## About

119

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Introduction

I graduated from Shandong University
of Technology in mathematic in 1980 and jointed
Shandong University of Technology in 1981. I am currently a full professor in School of Sciences
at Shandong University of Technology since 2000.
I was honored as National Outstanding Teacher
in 2001. I have authored or coauthored more than
80 refereed papers and my research interests include
semigroup, ﬁnite ﬁelds, coding theory and cryptog-
raphy.

Additional affiliations

August 2000 - present

July 1994 - July 2000

January 1980 - June 1994

## Publications

Publications (119)

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
left metacyclic codes over $\mathbb{F}_q$ of length $3m$, and abbreviated as left $G_{(m,3,r)}$-codes.
A system...

For R a Galois ring and m
1, . . . , m
l
positive integers, a generalized quasi-cyclic (GQC) code over R of block lengths (m
1, m
2, . . . , m
l
) and length $${\sum_{i=1}^lm_i}$$ is an R[x]-submodule of $${R[x]/(x^{m_1}-1)\times\cdots \times R[x]/(x^{m_l}-1)}$$. Suppose m
1, . . . , m
l
are all coprime to the characteristic of R and let {g
1, ....

Let F
q
be a finite field of cardinality q, m
1, m
2, . . . , m
l
be any positive integers, and $${A_i=F_q[x]/(x^{m_i}-1)}$$ for i = 1, . . . , l. A generalized quasi-cyclic (GQC) code of block length type (m
1, m
2, . . . , m
l
) over F
q
is defined as an F
q
[x]-submodule of the F
q
[x]-module $${A_1\times A_2\times\cdots\times A_l}$$. By t...

Let R be an arbitrary commutative finite chain ring, γ a generator of the maximal ideal and R×R× the multiplicative group of units of R. For any w∈R×w∈R×, the structural properties and dual codes of (1+wγ)(1+wγ)-constacyclic codes of arbitrary length over R are given. As corollaries, self-dual constacyclic codes over the finite chain ring F2m+uF2mF...

Let F
q
be a finite field of cardinality q, l and m be positive integers and Ml
(F
q
) the F
q
-algebra of all l × l matrices over F
q
. We investigate the relationship between monic factors of X
m
− 1 in the polynomial ring Ml
(F
q
)[X] and quasi-cyclic (QC) codes of length lm and index l over F
q
. Then we consider the idea of constructing QC cod...

Let Fq be the finite field of q elements and let D2n=〈x,y|xn=1,y2=1,yxy=xn−1〉 be the dihedral group of 2n elements. Left ideals of the group algebra Fq[D2n] are known as left dihedral codes over Fq of length 2n, and abbreviated as left D2n-codes. Let gcd(n,q)=1. In this paper, we give an explicit representation for the Euclidean hull of every left...

Let p be an odd prime and m and s positive integers, with m even. Let further Fpm be the finite field of pm elements and R=Fpm+uFpm (u2=0). Then R is a finite chain ring of p2m elements, and there is a Gray map from RN onto Fpm2N which preserves distance and orthogonality, for any positive integer N. It is an interesting approach to obtain self-dua...

Self-dual codes over the ring Z4 are related to combinatorial designs and unimodular lattices. First, we discuss briefly how to construct self-dual cyclic codes over Z4 of arbitrary even length. Then we focus on solving one key problem of this subject: for any positive integers k and m such that m is even, we give a direct and effective method to c...

For any odd positive integer n, we express cyclic codes over \({\mathbb {Z}}_4\) of length 4n in a new way. Based on the expression of each cyclic code \({\mathcal {C}}\), we provide an efficient encoder and determine the type of \({\mathcal {C}}\). In particular, we give an explicit representation and enumeration for all distinct self-dual cyclic...

Let \begin{document}$ \mathbb{F}_{p^m} $\end{document} be a finite field of \begin{document}$ p^m $\end{document} elements, where \begin{document}$ p $\end{document} is a prime number and \begin{document}$ m $\end{document} is a positive integer. Let \begin{document}$ e\geq 2 $\end{document} be an integer and set \begin{document}$ R = \mathbb{F}_{p...

