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.

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.

A set $$T\subset {GF(q)}$$ , $$q=p^h$$ is a super-Vandermonde set if $$\sum _{y\in T} y^k=0$$ for $$0< k <|T|$$ . We determine the structure of super-Vandermonde sets of size $$p+1$$ (almost small) and size $$q/p-1$$ (almost large).