On the weight distribution of the cosets of MDS codes
share
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 ≤ 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 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 PG(N,q).
READ FULL TEXT
