Quantum codes from $(1-2u_1-2u_2-\cdots -2u_m)$-skew constacyclic codes over the ring $F_q+u_1F_{q}+\cdots +u_{2m}F_{q}$

Abstract

In this paper, we study a class of skew constacyclic codes over the ring $R=F_q+u_1F_{q}+\cdots +u_{2m}F_{q}$, where $u_i^2=u_i$, $u_iu_j=u_ju_i=0$, for $i,j=1,2,\ldots ,2m ~,~ i \ne j$ and $q=p^s$, and derive the generator polynomials of this class of codes over R. Also, by using Calderbank–Shor–Steane construction, some new non-binary quantum codes have been obtained. Moreover, new quantum codes $[[225, 201, 5]]_9$, $[[351, 333, 4]]_9$, $[[405, 393, 3]]_9$, $[[405, 381, 5]]_{9}$ have been constructed.

1 Introduction

Linear codes over finite rings have recently raised a great interest for their new role in algebraic coding theory and for their successful application in combined coding and modulation. Recent developments have contributed toward achieving the reliability required by today’s high-speed digital systems, and the use of coding for error control has become an integral part in the design of modern communication systems and digital computers.

Constacyclic codes consist of an algebraically rich family of error-correcting codes and are generalizations of cyclic and negacyclic codes. These codes can be easily encoded using shift registers and can be easily decoded due to their rich algebraic structure, which justify their preferred role from engineering perspective.

Boucher et al. [11] generalized the notion of cyclic codes to skew cyclic codes by using generator polynomials in (non-commutative) skew polynomial rings. Since skew polynomial rings are left and right Euclidean, the obtained codes share most properties of cyclic codes. Since there are much more skew cyclic codes, this new class of codes allows to systematically search for codes with good properties.

Jitman et al. [22] defined skew constacyclic codes by defining the skew polynomial ring with coefficients from finite chain rings, especially the ring $F_{p^{m}}+uF_{p^{m}}$ where $u^2=0$. Further, the structural properties of skew cyclic codes through the decomposition method over $F_q+vF_q$, where $v^2=v$ and $q=p^m$, were given by Gursoy et al. [18]. Ashraf and Mohammad [1] studied the skew cyclic codes over the ring $F_q+vF_q$ with $v^2=1$ by taking the automorphism as $\theta ~:~v~\mapsto -v$. Recently, AL-Ashker and Abu-Jafar [6] investigated the structure of skew constacyclic codes over the ring $F_p+vF_p$ with $v^2=v$. Later on, Ashraf and Mohammad [2] gave the construction of skew constacyclic codes over the ring $F_q+vF_q+v^2F_q$, where $v^3=v$. Motivated by this study, in this paper, we study $(1-2u_1-2u_2-\cdots -2u_m)$-skew constacyclic codes over the ring $F_q+u_1F_{q}+\cdots +u_{2m}F_{q}$, where $u_i^2=u_i$, $u_iu_j=u_ju_i=0$, for $i,j=1,2,\ldots ,2m ~;~ i \ne j$ and $q=p^s$.

Quantum error-correcting codes play a prominent role in both quantum communication and quantum computation. Quantum error-correcting codes provide an efficient way to overcome decoherence. Shor discovered the first quantum error-correcting code in [31]. Later on, a method to obtain quantum error-correcting codes from classical error-correcting codes was given by Calderbank et al. [14]. Recently, the theory of quantum error-correcting codes has been developed rapidly. Many good quantum error-correcting codes have been constructed by using classical cyclic codes over finite field $F_q$ (q is a power of prime number) with self-orthogonal or dual containing properties (for references see [7, 19,20,21, 24,25,26, 32]).

