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].

A fundamental problem in the theory of linearized and projective polynomials over finite fields is to characterize the number of roots in the coefficient field directly from the coefficients. We prove results of this type, of a recursive nature. These results follow from our main theorem which characterizes the number of roots using the rank of a matrix that is smaller than the Dickson matrix.

We present the theory of rank-metric codes with respect to the 3-tensors that generate them. We define the generator tensor and the parity check tensor of a matrix code, and describe the properties of a code through these objects. We define the tensor rank of a code to be the tensor rank of its generating tensors, and propose that this quantity is a significant coding theoretic parameter. By a result on the tensor rank of Kruskal from the 1970s, the tensor rank of a rank-metric code of dimension k and minimum rank distance d is at least k + d − 1. We call codes that meet this bound minimal tensor rank (MTR) codes. It is known from results in algebraic complexity theory that an MTR code implies the existence of an MDS code. In this paper, we also address the converse problem, that of the existence of an MTR code, given an MDS code. We identify several parameters for which the converse holds and give explicit constructions of MTR codes using MDS codes. We furthermore define generalized tensor ranks, which give a refinement of the tensor rank as a code invariant. Moreover, we use these to distinguish inequivalent rank-metric codes.

Rank-metric codes are codes consisting of matrices with entries in a finite field, with the distance between two matrices being the rank of their difference. Codes with maximum size for a fixed minimum distance are called Maximum Rank Distance (MRD) codes. Such codes were constructed and studied independently by Delsarte (1978), Gabidulin (1985), Roth (1991), andCooperstein (1998). Rank-metric codes have seen renewed interest in recent years due to their applications in random linear network coding.MRD codes also have interesting connections to other topics such as semifields (finite nonassociative division algebras), finite geometry, linearized polynomials, and cryptography. In this chapter we will survey the known constructions and applications of MRD codes, and present some open problems.The most well-known and widely-used example are codes in the Hamming metric; codewords are taken from a vector-space over a finite field, and distance between two vectors in F n q is defined as the number of positions in which they differ. We refer to [65] for a detailed reference on this well-studied topic.In rank-metric coding, codewords are instead taken from the set of matrices over a finite field, with the distance between two matrices defined as the rank of their difference.Definition 1. Let F q denote the finite field with q elements, and M n×m (F q ) the set of n × m matrices with entries in F q , with m ≤ n. The rank-weight of a matrix X is defined to be its usual column rank, and denoted by rank(X). The rank-distance between two matrices X, Y ∈ M n×m (F q ) is denoted by d(X, Y ) and defined asA rank-metric code is then a subset C of M n×m (F q ). The minimum distance of a set C with at least two elements, denoted d(C) is MRD codesA main goal in coding theory is to find codes of maximum possible size for a given minimum distance. While there are a variety of bounds for codes in the Hamming metric [65], for rank metric codes we need only the following simple analogue of the Singleton bound, first proved by Delsarte [24]. We include a short proof for illustrative purposes.Theorem 1 (Singleton-like bound, Delsarte). Suppose C ⊂ M n×m (F q ) is such that d(C) = d. Then |C| ≤ q n(m−d+1) .(1)Proof. Suppose |C| > q n(m−d+1) . By the pigeonhole principle, there must exist two distinct elements X, Y of C which coincide in every entry of their first m − d + 1 rows. Thus X − Y has at least m − d + 1 zero rows, and thus rank(X − Y ) ≤ d − 1 < d, a contradiction.is said to be a Maximum Rank Distance code, or MRD-code for short. We say C is a [n × m, d(C)]-MRD code.MRD-codes are the rank-metric analogue of MDS-codes in the Hamming metric. Unlike MDS codes, MRD codes in fact exist for all values of q, n, m, and all d. This was first shown by Delsarte [24]; indeed, he showed that the bound can be met by an additively-closed set.Theorem 2 ([24]). There exists an MRD code in M n×m (F q ) with minimum distance d for all q, n, m, and all d.This article with focus on the case where C is an MRD-code which is additively-closed, which we wi...

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