  We calculate the almost sure Hausdorff dimension of the random covering set
$\limsup_{n\to\infty}(g_n + \xi_n)$ in $d$-dimensional torus $\mathbb T^d$,
where the sets $g_n\subset\mathbb T^d$ are parallelepipeds, or more generally,
linear images of a set with nonempty interior, and $\xi_n\in\mathbb T^d$ are
independent and uniformly distributed random points. The dimension formula,
derived from the singular values of the linear mappings, holds provided that
the sequences of the singular values are decreasing.
