Abstract. By exploiting the connection between scattered [Formula: see text]-subspaces of [Formula: see text] and minimal nondegenerate 3-dimensional rank-metric codes of [Formula: see text], [Formula: see text], described in [ G. N. Alfarano et al., J. Combin. Theory Ser. A, 192 (2022), 105658 ], we will exhibit a new class of codes with parameters [Formula: see text] for infinite values of [Formula: see text] and [Formula: see text] odd. Moreover, by studying the geometric structures of these scattered subspaces, we determine the rank weight distribution of the associated codes.

Abstract In this article, constant dimension subspace codes whose codewords have subspace distance in a prescribed set of integers, are considered. The easiest example of such an object is a junta (Combin Probab Comput 18(1–2):107–122, 2009); i.e. a subspace code in which all codewords go through a common subspace. We focus on the case when only two intersection values for the codewords, are assigned. In such a case we determine an upper bound for the dimension of the vector space spanned by the elements of a non-junta code. In addition, if the two intersection values are consecutive, we prove that such a bound is tight, and classify the examples attaining the largest possible dimension as one of four infinite families.

Let k, n, m ∈ ℤ+be integers such that k ≤ n ≤ m, let Gn,k∈ Fn(qm) be a Delsarte-Gabidulin code. Recently, Wachter-Zeh proved that codes belonging to this family cannot be efficiently list decoded for any radius τ, providing τ is large enough. This achievement essentially relies on proving a lower bound for the list size of some specific words in Fn(qm). Some years later, Raviv and Wachter-Zeh improved this bound in a special case, i.e. when n|m. As a consequence, they were able to detect infinite families of Delsarte-Gabidulin codes that cannot be efficiently list decoded at all. In this article we determine similar lower bounds for Maximum Rank Distance codes belonging to a wider class of examples, containing Generalized Gabidulin codes, Generalized Twisted Gabidulin codes, and examples recently described by Trombetti and Zhou. By exploiting arguments such as those used by Raviv and Wachter-Zeh, when n|m, we also show infinite families of generalized Gabidulin codes that cannot be list decoded efficiently at any radius greater than or equal to ⌊d-1/2⌋ + 1, where d is its minimum distance. Nonetheless, in all the examples belonging to above mentioned class, we detect infinite families that cannot be list decoded efficiently at any radius greater than or equal to ⌊d-1/2⌋ + 2, where d is its minimum distance. In particular, this leads to show infinite families of Gabidulin codes, with underlying parameters not already covered by the result of Raviv and Wachter-Zeh, having this decodability defect. Finally, relying on the properties of a set of subspace trinomials recently presented by McGuire and Mueller, we are able to prove our main result, that is any rank-metric code of Fn(qm) of order qknwith n dividing m, such that 4n - 3 is a square in ℤ and containing Gn,2, is not efficiently list decodable at some values of the radius τ.

In this paper, we present a new family of maximum rank-distance (MRD) codes in$\mathbb F_{q}^{2n\times 2n}$of minimum distance$2\leq d\leq 2n$. In particular, when$d=2n$, we can show that the corresponding semifield is exactly a Hughes–Kleinfeld semifield. The middle and right nuclei of these MRD codes are both equal to$\mathbb F_{q^{n}}$. We also prove that the MRD codes of minimum distance$2< d< 2n$in this family are inequivalent to all known ones. The equivalence between any two members of this new family is also determined.