The construction for finding quantum codes from cyclic codes of odd length over the finite chain ring $F_2+uF_2$ with $u^2=0$ was first given by Qian et al. [27]. Later on, Kai and Zhu [23] gave a construction for obtaining quantum codes from cyclic codes of odd length over the finite chain ring $F_4+uF_4$ with $u^2=0$. Further, Qian [28] provided a new method of constructing quantum error-correcting codes from cyclic codes over the finite non-chain ring $F_2+vF_2$ with $v^2=v$ of arbitrary length. Motivated by this study, Ashraf and Mohammad [3,4,5] obtained non-binary quantum codes from cyclic codes over different types of finite rings. A lot of work has been done in this direction (see [9, 15,16,17, 29]). In this paper, we obtain quantum codes over $F_q$ from skew constacyclic codes over the ring $R=F_q+u_1F_{q}+\cdots +u_{2m}F_{q}$ using decomposition method.

2 Preliminaries

Let R be the ring $F_q+u_1F_{q}+\cdots +u_{2m}F_{q}$, where $u_i^2=u_i$, $u_iu_j=u_ju_i=0$, for $i,j=1,2,\ldots ,2m ~;~ i \ne j$ and $q=p^s$. It is a commutative ring with $q^{2m+1}$ elements. Here, $F_q$ denotes the finite field with q elements.

Recall that a linear code C of length n over R is a R-submodule of $R^n$. Elements of C are called code words. Let $x=(x_0, x_1, \ldots , x_{n-1})$ and $y=(y_0, y_1, \ldots , y_{n-1})$ be two elements of $R^n$. Then, the Euclidean inner product of x and y is defined as $x\cdot y =x_0y_0+x_1y_1+\cdots +x_{n-1}y_{n-1}.$ The dual code $C^\perp$ of C is defined as $C^\perp =\{x\in R^n|~x\cdot y=0,~\forall ~y\in C\}.$ A code C is called self-orthogonal if $C \subseteq C^\perp$ and self-dual if $C=C^\perp$.

Any element of R can be written as $a_0+u_1a_1+u_2a_2+\cdots +u_{2m}a_{2m} = a_0(1-u_1 -u_2- \cdots - u_{2m})+u_1(a_0 +a_1) + u_2(a_0 +a_2)+\cdots +u_{2m}(a_0 +a_{2m})$.

Let

$$\begin{aligned} \eta _0= & {} 1-u_1 - u_2-\cdots - u_{2m},\\ \eta _1= & {} u_1,~~ \eta _2=u_2 , \ldots , ~\eta _{2m}=u_{2m}. \end{aligned}$$

It is easy to see

$$\begin{aligned} \sum _{i=0}^{2m}\eta _i = 1, \quad \eta _i ^ 2=\eta _i \quad \hbox {and} \quad \eta _i.\eta _j=0\quad \hbox {for} \quad i,j=0,1,2,\ldots ,{2m} \quad \hbox {and}\quad i \ne j. \end{aligned}$$

Thus, $R=\eta _0R\oplus \eta _1R\oplus \cdots \oplus \eta _{2m}R.$ Therefore, any arbitrary element of R can be uniquely expressed as $x=\eta _0a_0 +\eta _1a_1+\cdots +\eta _{2m}a_{2m}$, where $a_0,a_1,\ldots ,a_{2m} \in F_q$. Now, we define a gray map $\Phi$ from R to $F_q^{2m+1}$ defined as

$$\begin{aligned} \Phi (x) = (a_0,a_1,\ldots ,a_{2m}). \end{aligned}$$

It is easy to see that this is a linear map and can be extended component-wise.

For any element $x=\eta _0a_0 +\eta _1a_1+\cdots +\eta _{2m}a_{2m} \in R,$ we define the Lee weight of x as $w_{\mathrm{L}}(x)=w_{\mathrm{H}}(\Phi (x)),$ where $w_{\mathrm{H}}(\Phi (x))$ denotes the Hamming weight of $\Phi (x)$ over $F_q$, where the Hamming weight of any elements is defined as the number of nonzero components. We define the Lee weight of x as $w_{\mathrm{L}} (x) =\sum _{i=0} ^ {n-1} w_{\mathrm{L}} (x_i)$. The Lee distance between $x=(x_0, x_1, \ldots , x_{n-1})$ and $y=(y_0, y_1, \ldots , y_{n-1})\in R^n$ is defined by $d_\mathrm{L}(x, y)=w_{\mathrm{L}}(x-y)=\sum _{i=0}^{n-1}w_{\mathrm{L}}(x_i-y_i).$ The Lee distance of C is defined as $d_\mathrm{L}(C)=\hbox {min}$ $d_\mathrm{L}(x,y)$ for any $x \ne y$.

