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Lifted maximum rank distance (MRD) codes, which are constant dimension codes, are considered. It is shown that a lifted MRD code can be represented in such a way that it forms a block design known as a transversal design. A slightly different representation of this design makes it similar to a q -analog of a transversal design. The structure of these designs is used to obtain upper bounds on the sizes of constant dimension codes which contain a lifted MRD code. Codes that attain these bounds are constructed. These codes are the largest known constant dimension codes for the given parameters. These transversal designs can also be used to derive a new family of linear codes in the Hamming space. Bounds on the minimum distance and the dimension of such codes are given.