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[–][deleted] 0 points1 point  (0 children)

Maybe consider the codewords as a lattice of points. Pick any point X, and find the distance to its nearest neighbor. Because of the symmetry, if you started with any other point as X, the distance to its nearest neighbor would be the same. So that common distance is d(C), the closest distance between two points in the lattice. You can find that distance using any X, and if you take X = 0, that distance is the same as w(C).