  Let $\on_{k \in \nz}$ be an orthonormal system on some $\sigma$-finite
measure space $(\Om,p)$. We study the notion of cotype with respect to $\Phi$
for an operator $T$ between two Banach spaces $X$ and $Y$, defined by $\fco T
:= \inf$ $c$ such that \[ \Tfmm \pl \le \pl c \pll \gmm \hspace{.7cm}\mbox{for
all}\hspace{.7cm} (x_k)\subset X \pl,\] where $(g_k)_{k\in \nz}$ is a sequence
of independent and normalized gaussian variables. It is shown that this
$\Phi$-cotype coincides with the usual notion of cotype $2$ iff \linebreak
$\fco {I_{\lin}} \sim \sqrt{\frac{n}{\log (n+1)}}$ uniformly in $n$ iff there
is a positive $\eta>0$ such that for all $n \in \nz$ one can find an
orthonormal $\Psi = (\psi_l)_1^n \subset {\rm span}\{ \phi_k \p|\p k \in \nz\}$
and a sequence of disjoint measurable sets $(A_l)_1^n \subset \Om$ with \[
\int\limits_{A_l} \bet \psi_l\rag^2 d p \pl \ge \pl \eta \quad \mbox{for
all}\quad l=1,...,n \pl. \] A similar result holds for the type situation. The
study of type and cotype with respect to orthonormal systems of a given length
provides the appropriate approach to this result. We intend to give a quite
complete picture for orthonormal systems in measure space with few atoms.