Let $A_i;~i=0,1,2,\ldots ,{2m}$ be code over R. We denote $A_0\oplus A_1\oplus \cdots \oplus A_{2m}=\{a_0+a_1+\cdots +a_{2m}~|~a_i\in A_i,~i=0,1,2,\ldots ,{2m}\}$ and $A_0\otimes A_1\otimes \cdots \otimes A_{2m}=\{(a_0,a_1,\ldots ,a_{2m})~|~a_i\in A_i,~i=0,1,2,\ldots ,{2m}\}.$ For a linear code C of length n over R, define

$$\begin{aligned} C_0= & {} \{a_0\in F_{q}^{n}~|~\eta _0 a_0+\eta _1 a_1+\cdots +\eta _{2m}a_{2m}\in C,~ ~a_{i}\in F_{q}^{n}, ~i=1,2,\ldots ,2m\},\\ C_1= & {} \{a_1\in F_{q}^{n}~|~\eta _0 a_0+\eta _1 a_1+\cdots +\eta _{2m}a_{2m}\in C,~ ~a_{i}\in F_{q}^{n}, ~i=0,2,\ldots ,2m\},\\&\ldots \ldots \ldots \\ C_{2m}= & {} \{a_{2m } \in F_{q}^{n}~|~\eta _0 a_0+\eta _1 a_1+\cdots +\eta _{2m}a_{2m}\in C,~ ~a_{i}\in F_{q}^{n},\\&~i=0,1,2,\ldots ,2m-1\}. \end{aligned}$$

Here, $C_i$ are linear codes over $F_{q}^{n}$ for $i=0,1,2,\ldots ,2m$. Then, $C_i$ are q-ary linear codes of length n. Hence, a linear code C of length n over R can be uniquely expressed as $C=\eta _0C_0 \oplus \eta _1C_1 \oplus \cdots \oplus \eta _{2m}C_{2m}$ and $|C|=|C_0||C_1|\cdots |C_{2m}|$ and $d_{\mathrm{H}}(C)=\text{ min }\{d_{\mathrm{H}}(C_i),~i=0,1,2,\ldots ,2m\}.$

A matrix is called generator matrix of C if the rows generate C. If $M_i$ are the generator matrices of q-ary linear codes $C_i,~i=0,1,\ldots ,{2m}$, respectively, then the generator matrix of C is

$$\begin{aligned} M=\left( \begin{array}{cccccc} \eta _0 M_0\\ \eta _1 M_1\\ \ldots \\ \eta _{2m} M_{2m} \end{array}\right) \end{aligned}$$

and the generator matrix of $\Phi (C)$ is

$$\begin{aligned} \Phi (M)=\left( \begin{array}{cccccc} \Phi (\eta _0 M_0)\\ \Phi (\eta _1 M_1)\\ \ldots \\ \Phi (\eta _{2m} M_{2m}) \end{array}\right) . \end{aligned}$$

Here, we define an automorphism on R as

$$\begin{aligned} \theta _t:R\longrightarrow R \end{aligned}$$

defined by

$$\begin{aligned} \theta _t(a_0+ua_1+u_2a_2+\cdots +u_{2m}a_{2m})=(a_0^{p^t}+u_1a_1 ^{p^t}+u_2 a_2^{p^t}+\cdots +u_{2m} a_{2m}^{p^t}) \end{aligned}$$

for all $a_0,a_1,\ldots ,a_{2m} \in F_{q}$. Also, the automorphism $\theta _t$ acts on $F_q$ as follows:

$$\begin{aligned}&\theta _t:F_{q}\longrightarrow F_{q}\\&\theta _t(a)=a^{p^t}. \end{aligned}$$

The order of this automorphism is $|\langle \theta _t\rangle |=s/t$.

