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 $d$ 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 $\mathrm{PG}(N,q)$.