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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 ${\rm gcd}(n,q)=1$. In this paper, we give an explicit representation for the Euclidean hull of every left $D_{2n}$-code over $\mathbb{F}_{q}$. On this basis, we determine all distinct Euclidean LCD codes and Euclidean self-orthogonal codes which are left $D_{2n}$-codes over $\mathbb{F}_{q}$. In particular, we provide an explicit representation and a precise enumeration for these two subclasses of left $D_{2n}$-codes and self-dual left $D_{2n}$-codes, respectively. Moreover, we give a direct and simple method for determining the encoder (generator matrix) of any left $D_{2n}$-code over $\mathbb{F}_{q}$, and present several numerical examples to illustrative our applications.

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 ${\rm GR}(4,m)$, 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 efficient method to construct all these self-dual cyclic codes precisely. On this basis, we provide an explicit expression to accurately represent all distinct Euclidean self-dual cyclic codes of length $2^k$ over ${\rm GR}(4,m)$, using combination numbers. As an application, we list all distinct Euclidean self-dual cyclic codes over ${\rm GR}(4,m)$ of length $2^k$ explicitly, for $k=4,5,6$.

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 a mass formula to count the number of these codes are given first. Then a generator matrix is provided for the self-dual and $2$-quasi-cyclic code of length $4n$ over $\mathbb{F}_{2^m}$ derived by every self-dual cyclic code of length $2n$ over $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ and a Gray map from $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ onto $\mathbb{F}_{2^m}^2$. Finally, the hull of each cyclic code with length $2n$ over $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ is determined and all distinct self-orthogonal cyclic codes of length $2n$ over $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ are listed.

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 codes. Moreover, we obtain a clear formula to count the number of all these self-dual cyclic codes. As an application, self-dual and $2$-quasi-cyclic codes over $\mathbb{F}_{2^m}$ of length $2^{s+1}n$ can be obtained from self-dual cyclic code over $R$ of length $2^sn$ and by a Gray map preserving orthogonality and distances from $R$ onto $\mathbb{F}_{2^m}^2$., Comment: arXiv admin note: text overlap with arXiv:1811.11018, arXiv:1907.07107

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-dual cyclic codes of length $p^s$ over the finite chain ring $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ $(u^2=0)$. Moreover, We provide an efficient method to construct every self-dual cyclic code of length $p^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ precisely., Comment: arXiv admin note: substantial text overlap with arXiv:1811.11018

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 $(\delta+\alpha u^2)$-constacyclic codes over $R$ of length $2^kn$, and provide a clear formula to count the number of all these codes. As a corollary, we conclude that every $(\delta+\alpha u^2)$-constacyclic code over $R$ of length $2^kn$ is an ideal generated by at most $2$ polynomials in the residue class ring $R[x]/\langle x^{2^kn}-(\delta+\alpha u^2)\rangle$., Comment: arXiv admin note: text overlap with arXiv:1805.05595

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 explicit representation and enumeration for all distinct self-dual binary left $D_{8m}$-codes. These codes make up an important class of self-dual binary $[8m,4m]$-codes such that the dihedral group $D_{8m}$ is necessary a subgroup of the automorphism group of each code. In particular, we provide recursive algorithms to solve congruence equations over finite chain rings for constructing all distinct self-dual binary left $D_{8m}$-codes and obtain a Mass formula to count the number of all these self-dual codes. As a preliminary application, we obtain the extremal self-dual binary $[48,24,12]$-code and an extremal self-dual binary $[56,28,12]$-code from self-dual binary left $D_{48}$-codes and left $D_{56}$-codes respectively.

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 $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ $(u^2=0)$ is provided. On that basis, an explicit representation for every self-dual cyclic code of length $2^s$ over $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ and an exact formula to count the number of all these self-dual cyclic codes are given.

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 mass formula for the number of negacyclic codes of length $2^kn$ over $R$ is given and a mistake in [Cryptogr. Commun. (2017) 9: 241--272] is corrected.

Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ where $p$ is an odd prime, $n$ be a positive integer satisfying ${\rm gcd}(n,p)=1$, and denote $R=\mathbb{F}_{p^m}[u]/\langle u^e\rangle$ where $e\geq 4$ be an even integer. Let $\delta,\alpha\in \mathbb{F}_{p^m}^{\times}$. Then the class of $(\delta+\alpha u^2)$-constacyclic codes over $R$ is a significant subclass of constacyclic codes over $R$ of Type 2. For any integer $k\geq 1$, an explicit representation and a complete description for all distinct $(\delta+\alpha u^2)$-constacyclic codes over $R$ of length $np^k$ and their dual codes are given. Moreover, formulas for the number of codewords in each code and the number of all such codes are provided respectively. In particular, all distinct $(\delta+\alpha u^2)$-contacyclic codes over $\mathbb{F}_{p^m}[u]/\langle u^{e}\rangle$ of length $p^k$ and their dual codes are presented precisely.