Definition 1.1

The set $R[x, \theta _t]=\{a_0+a_1x+a_2x^2+\cdots +a_nx^n|~a_i\in R, n\ge 0\}$ forms a non-commutative ring under usual addition of polynomials, and multiplication is defined by the rule $(ax^i)(bx^j)=a\theta _{t}^{i}(b)x^{i+j}$. This ring is called skew polynomial ring. This ring was first introduced by Ore. The ring has no nonzero divisors; the units of this ring are exactly the units of the ring R.

Definition 1.2

A subset C of $R^n$ is called a $\lambda$-skew constacyclic code of length n if

• (1) C forms an R-submodule of $R^n$ and

• (2) If $c=(c_{0},c_{1}, \ldots , c_{n-1}) \in C$, then $\sigma _\lambda (c): =(\theta _t (\lambda c_{n-1}), \theta _t(c_{0}), \ldots , \theta _t (c_{n-2})) \in C$.

Defining a map, $\psi : R^n \longrightarrow R_n=R[x,\theta _t]/\langle x^{n}-\lambda \rangle$ as $c=(c_{0},c_{1}, \ldots , c_{n-1}) \longrightarrow c_{0}+c_{1}x+\cdots +c_{n-1}x^{n-1}.$ Each code word $c = (c_{0}, c_{1}, \ldots ,c_{n-1})\in R^{n}$ can be identified with the polynomial $c(x) = c_{0}+c_{1}x+\cdots +c_{n-1}x^{n-1}$ in $R_n$. This map is a R-module isomorphism. By this identification, C is a $\lambda$-skew constacyclic code over $R_n$ if and only if C is a left $R[x,\theta _t]$-submodule of $R[x,\theta _t]/\langle x^{n}-\lambda \rangle$.

3 Results on gray map and linear codes over R

Proposition 2.1

Let $\Phi$ be the gray map defined in the preliminary section. Then,

• (1) $\Phi$ is a $F_q$-linear distance preserving map from $R^n$ (Lee distance) to $F_q^{(2m+1)n}$ (Hamming distance).

• (2) If C is a $[n,k,d_\mathrm{L}]$ linear code over R, then $\Phi (C)$ is a $[(2m+1)n,k,d_{\mathrm{H}}]$ linear code over $F_q$, where $d_\mathrm{L}=d_{\mathrm{H}}$.

Proof

Let $x_1=a_0\eta _0+a_1\eta _1+\cdots +a_{2m}\eta _{2m}$ and $x_2=b_0\eta _0+b_1\eta _1+\cdots +b_{2m}\eta _{2m}\in R^n$. $\Phi (x_1+x_2)=(a_0+b_0,a_1+b_1,\ldots ,a_{2m}+b_{2m})=(a_0,a_1,\ldots ,a_{2m})+(b_0,b_1,\ldots ,b_{2m})=\Phi (x_1)+\Phi (x_2)$ and $\Phi (c x_1) =(c a_0,c a_1,\ldots ,c a_{2m})=c \Phi (x)$ for $c \in F_q$. Therefore, $\Phi$ is a $F_q$-linear map.

Since $d_{\mathrm{L}} (c_1,c_2)=w_{\mathrm{L}} (c_1-c_2)=w_{\mathrm{H}}( \Phi (c_1-c_2))=w_{\mathrm{H}}( \Phi (c_1)-\Phi (c_2))=d_{\mathrm{H}}( \Phi (c_1),\Phi (c_2))$, $\Phi$ is a $F_{q}$-linear distance preserving map.

By the first part, $\Phi$ is a distance preserving linear map so $d_{\mathrm{L}}=d_{\mathrm{H}}$. Also, as $\Phi$ is bijection, therefore $|C|=|\Phi (C)|=q^k$. Hence, the result follows. $\square$

Proposition 2.2

Let C be a linear code of length n over R.

