in 1991. By a computer search we find that this
property holds for $r=2, m=1, n=4$ too. That is an open problem for larger
$n,m$ and $r\geq 2$. The definitions may be easily extended to $q$-ary
functions.