Let \(\mathbb {F}_{2^m}\) be a finite field of \(2^m\) elements and denote \(R=\mathbb {F}_{2^m}[u]/\langle u^k\rangle \)\(=\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}+\cdots +u^{k-1}\mathbb {F}_{2^m}\) (\(u^k=0\)), where k is an integer satisfying \(k\ge 2\). For any odd positive integer n, an explicit representation for every self-dual cyclic code over...

For any positive integers m and k, existing literature only determines the number of all Euclidean self-dual cyclic codes of length 2k over the Galois ring GR(4,m), such as in Kiah et al. (2012) [17]. Using properties for Kronecker products of matrices of a specific type and column vectors of these matrices, we give a simple and efficient method to...

Let p be any odd prime number and let m, k be arbitrary positive integers. The construction for self-dual cyclic codes of length \(p^k\) over the Galois ring \(\mathrm{GR}(p^2,m)\) is the key to construct self-dual cyclic codes of length \(p^kn\) over the integer residue class ring \({\mathbb {Z}}_{p^2}\) for any positive integer n satisfying \(\ma...

Let $\mathbb{F}_{q}$ be the finite field of $q$ elements and let $D_{2n}=\langle x,y\mid x^n=1, y^2=1, yxy=x^{n-1}\rangle$ be the dihedral group of order $n$. Left ideals of the group algebra $\mathbb{F}_{q}[D_{2n}]$ are known as left dihedral codes over $\mathbb{F}_{q}$ of length $2n$, and abbreviated as left $D_{2n}$-codes. Let ${\rm gcd}(n,q)=1$...

Let F 2 m be the finite field of 2 m elements and s be any positive integer. The existing literature only gives an effective calculation method to represent all distinct Euclidean self-dual cyclic codes of length 2 s over the finite chain ring F 2 m + uF 2 m (u 2 = 0), such as in Cao et al., (2019). As a development of this topic, we provide an exp...

In this paper, we are interested in finding an algebraic structure of conjucyclic codes of length n over the finite field F4. We show that conjucyclic codes of length n over F4 are related to binary cyclic codes of length 2n and show that there is a canonical bijective correspondence between the two sets. We illustrate how the factorization of the...

For any positive integers m and k, existing literature only determines the number of all Euclidean self-dual cyclic codes of length 2 k over the Galois ring GR(4, m), such as in [Des. Codes Cryptogr. (2012) 63:105-112]. Using properties for Kronecker products of matrices of a specific type and column vectors of these matrices, we give a simple and...

https://doi.org/10.1016/j.disc.2019.111768

We correct some mistakes in the paper “A mass formula for negacyclic codes of length 2k and some good negacyclic codes over \(\mathbb {Z}_{4}+u\mathbb {Z}_{4}\)” (Bandi et al. Cryptogr. Commun. 9, 241–272, 2017).

Let F2m be a finite field of 2m elements, λ and k be integers satisfying λ k ≥ 2 and denote R = F2m[u]/⟨u2λ⟩.Let δ,α ∈ F×2m. For any odd positive integer n, we give an explicit representation and enumeration for all distinct (δ + αu2)-constacyclic codes over R of length 2kn, and provide a clear formula to count the number of all these codes. In par...

Let p be any odd prime number, and be arbitrary positive integers, and let be the finite field of cardinality . Existing literature only determines the number of all (Euclidean) self-dual cyclic codes of length over finite chain ring , such as Dinh et al. (2018). Using some combinatorial identities, we obtain certain properties for Kronecker produc...

Let m be an arbitrary positive integer and D8m be the dihedral group of order 8m, i.e., D8m = [x; y | x4m = 1; y2 = 1,yxy = x-1.]. Left ideals of the dihedral group algebra F2[D8m] are called binary left dihedral codes of length 8m, and abbreviated as binary left D8m-codes. In this paper, we give an explicit representation and enumeration for all d...

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 positive integer $n$, an explicit representation for every self-dual cyclic code over $R$ of length $2n$
and...

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 positive integer $n$, an explicit representation for every self-dual cyclic code over $R$ of length $2n$ and...

Let F 2 m be a finite field of cardinality 2 m , R = F 2 m + uF 2 m (u 2 = 0) and s, n be positive integers such that n is odd. In this paper, we give an explicit representation for every self-dual cyclic code over the finite chain ring R of length 2 s n and provide a calculation method to obtain all distinct codes. Moreover, we obtain a clear form...

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 $\mathbb{F}_p$ with a specific type.
On that basis, we give an explicit representation and enumeration for all distinct self-du...

Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$, $R=\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ $(u^2=0)$ and $s,n$ be positive integers such that $n$ is odd. In this paper, we give an explicit representation for every self-dual cyclic code over the finite chain ring $R$ of length $2^sn$ and provide a calculation method to obtain all distinct...

Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$, $\lambda$ and $k$ be integers satisfying $\lambda,k\geq 2$ and denote $R=\mathbb{F}_{2^m}[u]/\langle u^{2\lambda}\rangle$. Let $\delta,\alpha\in \mathbb{F}_{2^m}^{\times}$. For any odd positive integer $n$, we give an explicit representation and enumeration for all distinct $(\delta+\al...

Let F 2 m be a finite field of cardinality 2 m , λ and k be integers satisfying λ, k ≥ 2 and denote R = F 2 m [u]/u 2λ. Let δ, α ∈ F × 2 m. For any odd positive integer n, we give an explicit representation and enumeration for all distinct (δ +αu 2)-constacyclic codes over R of length 2 k n, and provide a clear formula to count the number of all th...

Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ where $p$ is an odd prime, $k,\lambda$ be positive integers satisfying $\lambda\geq 2$, and denote $\mathcal{K}=\mathbb{F}_{p^m}[x]/\langle f(x)^{\lambda p^k}\rangle$ where $f(x)$ is an irreducible polynomial in $\mathbb{F}_{p^m}[x]$. In this paper, for any fixed invertible element $\ome...

For any positive odd integer n, a precise representation for cyclic codes over \({\mathbb {Z}}_4\) of length 2n is given in terms of the Chinese Remainder Theorem. Using this representation, an efficient encoder for each of these codes is described. Then the dual codes are determined precisely and this is used to study codes which are self-dual. In...

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 left dihedral codes of length $8m$, and abbreviated as binary left $D_{8m}$-codes.
In this paper, we give an ex...

In this paper, an explicit representation and enumeration for negacyclic codes of length 2^kn over the local non-principal ideal ring R=Z_4+uZ_4 (u^2=0) is provided, where k, n are any positive integers and n is odd. In particular, all distinct negacyclic codes of length 2^k over R are listed precisely. Moreover, an exact mass formula for the numbe...

In this paper, an explicit representation and enumeration for nega-cyclic codes of length 2 k n over the local non-principal ideal ring R = Z 4 + uZ 4 (u 2 = 0) is provided, where k, n are any positive integers and n is odd. As a corollary, all distinct negacyclic codes of length 2 k over R are listed precisely. Moreover, a mass formula for the num...

Let F 2 m be a finite field of cardinality 2 m and s a positive integer. Using properties for Kronecker product of matrices and calculation for linear equations over F 2 m , an efficient method for the construction of all distinct self-dual cyclic codes with length 2 s over the finite chain ring F 2 m + uF 2 m (u 2 = 0) is provided. On that basis,...

Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$ and $s$ a positive integer. Using properties for Kronecker product of matrices and calculation for linear equations over $\mathbb{F}_{2^m}$, an efficient method for the construction of all distinct self-dual cyclic codes with length $2^s$ over the finite chain ring $\mathbb{F}_{2^m}+u\ma...

In this paper, an explicit representation and enumeration for negacyclic codes of length $2^kn$ over the local non-principal ideal ring $R=\mathbb{Z}_4+u\mathbb{Z}_4$ $(u^2=0)$ is provided, where $k, n$ are any positive integers and $n$ is odd. As a corollary, all distinct negacyclic codes of length $2^k$ over $R$ are listed precisely. Moreover, a...

Let F p m be a finite field of cardinality p m where p is an odd prime, n be a positive integer satisfying gcd(n, p) = 1, and denote R = F p m [u]/u e where e ≥ 4 be an even integer. Let δ, α ∈ F × p m. Then the class of (δ + αu 2)-constacyclic codes over R is a significant subclass of constacyclic codes over R of Type 2. For any integer k ≥ 1, an...

Let F p m be a finite field of cardinality p m where p is an odd prime, k, λ be positive integers satisfying λ ≥ 2, and denote K = F p m [x] / ⟨ f (x) λp k ⟩ where f (x) is an irreducible polynomial in F p m [x]. In this note, for any fixed invertible element ω ∈ K × we present all distinct linear codes S over K of length 2 satisfying the condition...