• (1) Then, C is self-orthogonal, if and only if $C_j$ are self-orthogonal over $F_q$, for $j=0,1,2,\ldots ,{2m}$.

• (2) If C is a self-orthogonal, then $\Phi (C)$ is also self-orthogonal.

Proof

(1) Let C be a self-orthogonal linear code over R and $x=a_0\eta _0+a_1\eta _1+\cdots +a_{2m}\eta _{2m}$ be any element of C, where $a_j \in C_j$ for $j=0,1,2,\ldots ,{2m}$. Since C is self orthogonal,

$$\begin{aligned} x\cdot x= & {} (a_0\eta _0+a_1\eta _1+\cdots +a_{2m}\eta _{2m})\cdot (a_0\eta _0+a_1\eta _1+\cdots +a_{2m}\eta _{2m})\\= & {} a_0^2 \eta _0+a_1^2\eta _1+\cdots +a_{2m}^2\eta _{2m}=0. \end{aligned}$$

This implies $a_0^2 = a_1^2 =\cdots = a_{2m}^2=0$. Hence, $a_j \in C^{\perp }_j$, for $j=0,1,2,\ldots ,2m$. Therefore, $C_j$ are self orthogonal over $F_q$, for $j=0,1,2,\ldots ,2m$.

(2) Let $x_1=a_0\eta _0+a_1\eta _1+\cdots +a_{2m}\eta _{2m}$ and $x_2=b_0\eta _0+b_1\eta _1+\cdots +b_{2m}\eta _{2m}\in C$, where $a_j, b_j \in F_q$, for $j=0,1,2,\ldots ,{2m}$. Now by inner product of $x_1$ and $x_2$, we have $x_1\cdot x_2=a_0b_0\eta _0+a_1b_1\eta _1+\cdots +a_{2m}b_{2m}\eta _{2m}$. Since C is self-orthogonal, $a_0b_0=a_1b_1=\cdots =a_{2m}b_{2m}=0$.

On the other hand, $\Phi (x_1)\cdot \Phi (x_2)=(a_0,a_1,\ldots ,a_{2m})\cdot (b_0,b_1,\ldots ,b_{2m})=a_0b_0+a_1b_1+\cdots +a_{2m}b_{2m}=0$. Hence, $\Phi (C)$ is self-orthogonal. $\square$

Theorem 2.1

Let C be a linear code of length n over R. Then, $\Phi (C) = C_0 \otimes C_1 \otimes \cdots C_{2m}$.

4 Properties of skew constacyclic codes over R

Theorem 3.1

Let $C=\oplus _{j} \eta _jC_j$ be a linear code of length n over R for $j=0,1,2,\ldots ,{2m}$.

• (1) Then, C is a $(1-2u_1-2u_2- \cdots -2u_m)$-skew constacyclic code of length n over R if and only if $C_{0},C_{m+1},C_{m+2},\ldots ,C_{2m}$ are skew cyclic and $C_{1},C_{2},C_{3},\ldots ,C_{m}$ are skew negacyclic codes of length n over $F_{q}$.

• (2) Let the order of the automorphism divides n. If C is a $(1-2u_1-2u_2-\cdots -2u_m)$-skew constacyclic code of length n over R, then the dual $C^{\perp }=\oplus _{j} \eta _jC_j ^{\perp }$ is a $(1-2u_1-2u_2-\cdots -2u_m)$-skew constacyclic code of length n over R, for $j=0,1,\ldots ,{2m}$.

