Algebraic structures of MRD codes

Let Fq denote a finite field with q elements and let V = (Fq)m,n be the Fq-vector space of matrices over Fq of type (m,n). On V we define the so-called rank metric distance by d(A,B) = rank(A−B) for A,B ∈ V . Clearly, the distance d is a translation invariant metric on V . A subset C ⊆ V endowed with the metric d is called a rank metric code with minimum distance d(C) = min {d(A,B) | A 6= B ∈ V }. For m ≥ n, an MRD (maximum rank distance) code C ⊆ V satisfies the following two conditions: (i) |C| = q and (ii) d(C) = n− k + 1. Note that an MRD code is a rank metric code which is maximal in size given the minimum distance, or in other words it achieves the Singleton bound for the rank metric distance (see [5, 8]).