A family of constacyclic codes over F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}} and its application to quantum codes

We introduce a Gray map from $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ to $\mathbb{F}_{2}^{2m}$ and study $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}},$ where $u^{2}=0.$ It is proved that the image of a $(1+u)$-constacyclic code length $n$ over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ under the Gray map is a distance-invariant quasi-cyclic code of index $m$ and length $2mn$ over $\mathbb{F}_{2}.$ We also prove that every code of length $2mn$ which is the Gray image of cyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ of length $n$ is permutation equivalent to a binary quasi-cyclic code of index $m.$ Furthermore, a family of quantum error-correcting codes obtained from the Calderbank-Shor-Steane (CSS) construction applied to $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}.

(1−2uk)(1-2u^k)-constacyclic codes over Fp+uFp+u2F+u3Fp+⋯+ukFp\mathbb{F}_p+u\mathbb{F}_p+u^2\mathbb{F}_+u^{3}\mathbb{F}_{p}+\dots+u^{k}\mathbb{F}_{p}

Let $\mathbb{F}_p$ be a finite field and $u$ be an indeterminate. This article studies $(1-2u^k)$-constacyclic codes over the ring $\mathcal{R}=\mathbb{F}_p+u\mathbb{F}_p+u^2\mathbb{F}_p+u^{3}\mathbb{F}_{p}+\cdots+u^{k}\mathbb{F}_{p}$ where $u^{k+1}=u$. We illustrate the generator polynomials and investigate the structural properties of these codes via decomposition theorem

(1+2u)(1+2u)-constacyclic codes over Z4+uZ4\mathbb{Z}_4+u\mathbb{Z}_4

Let $R=\mathbb{Z}_4+u\mathbb{Z}_4,$ where $\mathbb{Z}_4$ denotes the ring of integers modulo $4$ and $u^2=0$. In the present paper, we introduce a new Gray map from $R^n$ to $\mathbb{Z}_{4}^{2n}.$ We study $(1+2u)$-constacyclic codes over $R$ of odd lengths with the help of cyclic codes over $R$. It is proved that the Gray image of $(1+2u)$-constacyclic codes of length $n$ over $R$ are cyclic codes of length $2n$ over $\mathbb{Z}_4$. Further, a number of linear codes over $\mathbb{Z}_4$ as the images of $(1+2u)$-constacyclic codes over $R$ are obtained

σ\sigma-self-orthogonal constacyclic codes of length psp^s over Fpm+uFpm\mathbb F_{p^m}+u\mathbb F_{p^m}

In this paper, we study the $\sigma$-self-orthogonality of constacyclic codes of length $p^s$ over the finite commutative chain ring $\mathbb F_{p^m} + u \mathbb F_{p^m}$, where $u^2=0$ and $\sigma$ is a ring automorphism of $\mathbb F_{p^m} + u \mathbb F_{p^m}$. First, we obtain the structure of $\sigma$-dual code of a $\lambda$-constacyclic code of length $p^s$ over $\mathbb F_{p^m} + u \mathbb F_{p^m}$. Then, the necessary and sufficient conditions for a $\lambda$-constacyclic code to be $\sigma$-self-orthogonal are provided. In particular, we determine the $\sigma$-self-dual constacyclic codes of length $p^s$ over $\mathbb F_{p^m} + u \mathbb F_{p^m}$. Finally, we extend the results to constacyclic codes of length $2 p^s$

Constacyclic Codes over Fp+vFpF_p+vF_p

In this paper, we study constacyclic codes over $F_p+vF_p$, where $p$ is an odd prime and $v^2=v$. The polynomial generators of all constacyclic codes over $F_p+vF_p$ are characterized and their dual codes are also determined.Comment: 12 page

All α+uβ\alpha+u\beta-constacyclic codes of length npsnp^{s} over Fpm+uFpm\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}}

Let $\mathbb{F}_{p^{m}}$ be a finite field with cardinality $p^{m}$ and $R=\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}}$ with $u^{2}=0$. We aim to determine all $\alpha+u\beta$-constacyclic codes of length $np^{s}$ over $R$, where $\alpha,\beta\in\mathbb{F}_{p^{m}}^{*}$, $n, s\in\mathbb{N}_{+}$ and $\gcd(n,p)=1$. Let $\alpha_{0}\in\mathbb{F}_{p^{m}}^{*}$ and $\alpha_{0}^{p^{s}}=\alpha$. The residue ring $R[x]/\langle x^{np^{s}}-\alpha-u\beta\rangle$ is a chain ring with the maximal ideal $\langle x^{n}-\alpha_{0}\rangle$ in the case that $x^{n}-\alpha_{0}$ is irreducible in $\mathbb{F}_{p^{m}}[x]$. If $x^{n}-\alpha_{0}$ is reducible in $\mathbb{F}_{p^{m}}[x]$, we give the explicit expressions of the ideals of $R[x]/\langle x^{np^{s}}-\alpha-u\beta\rangle$. Besides, the number of codewords and the dual code of every $\alpha+u\beta$-constacyclic code are provided.Comment: arXiv admin note: text overlap with arXiv:1512.01406 by other author

An explicit representation and enumeration for self-dual cyclic codes over F2m+uF2m\mathbb{F}_{2^m}+u\mathbb{F}_{2^m} of length 2s2^s

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

Skew constacyclic codes over Fq+uFq+vFq

In this paper skew constacyclic codes over finite non-chain ring R = F_q+uF_q+vF_q, where q= p^m, p is an odd prime and u^{2}=u, v^{2}=v, uv = vu = 0 are studied. We show that Gray image of a skew alpha-constacyclic cyclic code of length n over R is a skew alpha-quasi-cyclic code of length $3n$ over F_{q} of index 3. It is also shown that skew alpha-constacyclic codes are either equivalent to alpha-constacyclic codes or alpha-quasi-twisted codes over R. Further, the structural properties of skew constacyclic over R are obtained by decomposition method.Comment: 10 pages paper communicated to the Journal of Algebra and its Application

Constacyclic codes over F_q + u F_q + v F_q + u v F_q

Let q be a prime power and F_q be a finite field. In this paper, we study constacyclic codes over the ring F_q+ u F_q +v F_q+ u v F_q, where u^2=u, v^2=v and uv=vu. We characterized the generator polynomials of constacyclic codes and their duals using some decomposition of this ring. We also define a gray map and characterize the Gray images of self-dual cyclic codes over F_q+uF_q+vF_q+uvF_q

Cyclic DNA codes over F2+uF2+vF2+uvF2

In this work, we study the structure of cyclic DNA codes of arbitrary lengths over the ring R=F2+uF2+vF2+uvF2 and establish relations to codes over R1=F2+uF2 by defining a Gray map between R and R1^2 where R1 is the ring with 4 elements. Cyclic codes of arbitrary lengths over R satisfied the reverse constraint and the reverse-complement constraint are studied in this paper. The GC content constraint is considered in the last