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Universally decodable matrices (UDMs) can be used for coding purposes when transmitting over slow fading channels. These matrices are parameterized by positive integers L and N and a prime power q. The main result of this correspondence is that the simple condition L = q + 1 is both necessary and sufficient for (L, N, q)-VDMs to exist. The existence proof is constructive and yields a coding scheme that is equivalent to a class of codes that was proposed by Rosenbloom and Tsfasman. Our work resolves an open problem posed recently in the literature.