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On the weight distribution of the cosets of MDS codes

  • * Corresponding author: Alexander A. Davydov

    * Corresponding author: Alexander A. Davydov 
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  • 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) $.

    Mathematics Subject Classification: Primary: 94B05; Secondary: 51E21, 51E22.


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