# Yonglin CaoShandong University of Technology · School of Mathematics and Statistics

None

119
Publications
111,264
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572
Citations
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.
August 2000 - present
Position
• Professor (Full)
July 1994 - July 2000
Position
• Professor (Associate)
January 1980 - June 1994
Position
• Professor (Assistant)

## Publications

Publications (119)
Article
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...
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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, ....
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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...
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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...
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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...
Article
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...
Article
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...
Article
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...
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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...
Article
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... Article Full-text available 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... Article 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... Article 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... Preprint 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... Article 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... Article 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... Preprint Full-text available 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... Article 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). Article Full-text available 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... Article 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... Article 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... Preprint Full-text available 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... Preprint Full-text available 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... Article Full-text available 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... Preprint Full-text available 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... Preprint Full-text available 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... Preprint Full-text available 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...
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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...
Article
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... Article 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... Preprint Full-text available 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... Article 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... Article Full-text available 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... Article Full-text available 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,... Preprint Full-text available 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...
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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...
Article
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...
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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...
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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...
Preprint
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,...
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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...
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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...
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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...
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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 \...
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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...
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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...
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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...
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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...
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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...
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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...
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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...
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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...
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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...
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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...
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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$...
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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...
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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...
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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... Working Paper Full-text available Article Full-text available 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... Article Full-text available 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... Article 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... Article Full-text available 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$$. For any odd positive integer n, it is known that cyclic codes over R of length 2n are i... Article Let be the dihedral group of order n. Left ideals of the group algebra are known as left dihedral codes over of length 2n, and abbreviated as left -codes. In this paper, a system theory for left -codes is developed only using finite field theory and basic theory of cyclic codes and skew cyclic codes. First, we prove that any left -code is a direct... Research Full-text available Let$\mathbb{F}_{2^m}$be a finite field of characteristic$2$and$R=\mathbb{F}_{2^m}[u]/\langle u^k\rangle=\mathbb{F}_{2^m} +u\mathbb{F}_{2^m}+\ldots+u^{k-1}\mathbb{F}_{2^m}$($u^k=0$) where$k\in \mathbb{Z}^{+}$satisfies$k\geq 2$. For any odd positive integer$n$, it is known that cyclic codes over$R$of length$2n$are identified with ideals... Research Full-text available On a class of$(\delta+\alpha u^2)$-constacyclic codes over$\mathbb{F}_{q}[u]/\langle u^4\rangle$Article Full-text available Let$\mathbb{F}_{p^m}$be a finite field of cardinality$p^m$and$R=\mathbb{F}_{p^m}[u]/\langle u^2\rangle=\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}(u^2=0)$, where$p$is a prime and$m$is a positive integer. For any$\lambda\in \mathbb{F}_{p^m}^{\times}$, an explicit representation for all distinct$\lambda$-constacyclic codes over$R$of length$p^...
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Let N=pkn where p is a prime, and k,n are positive integers satisfying gcd(p,n)=1. We present a canonical form decomposition for every cyclic code over Zp2 of length N, where each subcode is concatenated by a basic irreducible cyclic code over Zp2 of length n as the inner code and a constacyclic code over a Galois extension ring of Zp2 of length pk...
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Let $$\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}$$ are presented. Enumeration and construction of t...
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Let $$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}$$ are presented. By use of this decomposition, enumeration,...
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Let be a finite field of cardinality q where q is a power of an odd prime integer, and denote the generalized quaternion group by the presentation: where n is even and satisfies . Left ideals of the group algebra are called left quaternion codes over of length , and abbreviated as left -codes. In this paper, a system theory for left -codes is devel...
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Let R = GR(p(epsilon), l) be a Galois ring of characteristic p(epsilon) and cardinality p(epsilon l), where p and l are prime integers. First, we give a canonical form decomposition for additive cyclic codes over R. This decomposition is used to construct additive cyclic codes and count the number of such codes, respectively. Then we give the trace...
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Let $$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 \(...
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