Abstract We focus on two aspects of cyclic orbit codes: invariants under equivalence and quasi-optimality. Regarding the first aspect, we establish a connection between the cyclic orbit code generated by a subspace U of $${{\mathbb {F}}}_{q^n}$$ F q n and the associated linear set $$L_{U\times U}$$ L U × U . Relating the size of the linear set to the number of fractions formed by the elements of U allows us to derive new bounds on the parameters of the cyclic orbit code. In the second part, we study a particular family of (quasi-)optimal cyclic orbit codes. With the aid of these codes we establish the existence of quasi-optimal codes in even-dimensional vector spaces over finite fields of any characteristic. Finally, for the particular code family we determine the automorphism groups in various general linear group, depending on the assumed ground field, and their orbits under the Galois group over the prime field.

Subspace codes have recently been used for error correction in random network coding. In this work, we focus on one-orbit cyclic subspace codes. If$S$is an$\mathbb{F}_{q}$-subspace of$\mathbb{F}_{q^{n}}$, then the one-orbit cyclic subspace code defined by$S$is$\text{Orb} (S)=\{\alpha S:\alpha\in \mathbb{F}_{q^{n}}^{\ast}\}$, where$\alpha S=\{\alpha s:s\in S\}$. for any$\alpha\in \mathbb{F}_{q^{n}}^{\ast}$. Few classification results of subspace codes are known, therefore it is quite natural to initiate a classification of cyclic subspace codes, especially in the light of the recent classification of the isometries for cyclic subspace codes. We consider three-dimensional one-orbit cyclic subspace codes, which are divided into three families: the first one containing only Orb(Fq3); the second one containing the optimum-distance codes; and the third one whose elements are codes with minimum distance 2. We study inequivalent codes in the latter two families.

Two-weight linear codes are linear codes in which any nonzero codeword can have only two possible distinct weights. Those in the Hamming metric have proven to be very interesting for their connections with authentication codes, association schemes, strongly regular graphs, and secret sharing schemes. In this paper, we characterize two-weight codes in the rank metric, answering a recent question posed by Pratihar and Randrianarisoa.

Abstract. In this paper, we investigate the problem of determining the number of codewords with maximum rank weight in a linear rank-metric code. We also characterize the codes that reach the maximum or the minimum value for this parameter.

Abstract. We investigate two fundamental questions intersecting coding theory and combinatorial geometry, with emphasis on their connections. These are the problem of computing the asymptotic density of MRD codes in the rank metric, and the Critical Problem for combinatorial geometries by Crapo and Rota. In the first part of the paper, we use methods from semifield theory to derive two lower bounds for the density function of full-rank, square MRD codes. The first bound is sharp when the matrix size is a prime number and the underlying field is sufficiently large, while the second bound applies to the binary field. We then take a new look at the Critical Problem for combinatorial geometries, approaching it from a qualitative, often asymptotic, viewpoint. We illustrate the connection between this very classical problem and that of computing the asymptotic density of MRD codes. Finally, in the third part of the paper we study the asymptotic density of some special families of codes in the rank metric, including the symmetric, alternating, and Hermitian ones. In particular, we show that the optimal codes in these three contexts are sparse.

Scattered polynomials of a given index over finite fields are intriguing rare objects with many connections within mathematics. Of particular interest are the exceptional ones, as defined in 2018 by the first author and Zhou, for which partial classification results are known. In this paper we propose a unified algebraic description of$\mathbb {F}_{q^{n}}$-linear maximum rank distance codes, introducing the notion of exceptional linear maximum rank distance codes of a given index. Such a connection naturally extends the notion of exceptionality for a scattered polynomial in the rank metric framework and provides a generalization of Moore sets in the monomial MRD context. We move towards the classification of exceptional linear MRD codes, by showing that the ones of index zero are generalized Gabidulin codes and proving that in the positive index case the code contains an exceptional scattered polynomial of the same index.

A subspace of matrices in${\mathbb F}_{q^{e}}^{m\times n}$can be naturally embedded as a subspace of matrices in${\mathbb F}_{q}^{em\times en}$with the property that the rank of any of its matrix is a multiple of$e$. It is quite natural to ask whether or not all subspaces of matrices with such a property arise from a subspace of matrices over a larger field. In this paper we explore this question, which corresponds to studying divisible codes in the rank metric. We determine some cases for which this question holds true, and describe counterexamples by constructing subspaces with this property which do not arise from a subspace of matrices over a larger field.

In this paper we present an interpolation-based decoding algorithm to decode a family of maximum rank distance codes proposed recently by Trombetti and Zhou. We employ the properties of the Dickson matrix associated with a linearized polynomial with a given rank and the modified Berlekamp-Massey algorithm in decoding. When the rank of the error vector attains the unique decoding radius, the problem is converted to solving a quadratic polynomial, which ensures that the proposed decoding algorithm has polynomial-time complexity.

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.