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Linear codes with few weights have been a significant area of research in coding theory for many years, due to their applications in secret sharing schemes, authentication codes, association schemes, and strongly regular graphs. Inspired by…
Linear codes play a central role in coding theory and have applications in several branches of mathematics. For error correction purposes the minimum Hamming distance should be as large as possible. Linear codes related to applications in…
We consider the class of linear antipodal two-weight rank metric codes and discuss their properties and characterization in terms of $t$-spreads. It is shown that the dimension of such codes is $2$ and the minimum rank distance is at least…
A linear code of length $n$ over a finite chain ring $R$ with residue field $\F_q$ is a $R$-submodule of $R^n$. A $R$-linear code is a code over $\F_q$ (not necessarily linear) which is the generalized Gray map image of a linear code over…
In this article, we consider Schubert codes, linear codes associated to Schubert varieties, and discuss minimum weight codewords for dual Schubert codes. The notion of lines in Schubert varieties is looked closely at, and it has been proved…
It is known that for binary codes one can use Gr\"obner bases to obtain a subset of codewords of minimal support that can be used to determine the second generalized Hamming weight of the code. In this paper we establish conditions on a…
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,…
Bounds on linear codes play a central role in coding theory, as they capture the fundamental trade-off between error-correction capability (minimum distance) and information rate (dimension relative to length). Classical results…
We give an upper bound that relates the minimum weight of a nonzero componentwise product of codewords from some given number of linear codes, with the dimensions of these codes. Its shape is a direct generalization of the classical…
In this paper, we propose and study $r$-minimal codes, a natural extension of minimal codes which have been extensively studied with respect to Hamming metric, rank metric and sum-rank metric. We first propose $r$-minimal codes in a general…
Let ${\mathcal C}(\Omega)$ be the linear code arising from a projective system $\Omega$ of $\mathrm{PG}(V).$ Consider the point-line geometry $\Gamma=({\mathcal P},{\mathcal L})$ and a projective embedding $\varepsilon\colon…
The main purpose of this paper is to further study the structure, parameters and constructions of the recently introduced minimal codes in the sum-rank metric. These objects form a bridge between the classical minimal codes in the Hamming…
We study affine cartesian codes, which are a Reed-Muller type of evaluation codes, where polynomials are evaluated at the cartesian product of n subsets of a finite field F_q. These codes appeared recently in a work by H. Lopez, C.…
We discuss the problem of determining the complete weight hierarchy of linear error correcting codes associated to Grassmann varieties and, more generally, to Schubert varieties in Grassmannians. The problem is partially solved in the case…
Linear codes have been an interesting topic in both theory and practice for many years. In this paper, a class of $q$-ary linear codes with few weights are presented and their weight distributions are determined using Gauss periods. Some of…
This text contains some notes on the Griesmer bound. In particular, we give a geometric proof of the Griesmer bound for the generalized weights and show that a Solomon--Stiffler type construction attains it if the minimum distance is…
The error coefficient of a linear code is defined as the number of minimum-weight codewords. In an additive white Gaussian noise channel, optimal linear codes with the smallest error coefficients achieve the best possible asymptotic frame…
Subfield codes of linear codes over finite fields have recently received much attention. Some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, the $q$-ary…
Minimal codes are a class of linear codes which gained interest in the last years, thanks to their connections to secret sharing schemes. In this paper we provide the weight distribution and the parameters of families of minimal codes…
We consider binary abelian codes of length $p^m q^n$, where $p$ and $q$ are prime rational integers under some restrictive hypotheses. In this case, we determine the idempotents generating minimal codes and either the respective weights or…