Let F p m be a finite field of cardinality p m where p is an odd prime, n be a positive integer satisfying gcd(n, p) = 1, and denote R = F p m [u]/u e where e ≥ 4 be an even integer. Let δ, α ∈ F × p m. Then the class of (δ + αu 2)-constacyclic codes over R is a significant subclass of constacyclic codes over R of Type 2. For any integer k ≥ 1, an...

Let F p m be a finite field of cardinality p m where p is an odd prime, n be a positive integer satisfying gcd(n, p) = 1, and denote R = F p m [u]/⟨u e ⟩ where e ≥ 4 be an even integer. Let δ, α ∈ F × p m. Then the class of (δ + αu 2)-constacyclic codes over R is a significant subclass of constacyclic codes over R of Type 2. For any integer k ≥ 1,...

Let F 2 m be a finite field of cardinality 2 m , n be an odd positive integer, and denote R = F 2 m [u]/⟨u 3 ⟩. Let δ, α ∈ F × 2 m. Then (δ + αu 2)-constacyclic codes over R are called constacyclic codes over R of Type 2. In this paper , an explicit representation and a complete description for all distinct (δ + αu 2)-constacyclic codes over R of l...

In this paper, we study cyclic codes over Z9 of length 3n, where n is a positive integer satisfying gcd(3,n)=1. First, a canonical form decomposition of any cyclic code over Z9 of length 3n are given and a unique set of generators for each subcode is presented. Hence the structure of any cyclic code over Z9 of length 3n is determined. From this dec...

Let $R=\mathbb{Z}_{4}[v]/\langle v^2+2v\rangle=\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ ($v^2=2v$) and $n$ be an odd positive integer.
Then $R$ is a local non-principal ideal ring of $16$ elements and there is a $\mathbb{Z}_{4}$-linear Gray map from $R$ onto $\mathbb{Z}_{4}^2$ which preserves Lee distance and orthogonality.
First, a canonical form decomposi...

Let m, e be positive integers, p a prime number, \(\mathbb {F}_{p^m}\) be a finite field of \(p^m\) elements and \(R=\mathbb {F}_{p^m}[u]/\langle u^e\rangle \) which is a finite chain ring. For any \(\omega \in R^\times \) and positive integers k, n satisfying \(\mathrm{gcd}(p,n)=1\), we prove that any \((1+\omega u)\)-constacyclic code of length \...

Let m, e be positive integers, p a prime number, F p m be a finite field of p m elements and R = F p m [u]/u e which is a finite chain ring. For any ω ∈ R × and positive integers k, n satisfying gcd(p, n) = 1, we prove that any (1 + ωu)-constacyclic code of length p k n over R is monomially equivalent to a matrix-product code of a nested sequence o...

Let $\mathbb{F}_q$ be a finite field of q elements, $R=\mathbb{F}_q+u\mathbb{F}_q$ (u²=0) and D2n=<x, y | xⁿ=1, y²=1, yxy=x⁻¹> be a dihedral group of order n. Left ideals of the group ring R[D2n] are called left dihedral codes over R of length 2n, and abbreviated as left D2n-codes over R. Let n be a positive factor of qe+1 for some positive integer...

Let D2n= x,y xn=1,y2=1,yxy=x-1 be a dihedral group, and R=GR(p2,m) be a Galois ring of characteristic p2 and cardinality p2m where p is a prime. Left ideals of the group ring R[D2n] are called left dihedral codes over R of length 2n, and abbreviated as left D2n-codes over R. Let gcd(n,p)=1 in this paper. Then any left D2n-code over R is uniquely de...

Let p be a prime integer, n, s ≥ 2 be integers satisfying gcd(p, n) = 1, and denote R = Z p s [v]/⟨v 2 − pv⟩. Then R is a local non-principal ideal ring of p 2s elements. First, the structure of any cyclic code over R of length n and a complete classification of all these codes are presented. Then the cardinality of each code and dual codes of thes...

Let $p$ be a prime integer, $n,s\geq 2$ be integers satisfying ${\rm gcd}(p,n)=1$, and denote $R=\mathbb{Z}_{p^s}[v]/\langle v^2-pv\rangle$. Then $R$ is a local non-principal ideal ring of $p^{2s}$ elements. First, the structure of any cyclic code over $R$ of length $n$ and a complete classification of all these codes are presented. Then the cardin...

