相关论文: Do non-free LCD codes over finite commutative Frob…

Let $\texttt{R}$ be a commutative finite chain ring of invariants $(q,s)$ and $\Gamma(\texttt{R})$ the Teichm\"uller's set of $\texttt{R}.$ In this paper, the trace representation cyclic $\texttt{R}$-linear codes of length $\ell,$ is…

A method to construct and count all the linear codes (of arbitrary length) in $\mathbb{F}_{4}$ that are invariant under reverse permutation and that contain the repetition code is presented. These codes are suitable for constructing DNA…

Let $ \mathbb F_2[u]/ \langle u^k \rangle= \mathbb F_2+u\mathbb F_2+u^2\mathbb F_2+\cdots+u^{k-1}\mathbb F_2 ,$ where $u^k=0$ for a positive integer $k$, and $\mathcal{R}=M_4 (\mathbb F_2( u)/ \langle u^k \rangle)$ be the finite…

The main aim of this paper is to study $LCD$ codes. Linear code with complementary dual($LCD$) are those codes which have their intersection with their dual code as $\{0\}$. In this paper we will give rather alternative proof of Massey's…

The purpose of this article is to study polycyclic codes over the ring $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}, \,t \geq 1$, and their associated torsion codes. It is shown that if $\phi$ is a surjective ring homomorphism from a…

LDPC convolutional codes have been shown to be capable of achieving the same capacity-approaching performance as LDPC block codes with iterative message-passing decoding. In this paper, asymptotic methods are used to calculate a lower bound…

This paper gives lower and upper bounds on the covering radius of codes over $\mathbb{Z}_{p^2}$ with respect to Lee distance. We also determine the covering radius of various Repetition codes over $\mathbb{Z}_{p^2}.$

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…

Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields are studied. An algorithm for factorizing $x^n-\lambda$ over $\mathbb{F}_{q^2}$ is given, where $\lambda$ is a unit in $\mathbb{F}_{q^2}$. Based on this…

In this paper, we study some repeated-root two-dimensional cyclic and constacyclic codes over a finite field $F=\mathbb{F}_q$. We obtain the generator matrices and generator polynomials of these codes and their duals. We also investigate…

In this paper, we give a method for constructing linear codes with small hulls by generalizing the method in \cite{LCD-T-matric}. As a result, we obtain many optimal Euclidean LCD codes and Hermitian LCD codes, which improve the previously…

In this paper, we study the theory for constructing DNA cyclic codes of odd length over $\Z_4[u]/\langle u^2 \rangle$ which play an important role in DNA computing. Cyclic codes of odd length over $\Z_4 + u \Z_4$ satisfy the reverse…

We determine the asymptotic proportion of free modules over finite chain rings with good distance properties and treat the asymptotics in the code length n and the residue field size q separately. We then specialize and apply our technique…

Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ and $R=\mathbb{F}_{p^m}[u]/\langle u^2\rangle=\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ $(u^2=0)$, where $p$ is an odd prime and $m$ is a positive integer. For any $\alpha,\beta\in…

Locally repairable codes (LRCs) have gained significant interest for the design of large distributed storage systems as they allow a small number of erased nodes to be recovered by accessing only a few others. Several works have thus been…

In this paper, we give a matrix construction method for designing DNA codes that come from group matrix rings. We show that with our construction one can obtain reversible $G^k$-codes of length $kn,$ where $k, n \in \mathbb{N},$ over the…

Let $\mathbb{F}_q$ be a finite field of $q=p^m$ elements where $p$ is a prime and $m$ is a positive integer. This paper considers $(\gamma,\Delta)$-cyclic codes over a class of finite non-chain commutative rings…

Locally Decodable Codes (LDCs) are error-correcting codes $C\colon\Sigma^n\rightarrow \Sigma^m,$ encoding \emph{messages} in $\Sigma^n$ to \emph{codewords} in $\Sigma^m$, with super-fast decoding algorithms. They are important mathematical…

We study the secure decentralized Pliable Index CODing (PICOD) problem with circular side information sets at the users. The security constraint forbids every user to decode more than one message while a decentralized setting means there is…

We construct codes over the ring $\mathbb{F}_2+u\mathbb{F}_2$ with $u^2=0$. These code are designed for use in DNA computing applications. The codes obtained satisfy the reverse complement constraint, the $GC$ content constraint and avoid…