  The weight distribution of the cosets of maximum distance separable (MDS)
codes is considered. In 1990, P.G. Bonneau proposed a relation to obtain the
full weight distribution of a coset of an MDS code with minimum distance $d$
using the known numbers of vectors of weights $\le d-2$ in this coset. In this
paper, the Bonneau formula is transformed to a more structured and convenient
form. The new version of the formula allows to consider effectively cosets of
distinct weights $W$. (The weight $W$ of a coset is the smallest Hamming weight
of any vector in the coset.) For each of the considered $W$ or regions of $W$,
special relations more simple than the general one are obtained. We proved that
all the MDS code cosets of weight $W=1$ (as well as $W=d-1$) have the same
weight distribution. The cosets of weight $W=2$ or $W=d-2$ may have
non-identical weight distributions; in this case, we proved that the
distributions are symmetrical in some sense. The weight distributions of cosets
of MDS codes corresponding to arcs in the projective plane $\mathrm{PG}(2,q)$
are also considered. For MDS codes of covering radius $R=d-1$ we obtained the
number of the weight $W=d-1$ cosets and their weight distribution that gives
rise to a certain classification of the so-called deep holes. We show that any
MDS code of covering radius $R=d-1$ is an almost perfect multiple covering of
the farthest-off points (deep holes); moreover, it corresponds to a multiple
saturating set in the projective space $\mathrm{PG}(N,q)$.