For any prime number $p$, positive integers $m, k, n$
satisfying ${\rm gcd}(p,n)=1$ and $\lambda_0\in \mathbb{F}_{p^m}^\times$, we prove that any $\lambda_0^{p^k}$-constacyclic code of length $p^kn$ over the finite field $\mathbb{F}_{p^m}$ is monomially equivalent
to a matrix-product code of a nested sequence of $p^k$ $\lambda_0$-constacyclic codes...

Let F2m be a finite field of cardinality 2m, R=F2m[u]∕〈u4〉 and n be an odd positive integer. For any δ,α∈F2m×, ideals of the ring R[x]∕〈x2n−(δ+αu2)〉 are identified as (δ+αu2)-constacyclic codes of length 2n over R. In this paper, an explicit representation and enumeration for all distinct (δ+αu2)-constacyclic codes of length 2n over R are presented...

For any prime number $p$, positive integers $m, k, n$ satisfying ${\rm gcd}(p,n)=1$ and $\lambda_0\in \mathbb{F}_{p^m}^\times$, we prove that any $\lambda_0^{p^k}$-constacyclic code of length $p^kn$ over the finite field $\mathbb{F}_{p^m}$ is monomially equivalent to a matrix-product code of a nested sequence of $p^k$ $\lambda_0$-constacyclic codes...

Let (Formula presented.) be a finite field of cardinality (Formula presented.), (Formula presented.) which is a finite chain ring, and n be a positive integer satisfying (Formula presented.). For any (Formula presented.), an explicit representation for all distinct (Formula presented.)-constacyclic codes over R of length 3n is given, formulas for t...

Complete classification of (δ + αu 2)-constacyclic codes over F 3 m [u]/u 4 of length 3n Abstract Let F 3 m be a finite field of cardinality 3 m , R = F 3 m [u]/u 4 which is a finite chain ring, and n be a positive integer satisfying gcd(3, n) = 1. For any δ, α ∈ F × 3 m , an explicit representation for all distinct (δ +αu 2)-constacyclic codes ove...

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 $\lambda=a_0+a_1u+\ldots+a_{k-1}u^{k-1}$ where $a_0, a_1,\ldots, a_{k-1}\in F_{p^m}$ satisfying $a_0\neq 0$ and $a_1=1$. Let $r$...

1+pw)-constacyclic codes of arbitrary length over the non-principal ideal ring Zps + uZps are studied, where p is a prime, w ∈ Z × p s and s an integer satisfying s ≥ 2. First, the structure of any (1 + pw)-constacyclic code over Zps + uZps are presented. Then enumerations for the number of all codes and the number of codewords in each code, and th...

Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$, $R=\mathbb{F}_{2^m}[u]/\langle u^4\rangle)$ and $n$ is an odd positive integer. For any $\delta,\alpha\in \mathbb{F}_{2^m}^{\times}$, ideals of the ring $R[x]/\langle x^{2n}-(\delta+\alpha u^2)\rangle$ are identified as $(\delta+\alpha u^2)$-constacyclic codes of length $2n$ over $R$. I...

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 called left dihedral codes over $R$ of length $2n$, and abbreviated as left $D_{2n}$-codes over $R$. Let ${\rm...

Let Fq be a finite field of cardinality q, R = Fq[u]/u4/Fq + uFq + u2Fq + u3Fq (u4/0) which is a finite chain ring, and n be a positive integer satisfying gcd(q, n) = 1. For any δ, α ϵ F×q, an explicit representation for all distinct (δ + αu2)-constacyclic codes over R of length n is given, and the dual code for each of these codes is determined. F...

Let $\mathbb{F}_{q}$ be a finite field of cardinality $q$, $R=\mathbb{F}_{q}[u]/\langle u^4\rangle=\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}+u^3\mathbb{F}_{q}$ $(u^4=0)$ which is a finite chain ring, and $n$ be a positive integer satisfying ${\rm gcd}(q,n)=1$. For any $\delta,\alpha\in \mathbb{F}_{q}^{\times}$, an explicit representation for...

Let (Formula presented.) where n is odd and k a positive integer. We present a canonical form decomposition for every cyclic code over (Formula presented.) of length N, where each subcode is concatenated by a basic irreducible cyclic code over (Formula presented.) of length n as the inner code and a constacyclic code over a Galois extension ring of...

