  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 into 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 ones are obtained. For the MDS
code cosets of weight $W=1$ and weight $W=d-1$ we obtain formulas of the weight
distributions depending only on the code parameters. This proves that all the
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 different weight distributions;
in this case, we proved that the distributions are symmetrical in some sense.
The weight distributions of the 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 obtain 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 an optimal multiple saturating set in the
projective space $\mathrm{PG}(N,q)$.
