Related papers: Rank-metric codes, linear sets, and their duality
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…
In this manuscript, we construct a class of projective three-weight linear codes and two classes of projective four-weight linear codes over F2 from the defining sets construction, and determine their weight distributions by using additive…
Linear codes with a few weights can be applied to communication, consumer electronics and data storage system. In addition, the weight hierarchy of linear codes has many applications such as on the type II wire-tap channel, dealing with…
A subspace of matrices over $\mathbb{F}_{q^e}^{m\times n}$ 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…
We consider linear codes over a finite field of odd characteristic, derived from determinantal varieties, obtained from symmetric matrices of bounded ranks. A formula for the weight of a code word is derived. Using this formula, we have…
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…
The projective space $\mathbb{P}_q(n)$, i.e. the set of all subspaces of the vector space $\mathbb{F}_q^n$, is a metric space endowed with the subspace distance metric. Braun, Etzion and Vardy argued that codes in a projective space are…
Saturating sets are combinatorial objects in projective spaces over finite fields that have been intensively investigated in the last three decades. They are related to the so-called covering problem of codes in the Hamming metric. In this…
We study spaces of realisations of linkages (weighted graphs) whose underlying graph is a series parallel graph. In particular, we describe an algorithm for determining whether or not such spaces are connected.
Linear codes are considered over the ring $\mathbb{Z}_4+v\mathbb{Z}_4$, where $v^2=v$. Gray weight, Gray maps for linear codes are defined and MacWilliams identity for the Gray weight enumerator is given. Self-dual codes, construction of…
The study of the generalized Hamming weight of linear codes is a significant research topic in coding theory as it conveys the structural information of the codes and determines their performance in various applications. However,…
It is well-known that few-weight linear codes have better applications in secret sharing schemes \cite{JY2006,CC2005}.In particular, projective two-weight codes are very precious as they are closely related to finite projective spaces,…
The aim of this paper is to develop a $(q,m)$-polymatroidal approach to higher supports and higher rank-weight enumerators of rank-metric codes. In this framework, we establish analogs of several fundamental results known for matroids and…
We present a structure associated to the class of linear codes. The properties of that structure are similar to some structures in the linear algebra techniques into the framework of the Gr\"obner bases tools. It allows to get some insight…
We describe new relations among conformal block divisors in $\operatorname{Pic}(\bar{\operatorname{M}}_{0,n})$. These relations appear from various rank-level dualities of conformal blocks on $\mathbb{P}^1$ with $n$ marked points. We also…
In this paper, we mainly study quaternary linear codes and their binary subfield codes. First we obtain a general explicit relationship between quaternary linear codes and their binary subfield codes in terms of generator matrices and…
We present the theory of linear rank-metric codes from the point of view of their fundamental parameters. These are: the minimum rank distance, the rank distribution, the maximum rank, the covering radius, and the field size. The focus of…
The aim of this paper is to investigate the intersection problem between two linear sets in the projective line over a finite field. In particular, we analyze the intersection between two clubs with eventually different maximum fields of…
The aim of this survey is to outline the state of the art in research on a class of linearized polynomials with coefficients over finite fields, known as scattered polynomials. These have been studied in several contexts, such as in [A.…
The equivalence problem of $\mathbb{F}_q$-linear sets of rank n of $PG(1,q^n)$ is investigated, also in terms of the associated variety, projecting configurations, $\mathbb{F}_q$-linear blocking sets of R\'edei type and MRD-codes.