Let \({\mathbb {F}}_{2^m}\) be a finite field of characteristic 2 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 \(k\in {\mathbb {Z}}^{+}\) satisfies \(k\ge 2\). For any odd positive integer n, it is known that cyclic codes over R of length 2n are i...

Let be the dihedral group of order n. Left ideals of the group algebra are known as left dihedral codes over of length 2n, and abbreviated as left -codes. In this paper, a system theory for left -codes is developed only using finite field theory and basic theory of cyclic codes and skew cyclic codes. First, we prove that any left -code is a direct...

Let $\mathbb{F}_{2^m}$ be a finite field of characteristic $2$ 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 $k\in \mathbb{Z}^{+}$ satisfies $k\geq 2$.
For any odd positive integer $n$, it is known that cyclic codes over $R$ of length $2n$ are identified with ideals...

On a class of $(\delta+\alpha u^2)$-constacyclic codes over $\mathbb{F}_{q}[u]/\langle u^4\rangle$

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 a prime and $m$ is a positive integer. For any
$\lambda\in \mathbb{F}_{p^m}^{\times}$, an explicit representation for all
distinct $\lambda$-constacyclic codes over $R$ of length $p^...

Let N=pkn where p is a prime, and k,n are positive integers satisfying gcd(p,n)=1. We present a canonical form decomposition for every cyclic code over Zp2 of length N, where each subcode is concatenated by a basic irreducible cyclic code over Zp2 of length n as the inner code and a constacyclic code over a Galois extension ring of Zp2 of length pk...

Let \(\mathbb {F}_q\) be a finite field of cardinality \(q\), \(l\) a prime number and \(\mathbb {F}_{q^l}\) an extension field of \(\mathbb {F}_q\) with degree \(l\). The structure and canonical form decompositions of semisimple multivariable \(\mathbb {F}_q\)-linear codes over \(\mathbb {F}_{q^l}\) are presented. Enumeration and construction of t...

Let \(F_q\) be a finite field of order \(q\), \(l\) a prime, \(\lambda \in F_q^{\times }=F_q-\{0\}\) and \(F_{q^l}\) an extension field of \(F_q\) with degree \(l\). First, the structure and a canonical form decomposition of any \(\lambda \)-constacyclic \(F_q\)-linear code over \(F_{q^l}\) are presented. By use of this decomposition, enumeration,...

Let be a finite field of cardinality q where q is a power of an odd prime integer, and denote the generalized quaternion group by the presentation: where n is even and satisfies . Left ideals of the group algebra are called left quaternion codes over of length , and abbreviated as left -codes. In this paper, a system theory for left -codes is devel...

Let R = GR(p(epsilon), l) be a Galois ring of characteristic p(epsilon) and cardinality p(epsilon l), where p and l are prime integers. First, we give a canonical form decomposition for additive cyclic codes over R. This decomposition is used to construct additive cyclic codes and count the number of such codes, respectively. Then we give the trace...

Let \(p\) be a prime number, \(f(x)\) a monic basic irreducible polynomial in \(\mathbb {Z}_{p^2}[x]\) and \(\overline{f}(x)=f(x)\) mod \(p\) . Set \(F=\mathbb {Z}_p[x]_{/\langle \overline{f}(x)\rangle }\) and \(R=\mathbb {Z}_{p^2}[x]_{/\langle f(x)\rangle }\) , and denote by \(\mathrm{End}(F\times R)\) the endomorphism ring of the \(R\) -module \(...

In this paper, we study the construction of cyclic DNA codes by cyclic codes over the finite chain ring F4[u]/⟨u 2 + 1⟩. First, we establish a 1-1 correspondence φ between DNA pairs and the 16 elements of the ring F4[u]/⟨u 2 +1⟩. Considering the biology features of DNA codes, we investigate the structure and properties of self-reciprocal complement...

Let be a finite field of cardinality q, where q is a power of a prime number p, n a positive multiple of p, l a prime number and an extension field of with degree l. First, the structure and a canonical form decomposition of any cyclic -linear code over of length n are presented. Then from this decomposition and by the theory of linear codes over f...

