An Assmus-Mattson Theorem for Rank Metric Codes
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A t-(n,d,λ) design over F_q, or a subspace design, is a collection of d-dimensional subspaces of F_q^n, called blocks, with the property that every t-dimensional subspace of F_q^n is contained in the same number λ of blocks. A collection of matrices in over F_q is said to hold a subspace design if the set of column spaces of its elements forms the blocks of a subspace design. We use notions of puncturing and shortening of rank metric codes and the rank-metric MacWilliams identities to establish conditions under which the words of a given rank in a linear rank metric code hold a subspace design.
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