Related papers: Cartesian square-free codes
We consider linear error correcting codes associated to higher dimensional projective varieties defined over a finite field. The problem of determining the basic parameters of such codes often leads to some interesting and difficult…
In this letter we consider the ensemble of codes formed by the serial concatenation of a Hamming code and two accumulate codes. We show that this ensemble is asymptotically good, in the sense that most codes in the ensemble have minimum…
In \cite{shi2022few-weight}, Shi and Li studied $\mathcal{C}_D$-codes over the ring $\mathcal{R}:=\mathbb{F}_2[x,y]/\langle x^2, y^2, xy-yx\rangle$ and their binary Gray images, where $D$ is derived using certain simplicial complexes. We…
It is well-known that the first generalized Hamming weight of a linear code, more commonly called \textit{the minimum distance} of the linear code, corresponds to the initial degree of the Stanley-Reisner ideal of the matroid of the dual…
The high-density data storage technology aims to design high-capacity storage at a relatively low cost. In order to achieve this goal, symbol-pair codes were proposed by Cassuto and Blaum \cite{CB10,CB11} to handle channels that output…
In this paper we consider completely regular codes, obtained from perfect (Hamming) codes by lifting the ground field. More exactly, for a given perfect code C of length n=(q^m-1)/(q-1) over F_q with a parity check matrix H_m, we define a…
This work develops new foundations for the theory of linear codes over local Artinian commutative rings. We use algebraic invariants such as the socle, type, length, and minimal number of generators to measure the size of codes. We prove a…
We introduce and analyze a natural geometric version of Renegar's condition number R for the homogeneous convex feasibility problem associated with a regular cone C subseteq R^n. Let Gr_{n,m} denote the Grassmann manifold of m-dimensional…
In this paper we study a special type of quasi-cyclic (QC) codes called skew QC codes. This set of codes is constructed using a non-commutative ring called the skew polynomial rings $F[x;\theta ]$. After a brief description of the skew…
The sizes of optimal constant-composition codes of weight three have been determined by Chee, Ge and Ling with four cases in doubt. Group divisible codes played an important role in their constructions. In this paper, we study the problem…
We introduce a consistent and efficient method to construct self-dual codes over $GF(q)$ with symmetric generator matrices from a self-dual code over $GF(q)$ of smaller length where $q \equiv 1 \pmod 4$. Using this method, we improve the…
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…
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…
The evaluation of the minimum distance of linear block codes remains an open problem in coding theory, and it is not easy to determine its true value by classical methods, for this reason the problem has been solved in the literature with…
In this paper, we propose a class of linear codes and obtain their weight distribution. Some of these codes are almost optimal. Moreover, several classes of constant composition codes(CCCs) are constructed as subcodes of linear codes.
The hull of a linear code (i.e., a finite field vector space)~\({\mathcal C}\) is defined to be the vector space formed by the intersection of~\({\mathcal C}\) with its dual~\({\mathcal C}^{\perp}.\) Constructing vector spaces with a…
We explicitly determine all the relative generalized Hamming weights of affine Cartesian codes using the notion of footprints and results from extremal combinatorics. This generalizes the previous works on the determination of relative…
It is shown in this paper that, if $R$ is a Frobenius ring, then the quaternion ring $\mathcal{H}_{a,b}(R)$ is a Frobenius ring for all units $a,b \in R$. In particular, if $q$ is an odd prime power then $\mathcal{H}_{a,b}(\mathbb{F}_q)$ is…
This work investigates the structure of rank-metric codes in connection with concepts from finite geometry, most notably the $q$-analogues of projective systems and blocking sets. We also illustrate how to associate a classical…
In this paper, a Roos like bound on the minimum distance for skew cyclic codes over a general field is provided. The result holds in the Hamming metric and in the rank metric. The proofs involve arithmetic properties of skew polynomials and…