In this article we construct a new family of linear maximum rank distance (MRD) codes for all parameters. This family contains the only known family, the Gabidulin codes, and contains codes inequivalent to the Gabidulin codes. We also calculate the automorphism group of these codes, including the automorphism group of the Gabidulin codes. Preliminaries Delsarte introduced rank metric codes in [8] . A rank metric code C is a subset of a matrix space M = M m×n (F), F a field, equipped with the distance function d(X, Y ) := rank(X − Y ). A rank metric code is called F 0 -linear if it forms an F 0 -subspace of M for some subfield F 0 ≤ F. A maximum rank distance code (MRD-code) is a rank metric code over a finite field F q meeting the Singleton-like bound |C| ≤ q n(m−d+1) , where d is the minimum distance of C. If C is an F q0 -linear MRD code in M m×n (F q ) with d = m − k + 1 for a subfield F q0 of F q , we say that C has parameters [nm, nk, m − k + 1] q0 , with the subscript omitted when there is no ambiguity. Delsarte [8] and Gabidulin [11] constructed linear MRD-codes over the finite field F q for every k, m and n. In the literature these are usually called (generalised) Gabidulin codes, although the first construction was by Delsarte. When n = m and k = 1, MRD-codes correspond to algebraic structures called quasifields, see Subsection 1.2. Cossidente-Marino-Pavese [6] recently constructed non-linear MRD-codes for n = m = 3, k = 2. When n = m and 1 < k < n − 1, no other linear MRD-codes were known. In this paper we will construct a new family of linear MRD-codes for each k, and we will show that they contain codes inequivalent to the generalised Gabidulin codes. A good overview of MRD-codes can be found in [28] . Similar problems for symmetric, alternating and hermitian matrices have been studied in for example [12]  A finite presemifield is a division algebra in which multiplication is not necessarily associative; if a multiplicative identity element exists, it is called a semifield. We refer to [20] for background, definitions, and terminology. Presemifield are studied in equivalence classes known as isotopy classes. Presemifields of order q n with centre containing F q0 ≤ F q and left nucleus containing F q are in one-to-one correspondence [n 2 , n, n] q0 MRD-codes in M n (F q ) (i.e. F q0 -linear MRD-codes with k = 1). In the theory of semifields, such spaces are called semifield spread sets. Many constructions for finite semifields are known, see for example [18] and [20] . There are two important operations defined on presemifields; the dual, which is the opposite algebra, and the transpose. These together form a chain of six (isotopy classes of) semifields, known as the Knuth orbit.
‚ 2016: Another generalization of Gabidulin codes, called twisted Gabidulin codes, were discovered in REF .
A new family of linear maximum rank distance codes
arXiv: Combinatorics
Computer Science Mathematics