  We show that a known condition for having rough basin boundaries in bistable
2D maps holds for high-dimensional bistable systems that possess a unique
nonattracting chaotic set embedded in their basin boundaries. The condition for
roughness is that the cross-boundary Lyapunov exponent $\lambda_x$ {\bfac on
the nonattracting set} is not the maximal one. Furthermore, we provide a
formula for the generally noninteger co-dimension of the rough basin boundary,
which can be viewed as a generalization of the Kantz-Grassberger formula. This
co-dimension that can be at most unity can be thought of as a partial
co-dimension, and, so, it can be matched with a Lyapunov exponent. We show
{\bfac in 2D noninvertible- and 3D invertible minimal models,} that, formally,
it cannot be matched with $\lambda_x$. Rather, the partial dimension
$D_0^{(x)}$ that $\lambda_x$ is associated with in the case of rough boundaries
is trivially unity. Further results hint that the latter holds also in higher
dimensions. This is a peculiar feature of rough fractals. Yet, $D_0^{(x)}$
cannot be measured via the uncertainty exponent along a line that traverses the
boundary. Indeed, one cannot determine whether the boundary is a rough or a
filamentary fractal by measuring fractal dimensions. Instead, one needs to
measure both the maximal and cross-boundary Lyapunov exponents numerically or
experimentally.
