Related papers: $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive cyclic cod…

We present a classification of the so-called "additive symmetric 2-cocycles" of arbitrary degree and dimension over Z/p, along with a partial result and some conjectures for m-cocycles over Z/p, m > 2. This expands greatly on a result…

This paper investigates the algebraic and graphical structure of the ring $\mathbb{Z}_{sp}$, with a focus on its decomposition into finite fields, kernels, and special subsets. We establish classical isomorphisms between $\mathbb{F}_s$ and…

In this paper we investigate repeated root cyclic and negacyclic codes of length $p^rm$ over $\mathbb{F}_{p^s}$ with $(m,p)=1$. In the case $p$ odd, we give necessary and sufficient conditions on the existence of negacyclic self-dual codes.…

Given two binary codes of length n, using Plotkin construction we obtain a code of length 2n. The construction works for linear and nonlinear codes. For the linear case, it is straightforward to see that the dimension of the final code is…

In this paper, we mainly study the theory of linear codes over the ring $R =\mathbb{Z}_4+u\mathbb{Z}_4+v\mathbb{Z}_4+uv\mathbb{Z}_4$. By the Chinese Remainder Theorem, we have $R$ is isomorphic to the direct sum of four rings…

Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimension and alphabet are given. An application to irreducible cyclic codes is considered. Sharper upper bounds are given for the special cyclic…

A new Gray map which is both an isometry and a weight preserving map from R=F_2+vF_2+v^2F_2 to (F_2)^3 is defined. A construction for quantum error correcting codes from cyclic codes over finite ring R=F_2+vF_2+v^2F_2, v^3=v is given. The…

In this paper, we have studied cyclic codes over the ring $R=\mathbb{Z}_4+u\mathbb{Z}_4$, $u^2=0$. We have considered cyclic codes of odd lengths. A sufficient condition for a cyclic code over $R$ to be a $\mathbb{Z}_4$-free module is…

Let $p$ be an odd prime number, $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ and $s$ a positive integer. Using some combinatorial identities, we obtain certain properties for Kronecker product of matrices over $\mathbb{F}_p$…

Cyclic codes have attracted a lot of research interest for decades. In this paper, for an odd prime $p$, we propose a general strategy to compute the complete weight enumerator of cyclic codes via the value distribution of the corresponding…

An additive code is an $\mathbb{F}_q$-linear subspace of $\mathbb{F}_{q^m}^n$ over $\mathbb{F}_{q^m}$, which is not a linear subspace over $\mathbb{F}_{q^m}$. Linear complementary pairs (LCP) of codes have important roles in cryptography,…

We examine the phenomenon of capitulation of the $p$-class group $H_K$ of a real number field $K$ in totally ramified cyclic p-extensions $L/K$ of degree $p^N$. Using an elementary property of the algebraic norm $\nu_{L/K}$, we show that…

Determining the weight distribution of a linear code is a classical and fundamental topic in coding theory that has been extensively investigated. Repeated-root cyclic codes, which form a significant subclass of error-correcting codes, have…

Sum-rank codes are an important class of codes which can be utilized for linear network coding, space-time coding and distributed storage. They can not only reduce the size of network alphabet but also detect and correct more errors. Based…

Let $\mathcal{R}=\mathbb{F}_{p^m}[u]/\langle u^3 \rangle $ be the finite commutative chain ring with unity, where $p$ is a prime, $m$ is a positive integer and $\mathbb{F}_{p^m}$ is the finite field with $p^m$ elements. In this paper, we…

An outstanding folklore conjecture asserts that, for any prime $p$, up to isomorphism the projective plane $PG(2,\mathbb{F}_p)$ over the field $\mathbb{F}_p := \mathbb{Z}/p\mathbb{Z}$ is the unique projective plane of order $p$. Let $\pi$…

In this article, we investigate properties of cyclic codes over a finite non-chain ring $\mathbb{F}_q+v\mathbb{F}_q+v^2\mathbb{F}_q+v^3\mathbb{F}_q+v^4\mathbb{F}_q,$ where $q=p^r,$ $r$ is a positive integer, $p$ is an odd prime, $4 \mid…

We investigate the $a$-numbers of $\mathbb{Z}/p^2\mathbb{Z}$-covers in characteristic $p>2$ and extend a technique originally introduced by Farnell and Pries for $\mathbb{Z}/p\mathbb{Z}$-covers. As an application of our approach, we…

Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$, where $p$ is a prime, and $k, N$ be any positive integers. We denote $R_k=F_{p^m}[u]/\langle u^k\rangle =F_{p^m}+uF_{p^m}+\ldots+u^{k-1}F_{p^m}$ ($u^k=0$) and…

Let $m\geq 3$ be an odd integer and $p$ be an odd prime. % with $p-1=2^rh$, where $h$ is an odd integer. In this paper, many classes of three-weight cyclic codes over $\mathbb{F}_{p}$ are presented via an examination of the condition for…