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• W3202962786 abstract "<p style='text-indent:20px;'>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 <inline-formula><tex-math id=M1>begin{document}$ d $end{document}</tex-math></inline-formula> using the known numbers of vectors of weights <inline-formula><tex-math id=M2>begin{document}$ le d-2 $end{document}</tex-math></inline-formula> in this coset. In this paper, the Bonneau formula is transformed into a more structured and convenient form. The new version of the formula allows to consider effectively cosets of distinct weights <inline-formula><tex-math id=M3>begin{document}$ W $end{document}</tex-math></inline-formula>. (The weight <inline-formula><tex-math id=M4>begin{document}$ W $end{document}</tex-math></inline-formula> of a coset is the smallest Hamming weight of any vector in the coset.) For each of the considered <inline-formula><tex-math id=M5>begin{document}$ W $end{document}</tex-math></inline-formula> or regions of <inline-formula><tex-math id=M6>begin{document}$ W $end{document}</tex-math></inline-formula>, special relations more simple than the general ones are obtained. For the MDS code cosets of weight <inline-formula><tex-math id=M7>begin{document}$ W = 1 $end{document}</tex-math></inline-formula> and weight <inline-formula><tex-math id=M8>begin{document}$ W = d-1 $end{document}</tex-math></inline-formula> we obtain formulas of the weight distributions depending only on the code parameters. This proves that all the cosets of weight <inline-formula><tex-math id=M9>begin{document}$ W = 1 $end{document}</tex-math></inline-formula> (as well as <inline-formula><tex-math id=M10>begin{document}$ W = d-1 $end{document}</tex-math></inline-formula>) have the same weight distribution. The cosets of weight <inline-formula><tex-math id=M11>begin{document}$ W = 2 $end{document}</tex-math></inline-formula> or <inline-formula><tex-math id=M12>begin{document}$ W = d-2 $end{document}</tex-math></inline-formula> may have different weight distributions; in this case, we proved that the distributions are symmetrical in some sense. The weight distributions of the cosets of MDS codes corresponding to arcs in the projective plane <inline-formula><tex-math id=M13>begin{document}$ mathrm{PG}(2,q) $end{document}</tex-math></inline-formula> are also considered. For MDS codes of covering radius <inline-formula><tex-math id=M14>begin{document}$ R = d-1 $end{document}</tex-math></inline-formula> we obtain the number of the weight <inline-formula><tex-math id=M15>begin{document}$ W = d-1 $end{document}</tex-math></inline-formula> cosets and their weight distribution that gives rise to a certain classification of the so-called deep holes. We show that any MDS code of covering radius <inline-formula><tex-math id=M16>begin{document}$ R = d-1 $end{document}</tex-math></inline-formula> is an almost perfect multiple covering of the farthest-off points (deep holes); moreover, it corresponds to an optimal multiple saturating set in the projective space <inline-formula><tex-math id=M17>begin{document}$ mathrm{PG}(N,q) $end{document}</tex-math></inline-formula>.</p>" @default.
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• W3202962786 date "2021-01-01" @default.
• W3202962786 modified "2023-10-13" @default.
• W3202962786 title "On the weight distribution of the cosets of MDS codes" @default.
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• W3202962786 doi "https://doi.org/10.3934/amc.2021042" @default.
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