Codes and Designs Related to Lifted MRD Codes

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 which attain these bounds are constructed. These codes are the largest known codes for the given parameters. These transversal designs can be also 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.