We consider linear codes associated to Schubert varieties in Grassmannians. Aformula for the minimum distance of these codes was conjectured in 2000 andafter having been established in various special cases, it was proved in 2008by Xiang. We give an alternative proof of this formula. Further, we propose acharacterization of the minimum weight codewords of Schubert codes byintroducing the notion of Schubert decomposable elements of certain exteriorpowers. It is shown that codewords corresponding to Schubert decomposableelements are of minimum weight and also that the converse is true in manycases. A lower bound, and in some cases, an exact formula, for the number ofminimum weight codewords of Schubert codes is also given. From a geometricpoint of view, these results correspond to determining the maximum number of$\mathbb{F}_q$-rational points that can lie on a hyperplane section of aSchubert variety in a Grassmannian with its nondegenerate embedding in aprojective subspace of the Pl\"ucker projective space, and also the number ofhyperplanes for which the maximum is attained.
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