Proof

(1) Let C be a $(1-2u_1-2u_2-\cdots -2u_m)$-skew constacyclic code of length n over R and let $c=(c_{0},c_{1}, \ldots , c_{n-1}) \in C$, where $c_{i}=c_{i}^0\eta _0+c_{i}^1\eta _1+\cdots +c_i^{2m} \eta _{2m}$, $c_{i}^j\in F_{q}$ for $i=0,1, \ldots , n-1$ and $j=0,1,2,\ldots ,2m$. So $(c_0^j,c_1^j,\ldots , c_{n-1}^j) \in C_j$. Since C is a $(1-2u_1-2u_2-\cdots -2u_m)$-skew constacyclic code of length n over R, we have

$$\begin{aligned} \sigma _\lambda (c)=( \theta _t(1-2u_1-2u_2-\cdots -2u_m)c_{n-1}),\theta _t(c_0),\ldots ,\theta _t(c_{n-2})) \in C. \end{aligned}$$

Note that

$$\begin{aligned} (1-2u_1-2u_2-\cdots -2u_m)\eta _l=-\eta _l \quad \hbox {for} \quad l=1,2,\ldots ,m \end{aligned}$$

and

$$\begin{aligned} (1-2u_1-2u_2-\cdots -2u_m)\eta _k=\eta _k \quad \hbox {for} \quad k=0,m+1,\ldots ,2m. \end{aligned}$$

Therefore,

$$\begin{aligned} \sigma _\lambda (c)= & {} \eta _0(\theta _t( c^0_{n-1}),\theta _t(c^0_0),\ldots ,\theta _t(c^0_{n-2}))\\&+ \sum _{j=1}^{m} \eta _j(-\theta _t(c^j_{n-1}),\theta _t(c^j_1),\ldots ,\theta _t(c^j_{n-2}))\\&\,+ \sum _{j=m+1}^{2m} \eta _j(\theta _t(c^j_{n-1}),\theta _t(c^j_0),\ldots ,\theta _t(c^j_{n-2})). \end{aligned}$$

Hence,

$$\begin{aligned} (-\theta _t(c^j_{n-1}),\theta _t(c^j_1),\ldots ,\theta _t(c^j_{n-2})) \in C_j \quad \hbox {for} \quad j=1,2,\ldots ,{m}, \end{aligned}$$

and

$$\begin{aligned} (\theta _t(c^j_{n-1}),\theta _t(c^j_1),\ldots ,\theta _t(c^j_{n-2})) \in C_j \quad \hbox {for} \quad j=0,m,m+1,m+2,\ldots ,{2m}. \end{aligned}$$

Therefore, $C_{0},C_{m+1},C_{m+2},\ldots ,C_{2m}$ are skew cyclic and $C_{1},C_{2},C_{3},\ldots ,C_{m}$ are skew negacyclic codes of length n over $F_{q}$.

Conversely, suppose that $C_{0},C_{m+1},C_{m+2},\ldots ,C_{2m}$ are skew cyclic and $C_{1},C_{2},C_{3},\ldots ,C_{m}$ are skew negacyclic codes of length n over $F_{q}$. Then, considering the above notations,

$$\begin{aligned} (-\theta _t(c^j_{n-1}),\theta _t(c^j_1),\ldots ,\theta _t(c^j_{n-2})) \in C_j \quad \hbox {for} \quad j=1,2,\ldots ,{m}, \end{aligned}$$

and

$$\begin{aligned} (\theta _t(c^j_{n-1}),\theta _t(c^j_1),\ldots ,\theta _t(c^j_{n-2})) \in C_j \quad \hbox {for} \quad j=0,m,m+1,m+2,\ldots ,{2m}. \end{aligned}$$

Since

$$\begin{aligned} \sigma _\lambda (c)= & {} (\theta _t( c^0_{n-1}),\theta _t(c^0_0),\ldots ,\theta _t(c^0_{n-2}))+ \sum _{j=1}^{m} (-\theta _t(c^j_{n-1}),\theta _t(c^j_1),\ldots ,\theta _t(c^j_{n-2}))\\&\,+ \sum _{j=m+1}^{2m} (\theta _t(c^j_{n-1}),\theta _t(c^j_0),\ldots ,\theta _t(c^j_{n-2})) \in C, \end{aligned}$$

we find that C is a $(1-2u_1-2u_2-\cdots -2u_m)$-skew constacyclic code of length n over R.

