This paper introduces a new metric, called the “rank metric”. For this metric, a theory of codes over \(\mathrm{GF}(q^N)\) analogous to the theory of MBD codes is formulated. The studies presented include codes with maximum rank distance, their spectrum, and encoding and decoding algorithms. Analogous to “generalized Reed-Solomon codes” an extensive class of MRD codes (maximum rank distance codes) are described.