Let
$q$
be a power of a prime integer
$p, m=p^em_0$
and
$|q|_{m_{0}}$
the order of
$q$
modulo
$m_0$
. By use of finite commutative chain ring theory, an algorithm to construct all distinct 1-generator quasi-cyclic codes with a fixed parity check polynomial over a finite field
$F_q$
of length
$mn$
and index
$n$
, under the condition...

Let θ be the Frobenius automorphism of the finite field FqlFql over its subfield FqFq, Fql[Y;θ]Fql[Y;θ] the skew polynomial ring and Fql[Y;θ]/〈Yl−1〉Fql[Y;θ]/〈Yl−1〉 the quotient ring of Fql[Y;θ]Fql[Y;θ] modulo its ideal 〈Yl−1〉〈Yl−1〉. We construct a specific FqFq-algebra isomorphism from Fql[Y;θ]/〈Yl−1〉Fql[Y;θ]/〈Yl−1〉 onto the matrix ring Ml(Fq)Ml(Fq...

Let R be an arbitrary commutative finite chain ring with
$1\ne 0$
. 1-generator quasi-cyclic (QC) codes over R are considered in this paper. Let
$\gamma $
be a fixed generator of the maximal ideal of R,
$F=R/\langle \gamma \rangle $
and
$|F|=q$
. For any positive integers m, n satisfying
$\mathrm{gcd}(q,n)=1$
, let
$\mathcal{R}_n=R[x]/\...

Let 𝑅=GR(𝑝𝑠,𝑝𝑠𝑚) be a Galois ring of characteristic 𝑝𝑠 and cardinality 𝑝𝑠𝑚, where 𝑠
and 𝑚
are positive integers and 𝑝
is an odd prime number. Two kinds of cogredient standard forms of symmetric matrices over 𝑅
are given, and an explicit formula to count the number of all distinct cogredient classes of symmetric matrices over 𝑅
is obtained.

Let R be a finite commutative chain ring, n a positive integer and R
n
the free R-module of rank n consisting of column vectors over R. The generalized affine transformation monoid Gaffn
(R) of R
n
is introduced, then Schützenberger groups of
-classes, principal factors and group
-classes of the monoid Gaffn
(R) are investigated. As corollaries, ba...

We show that every finitely presented, cancellative and commutative ordered monoid is determined by a finitely generated and cancellative pseudoorder on the monoid (ℕn
,+) for some positive integer n. Every cancellative pseudoorder on (ℕn
,+) is determined by a submonoid of the group (ℤn
,+), and we prove that the pseudoorder is finitely generated...

Let R be a finite commutative chain ring, 1 ≤ k ≤ n − 1, the set of right invertible k × n matrices, and GL n (R) the general linear group of degree n over R, respectively. It is clear that Q QU ( , U GL n (R)) is a transitive action on and induces the diagonal action of GL n (R) on defined by (Q 1, Q 2)U = (Q 1 U, Q 2 U) for and U GL n (R), which...

Let R be an Artinian chain ring with a principal maximal ideal. We investigate properties of matrices over R and give matrix representations of R-submodules of R(n) first, then consider Green's relations, Green's relation equivalent classes, Schutzenberger groups of D-classes, principal factors, and group H-classes of the multiplicative monoid M(n)...

For any prime number p and positive integer s, we consider the multiplicative monoid End(G) of endomorphisms of the abelian group G = Zps ⊕Z p s+1. We determine representatives and the number of all distinct D-classes of End(G) and show that each regular D-class is a completely regular subsemigroup of End(G) and precisely a rectangular band of grou...

Let R be a commutative, local, and principal ideal ring with maximal ideal m and residue class field F. Suppose that every element of 1+m is square. Then the problem of classifying arbitrary symmetric matrices over R by congruence naturally reduces, and is actually equivalent to, the problem of classifying invertible symmetric matrices over F by co...

For any positive integers m,n and a prime number p, we introduce the concept of invariant factors for matrices over Zpm first, then investigate Green's relations, Green's relation equivalence classes, Schützenberger groups of D-classes, group H-classes and regular principal factors of the matrix multiplicative monoid Mn(Zpm). As corollaries, we cha...

In this paper, we consider the generalized affine transformation monoid Gaff (A/I) of a residue class ring
A/I of a commutative ring A with identity modulo its nonzero ideal I. For the general case, we investigate the Green's relations,
Schutzenberger group for every D-class and, the structure of group H-classes, regular D-classes, the idempotent s...