(2) We have $C^{\perp }=\oplus _{j} \eta _jC_j ^{\perp }$ for $j=0,1,\ldots ,{2m}$. As the order of the automorphism divides n, the dual code of every skew constacyclic code over $F_{q}$ is also skew constacyclic [12, 13], by (1) of this theorem, $C^\perp$ is a $(1-2u_1-2u_2-\cdots -2u_m)$-skew constacyclic code. $\square$

Theorem 3.2

Let $C=\oplus _{i} \eta _iC_i$ be a $(1-2u_1-2u_2-\cdots -2u_m)$-skew constacyclic code of length n over R. Suppose $f_i$ are the monic generator polynomials of $C_i$ for $i=0,1,2,\ldots ,{2m}.$

• (1) Then, $C=\langle \eta _0f_0, \eta _1f_1,\ldots ,\eta _{2m} f_{2m} \rangle$ and $|C|=q^{({2m}+1)n-(\sum _{i=0}^{2m} \hbox {deg}(f_i))}$.

• (2) There exists a polynomial $f(x) \in R[x,\theta _t]$ such that $C=<f(x)>$, where $f(x)=\eta _0f_0 (x)+\eta _1f_2(x) +\cdots + \eta _{2m} f_{2m}(x)$ and $f(x)|(x^{n}- (1-2u_1-2u_2-\cdots -2u_m))$.

Proof

(1) Let C be a $(1-2u_1-2u_2-\cdots -2u_m)$-skew constacyclic code of length n over R. Then, by Theorem 3.1 we get, $C_{0},C_{m+1},C_{m+2},\ldots ,C_{2m}$ are skew cyclic and $C_{1},C_{2},C_{3},\ldots ,C_{m}$ are skew negacyclic codes of length n over $F_{q}$. So we can write $C_k=(f_k(x)) \subseteq F_q[x,\theta _t]/(x^n-1 )$, for $k=0,m+1,m+2,\ldots ,2m$, and $C_l=(f_l(x)) \subseteq F_q[x,\theta _t]/(x^n+1)$, for $l=1,2,\ldots ,m$. Also as $C=\oplus _{i} \eta _iC_i$, we can write C as

$$\begin{aligned}&C=\{f(x) \; : \; f(x) = \eta _0f_0 (x) + \eta _1f_1(x) +\cdots + \eta _{2m} f_{2m}(x), \\&\quad where \; f_i(x) \in C_i\quad \hbox {for} \quad i =0,1,\ldots ,{2m} \}. \end{aligned}$$

This implies $C \subseteq \; <\eta _0f_0, \eta _1f_1,\ldots ,\eta _{2m} f_{2m}> \; \subseteq \; R[x,\theta _t]/(x^n-(1-2u_1-2u_2-\cdots -2u_m))$.

On the other hand, let $\sum _{i=0}^{2m}\eta _if_i (x) g_i (x) \in <\eta _0f_0, \eta _1f_1,\ldots ,\eta _{2m} f_{2m}>$, where $g_i(x)$ are elements of $R[x,\theta _t]/(x^n-(1-2u_1-2u_2-\cdots -2u_m))$. Then, there exists

$$\begin{aligned} r_k(x) \in F_q[x,\theta _t]/(x^n-1) \quad \hbox {for} \quad k=0,m+1,m+2,\ldots ,2m \end{aligned}$$

and

$$\begin{aligned} r_l(x) \in \; F_q[x,\theta _t]/(x^n+1) \quad \hbox {for} \quad l=1,2,\ldots ,m \end{aligned}$$

such that $\eta _i g_i (x) = \eta _i r_i(x)$ for $i =0,1,\ldots ,{2m}$. Therefore, $<\eta _0f_0, \eta _1f_1,\ldots ,\eta _{2m} f_{2m} > \; \subseteq \; C$. Hence, $C = <\eta _0f_0, \eta _1f_1,\ldots ,\eta _{2m} f_{2m}>$.

