Theory of Codes with Maximum Rank Distance

The article considers codes over $GF(q^N)$. A new metric, called the rank metric, is introduced; the maximum number of coordinates of vector $\mathbf{x}=(x_1,\dots,x_n)$ that are linearly dependent over $GF(q)$ is called its norm. For this metric a theory analogous to the theory of MDS codes is formulated. Codes with maximum rank distance are described; their spectrum is obtained; and encoding and decoding algorithms are given.