Related papers: Construction and enumeration of left dihedral code…

Let $\mathbb{F}_q$ be a finite field of $q$ elements, for some prime power $q$, and let $G$ be a finite group. A (left) group code, or simply a $G$-code, is a (left) ideal of the group algebra $\mathbb{F}_q[G]$. In this paper, we provide a…

In this paper, we study the dihedral codes, i.e. the left ideals of $\mathbb{F}_qD_{n}$ in the case $\gcd(q, n) = 1$. An explicit algebraic description of the dihedral codes and their duals is obtained. In addition, a criterion for…

Let $D_{2n}=\langle x,y\mid x^n=1, y^2=1, yxy=x^{-1}\rangle$ be a dihedral group, and $R={\rm GR}(p^2,m)$ be a Galois ring of characteristic $p^2$ and cardinality $p^{2m}$ where $p$ is a prime. Left ideals of the group ring $R[D_{2n}]$ are…

Let $R$ be a finite commutative chain ring, $D_{2n}$ be the dihedral group of size $2n$ and $R[D_{2n}]$ be the dihedral group ring. In this paper, we completely characterize left ideals of $R[D_{2n}]$ (called left $D_{2n}$-codes) when ${\rm…

Let $m$ be an arbitrary positive integer and $D_{8m}$ be a dihedral group of order $8m$, i.e., $D_{8m}=\langle x,y\mid x^{4m}=1, y^2=1, yxy=x^{-1}\rangle$. Left ideals of the dihedral group algebra $\mathbb{F}_2[D_{8m}]$ are called binary…

We give a simple construction of codes from left ideals in group algebras of certain dihedral groups and give an example to show that they can produce codes with weights equal to those of the best known codes of the same length.

Let $k$ be an algebraically closed field of odd characteristic $p$, and let $D_n$ be the dihedral group of order $2n$ such that $p\mid 2n$. Let $D(kD_n)$ denote the quantum double of the group algebra $kD_n$. In this paper, we describe the…

Let $\mathbb{F}_{2^m}$ be a finite field of $2^m$ elements, and $R=\mathbb{F}_{2^m}[u]/\langle u^k\rangle=\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}+\ldots+u^{k-1}\mathbb{F}_{2^m}$ ($u^k=0$) where $k$ is an integer satisfying $k\geq 2$. For any odd…

The main focus of this paper is the complete enumeration of self-dual abelian codes in non-principal ideal group algebras $\mathbb{F}_{2^k}[A\times \mathbb{Z}_2\times \mathbb{Z}_{2^s}]$ with respect to both the Euclidean and Hermitian inner…

Linear codes with complementary duals are linear codes whose intersection with their duals are trivial, shortly named LCD codes. In this paper we outline a construction for LCD codes over finite fields of order $q$ using weighing matrices…

A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism from FG to Fn which maps G to the standard basis of Fn. Many classical linear codes have been shown to be group…

In this paper, a criterion of MDS Euclidean self-orthogonal codes is presented. New MDS Euclidean self-dual codes and self-orthogonal codes are constructed via this criterion. In particular, among our constructions, for large square $q$,…

In this paper,we show some $[2n,2n-2,3]$ and $[2n,2n-3,4]$ MDS codes over dihedral codes $F_qD_{2n}$,in the case $n$ is odd and char$F_q$$\nmid$$\lvert G \rvert$ and $F_q$ contains primitive root of exponent $\lvert G \rvert$ i.e $F_q$ is…

Let $G_{(m,3,r)}=\langle x,y\mid x^m=1, y^3=1,yx=x^ry\rangle$ be a metacyclic group of order $3m$, where ${\rm gcd}(m,r)=1$, $1<r<m$ and $r^3\equiv 1$ (mod $m$). Then left ideals of the group algebra $\mathbb{F}_q[G_{(m,3,r)}]$ are called…

Self-dual codes over finite fields and over some finite rings have been of interest and extensively studied due to their nice algebraic structures and wide applications. Recently, characterization and enumeration of Euclidean self-dual…

Let $\mathbb{F}_{q}$ denote the finite field of order $q,$ let $m_1,m_2,\cdots,m_{\ell}$ be positive integers satisfying $\gcd(m_i,q)=1$ for $1 \leq i \leq \ell,$ and let $n=m_1+m_2+\cdots+m_{\ell}.$ Let…

Double circulant codes of length $2n$ over the semilocal ring $R = \mathbb{F}_q + u\mathbb{F}_q,\, u^2=u,$ are studied when $q$ is an odd prime power, and $-1$ is a square in $\mathbb{F}_q.$ Double negacirculant codes of length $2n$ are…

Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual are trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault…

Let $\mathbb {F}_q$ be a finite field and $G$ a finte group with $(|G|,q)=1$. By a group code in $\mathbb {F}_q[G]$ we mean a two-sided ideal in $\mathbb {F}_q[G]$. We will prove a general criterion for the existence of group codes with…

In this article, we show the relation between the irreducible idempotents of the cyclic group algebra $\mathbb F_qC_n$ and the central irreducible idempotents of the group algebras $\mathbb F_qD_{2n}$, where $\mathbb F_q$ is a finite field…