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 ${\rm gcd}(p,n)=1$, we prove that any $(1+\omega u)$-constacyclic code of length $p^kn$ over $R$ is monomially equivalent to a matrix-product code of a nested sequence of $p^k$ cyclic codes with length $n$ over $R$ and a $p^k\times p^k$ matrix $A_{p^k}$ over $\mathbb{F}_p$. Using the matrix-product structures, we give an iterative construction of every $(1+\omega u)$-constacyclic code by $(1+\omega u)$-constacyclic codes of shorter lengths over $R$., Comment: arXiv admin note: text overlap with arXiv:1705.08819

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 decomposition and the structure for any negacyclic code over $R$ of length $2n$ are presented. From this decomposition, a complete classification of all these codes is obtained. Then the cardinality and the dual code for each of these codes are given, and self-dual negacyclic codes over $R$ of length $2n$ are presented. Moreover, all $23\cdot(4^p+5\cdot 2^p+9)^{\frac{2^{p}-2}{p}}$ negacyclic codes over $R$ of length $2M_p$ and all $3\cdot(4^p+5\cdot 2^p+9)^{\frac{2^{p-1}-1}{p}}$ self-dual codes among them are presented precisely, where $M_p=2^p-1$ is a Mersenne prime. Finally, $36$ new and good self-dual $2$-quasi-twisted linear codes over $\mathbb{Z}_4$ with basic parameters $(28,2^{28}, d_L=8,d_E=12)$ and of type $2^{14}4^7$ and basic parameters $(28,2^{28}, d_L=6,d_E=12)$ and of type $2^{16}4^6$ which are Gray images of self-dual negacyclic codes over $R$ of length $14$ are listed., Comment: arXiv admin note: text overlap with arXiv:1710.09236

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 cardinality of each code and dual codes of these codes are given. Moreover, self-dual cyclic codes over $R$ of length $n$ are investigated. Finally, we list some optimal $2$-quasi-cyclic self-dual linear codes over $\mathbb{Z}_4$ of length $30$ and extremal $4$-quasi-cyclic self-dual binary linear $[60,30,12]$ codes derived from cyclic codes over $\mathbb{Z}_{4}[v]/\langle v^2+2v\rangle$ of length $15$.

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 with length $n$ over $\mathbb{F}_{p^m}$.

$(1+pw)$-constacyclic codes of arbitrary length over the non-principal ideal ring $\mathbb{Z}_{p^s} +u\mathbb{Z}_{p^s}$ are studied, where $p$ is a prime, $w\in \mathbb{Z}_{p^s}^{\times}$ and $s$ an integer satisfying $s\geq 2$. First, the structure of any $(1+pw)$-constacyclic code over $\mathbb{Z}_{p^s} +u\mathbb{Z}_{p^s}$ are presented. Then enumerations for the number of all codes and the number of codewords in each code, and the structure of dual codes for these codes are given, respectively. Then self-dual $(1+2w)$-constacyclic codes over $\mathbb{Z}_{2^s} +u\mathbb{Z}_{2^s}$ are investigated, where $w=2^{s-2}-1$ or $2^{s-1}-1$ if $s\geq 3$, and $w=1$ if $s=2$.

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 $r$ be a positive integer satisfying $p^{r-1}+1\leq k\leq p^r$. We defined a Gray map from $R_k$ to $F_{p^m}^{p^r}$ first, then prove that the Gray image of any linear $\lambda$-constacyclic code over $R_k$ of length $N$ is a distance invariant linear $a_0^{p^r}$-constacyclic code over $F_{p^m}$ of length $p^rN$. Furthermore, the generator polynomials for each linear $\lambda$-constacyclic code over $R_k$ of length $N$ and its Gray image are given respectively. Finally, some optimal constacyclic codes over $F_{3}$ and $F_{5}$ are constructed.

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 $R$. In this paper, an explicit representation and enumeration for all distinct $(\delta+\alpha u^2)$-constacyclic codes of length $2n$ over $R$ are presented., Comment: arXiv admin note: text overlap with arXiv:1511.02369

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 ${\rm gcd}(n,p)=1$ in this paper. Then any left $D_{2n}$-code over $R$ is uniquely decomposed into a direct sum of concatenated codes with inner codes ${\cal A}_i$ and outer codes $C_i$, where ${\cal A}_i$ is a cyclic code over $R$ of length $n$ and $C_i$ is a skew cyclic code of length $2$ over an extension Galois ring or principal ideal ring of $R$, and a generator matrix and basic parameters for each outer code $C_i$ is given. Moreover, a formula to count the number of these codes is obtained, the dual code for each left $D_{2n}$-code is determined and all self-dual left $D_{2n}$-codes and self-orthogonal left $D_{2n}$-codes over $R$ are presented, respectively.

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 $p^sn$ is given by a canonical form decomposition for each code, where $s$ and $n$ are positive integers satisfying ${\rm gcd}(p,n)=1$. For any such code, using its canonical form decomposition the representation for the dual code of the code is provided. Moreover, representations for all distinct negacyclic codes and their dual codes of length $p^sn$ over $R$ are obtained, and self-duality for these codes are determined. Finally, all distinct self-dual negacyclic codes over $\mathbb{F}_5+u\mathbb{F}_5$ of length $2\cdot 5^s\cdot 3^t$ are listed for any positive integer $t$.

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 of the ring $R[x]/\langle x^{2n}-1\rangle$. In this paper, an explicit representation for each cyclic code over $R$ of length $2n$ is provided and a formula to count the number of codewords in each code is given. Then a formula to calculate the number of cyclic codes over $R$ of length $2n$ is obtained. Moreover, the dual code of each cyclic code and self-dual cyclic codes over $R$ of length $2n$ are investigated. (AAECC-1522), Comment: AAECC-1522

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 all distinct $(\delta+\alpha u^2)$-constacyclic codes over $R$ of length $n$ is given, and the dual code for each of these codes is determined. For the case of $q=2^m$ and $\delta=1$, all self-dual $(1+\alpha u^2)$-constacyclic codes over $R$ of odd length $n$ are provided.