On the weight distribution of the cosets of MDS codes

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