1 research outputs found
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
using the known numbers of vectors of weights 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 . (The weight of a coset is the smallest Hamming weight
of any vector in the coset.) For each of the considered or regions of ,
special relations more simple than the general ones are obtained. For the MDS
code cosets of weight and weight we obtain formulas of the weight
distributions depending only on the code parameters. This proves that all the
cosets of weight (as well as ) have the same weight distribution.
The cosets of weight or 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 are also considered. For MDS codes of
covering radius we obtain the number of the weight 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 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 .Comment: 32 pages, 45 references. The text is edited. The connections between
distinct parts of the paper are noted. Some transformations are simplified.
New results are added. Open problems are formulate