相关论文: Almost Affine Vector Rank-Metric Codes
Multilinear representability extends classical linear representability of matroids by assigning subspaces, rather than vectors, to ground elements. This notion is closely related to almost affine codes. In this paper, we introduce and study…
It is well known that linear rank-metric codes give rise to q-polymatroids. Analogously to matroid theory one may ask whether a given q-polymatroid is representable by a rank-metric code. We provide an answer by presenting an example of a…
We consider $q$-matroids and their associated classical matroids derived from Gabidulin rank-metric codes. We express the generalized rank weights of a Gabidulin rank-metric code in terms of Betti numbers of the dual classical matroid…
We revisit and extend the connections between $\mathbb{F}_{q^m}$-linear rank-metric codes and evasive $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^k$. We give a unifying framework in which we prove in an elementary way how the parameters…
This paper contributes to the study of rank-metric codes from an algebraic and combinatorial point of view. We introduce $q$-polymatroids, the $q$-analogue of polymatroids, and develop their basic properties. We associate a pair of…
Classical designs and their (projective) q-analogs can both be viewed as designs in matroids, using the matroid of all subsets of a set and the matroid of linearly independent subsets of a vector space, respectively. Another natural matroid…
By exploiting the connection between scattered $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^3$ and minimal non degenerate $3$-dimensional rank metric codes of $\mathbb{F}_{q^m}^{n}$, $n \geq m+2$, described in [2], we will exhibit a new…
We consider linear rank-metric codes in $\mathbb F_{q^m}^n$. We show that the properties of being MRD (maximum rank distance) and non-Gabidulin are generic over the algebraic closure of the underlying field, which implies that over a large…
We show how the theory of affine geometries over the ring ${\mathbb Z}/\langle q - 1\rangle$ can be used to understand the properties of toric and generalized toric codes over ${\mathbb F}_q$. The minimum distance of these codes is strongly…
We study properties of rank metric and codes in rank metric over finite fields. We show that in rank metric perfect codes do not exist. We derive an existence bound that is the equivalent of the Gilbert--Varshamov bound in Hamming metric.…
We define a class of automorphisms of rational function fields of finite characteristic and employ these to construct different types of optimal linear rank-metric codes. The first construction is of generalized Gabidulin codes over…
Over a finite field $\mathbb{F}_{q^m}$, the evaluation of skew polynomials is intimately related to the evaluation of linearized polynomials. This connection allows one to relate the concept of polynomial independence defined for skew…
For any admissible value of the parameters there exist Maximum Rank distance (shortly MRD) $\mathbb{F}_{q^n}$-linear codes of $\mathbb{F}_q^{n\times n}$. It has been shown in \cite{H-TNRR} (see also \cite{ByrneRavagnani}) that, if field…
It is shown that the Whitney function of a representable q-matroid and the collection of all higher weight enumerators of any representing rank-metric code determine each other via a monomial substitution. Moreover, the q-matroid itself and…
In this paper, we investigate the relation between a $q$-matroid and its associated matroid called the projectivization matroid. The latter arises by projectivizing the groundspace of the $q$-matroid and considering the projective space as…
We investigate the structure of intersecting error-correcting codes, with a particular focus on their connection to matroid theory. We establish properties and bounds for intersecting codes with the Hamming metric and illustrate how these…
We study the relationship between a q-analogue of matroids and linear codes with the rank metric in the vector space of matrices with entries in a finite field. We prove a Greene type identity for the rank generating function of these…
Constant-dimension codes have recently received attention due to their significance to error control in noncoherent random network coding. In this paper, we show that constant-rank codes are closely related to constant-dimension codes and…
In this article, we study polymatroids that are representable by means of linear restricted rank-metric codes, namely, by subspaces of the space of alternating, symmetric, or Hermitian square matrices endowed with the rank metric. More…
We study the interplay between the lattice of F_{q^m}-subspaces and the lattice of F_{q^m}-subspaces of an F_{q^m}-vector space. Introducing notions of weight and defect relative to an F_q-subspace, we analyze the sequence of maximum…