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 for general parameters, the Gabidulin codes, and contains codes inequivalent to the Gabidulin codes. This family also contains the well-known family of semifields known as Generalised Twisted Fields. We also calculate the automorphism group of these codes, including the automorphism group of the Gabidulin codes. Delsarte's duality theoremDefine the symmetric bilinear form b on M m,n (F) by b(X, Y ) := tr(Tr(XY T )),where Tr denotes the matrix trace, and tr denotes the absolute trace from F q to F p , where p is prime and q = p e . Define the Delsarte dual C ⊥ of an F p -linear code C by C ⊥ := {Y : Y ∈ M m,n (F q ), b(X, Y ) = 0 ∀X ∈ C}.

In this article, we construct a new family of semifields, containing and extending two well‐known families, namely Albert's generalised twisted fields and Petit's cyclic semifields (also known as Johnson–Jha semifields). The construction also gives examples of semifields with parameters for which no examples were previously known. In the case of semifields two dimensions over a nucleus and four‐dimensional over their centre, the construction gives all possible examples. Furthermore we embed these semifields in a new family of maximum rank‐distance codes, encompassing most known current constructions, including the (twisted) Delsarte–Gabidulin codes, and containing new examples for most parameters.

In this paper we investigate connections between linear sets and subspaces of linear maps. We give a geometric interpretation of the results of [18, Section 5] on linear sets on a projective line. We extend this to linear sets in arbitrary dimension, giving the connection between two constructions for linear sets defined in [9]. Finally, we then exploit this connection by using the MacWilliams identities to obtain information about the possible weight distribution of a linear set of rank n on a projective line PG(1, q n ).

Abstract. Skew polynomial rings were used to construct finite semifields by Petit in [20], following from a construction of Ore and Jacobson of associative division algebras. Johnson and Jha [10] later constructed the so-called cyclic semifields, obtained using irreducible semilinear transformations. In this work we show that these two constructions in fact lead to isotopic semifields, show how the skew polynomial construction can be used to calculate the nuclei more easily, and provide an upper bound for the number of isotopism classes, improving the bounds obtained by Kantor and Liebler in [13] and implicitly by Dempwolff in [2].