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doi: 10.3934/amc.2021042
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## On the weight distribution of the cosets of MDS codes

 1 Institute for Information Transmission Problems (Kharkevich institute), Russian Academy of Sciences, Moscow, 127051, Russian Federation 2 Department of Mathematics and Computer Science, University of Perugia, Perugia, 06123, Italy

* Corresponding author: Alexander A. Davydov

Received  January 2021 Revised  June 2021 Early access September 2021

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

Citation: Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco. On the weight distribution of the cosets of MDS codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2021042
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