For the other part, it is worth noting that $|\Phi (C)| =| C |$, and hence

$$\begin{aligned} | C |= & {} | C_0 || C_1 |\cdots | C_{2m} |= q^{n- \hbox {deg}(f_0)} q^{n- \hbox {deg}(f_1)} \ldots q^{n- \hbox {deg}(f_{2m})}\\= & {} q^{({2m}+1)n-\left( \sum _{i=0}^{2m} \hbox {deg}(f_i)\right) }. \end{aligned}$$

(2) Let C be a $(1-2u_1-2u_2-\cdots -2u_m)$-skew constacyclic code of length n over R and suppose $f_i$ are monic generator polynomials of $C_i$ for $i =0,1,\ldots ,{2m}$. Then, we can write $C = <\eta _0f_0, \eta _1f_1,\ldots ,\eta _{2m} f_{2m}>$. Suppose $C^\prime =<\eta _0f_0 (x) + \eta _1f_1(x) +\cdots + \eta _{2m} f_{2m}(x)>$, then it is obvious that $C^\prime \subseteq C$. As $\eta _i(\eta _0f_0 (x) + \eta _1f_1(x) +\cdots + \eta _{2m} f_{2m}(x))=\eta _i f_i$ for $i =0,1,\ldots ,{2m}$, this implies $C \subseteq C^\prime$. Therefore, $C = C^\prime ,$ and $C=<f(x)>$, where $f(x)=\eta _0f_0 (x) + \eta _1f_1(x) +\cdots + \eta _{2m} f_{2m}(x)$.

Now suppose $f_i$ is the monic generator polynomial of $C_i$ for $i =0,1,\ldots ,{2m}$. Then, $f_k$ divides $x^n-1$ for $k =0,m+1,\ldots ,{2m}$ and $f_l$ divides $x^n+1$ for $l =1,\ldots ,{m}$.

$$\begin{aligned} x^n -(1-2u_1-2u_2-\cdots -2u_m)=\left( \sum _{i=0}^{2m} \eta _ig_i (x)\right) \left( \sum _{i=0}^{2m} \eta _if_i (x) \right) \end{aligned}$$

for $g_i\in C_i$, for $i =0,1,\ldots ,{2m}$. Therefore, $x^{n}-(1-2u_1-2u_2-\cdots -2u_m)=(\eta _0g_0 (x) + \eta _1g_1(x) +\cdots + \eta _{2m} g_{2m}(x))f(x)$. Hence, $f(x)|(x^{n}- (1-2u_1-2u_2-\cdots -2u_m))$. $\square$

Corollary 3.1

Let the order of the automorphism divides n and $C=\oplus _{i} \eta _iC_i$ be a $(1-2u_1-2u_2-\cdots -2u_m)$-skew constacyclic code of length n over R and suppose $f_i$ are the generator polynomials of $C_i$ for $i=0,1,2,\ldots ,{2m}.$ Then,

• (1) $C^{\perp } = <\eta _0h_0^*,\eta _1h_1^*,\ldots ,\eta _{2m} h_{2m}^*>$ and $|C^\perp |={q}^{(\sum _{i=0}^{2m}\hbox {deg}(f_i))}$

• (2) There exists a polynomial $h^*(x)$ such that $C^{\perp }=<h^*(x)>$ where $h^*(x)=\eta _0h^*_0 (x) + \eta _1h^*_1(x) + \cdots +\eta _{2m} h^*_{2m}(x)$.

The polynomial $h_i(x)$ and $h_i^*(x)$ are defined as $x^n-1=h_i(x)f_i(x)$, where $f_i,h_i \in F_q[x,\theta _t]$. If $f_i(x)=a_0+a_1x+\cdots +a_s x^s$ and $h(x)=b_0+b_1x+\cdots +b_{n-s} x^{n-s}$, then the dual code of C is generated by $h_i^*(x)$, where $h_i^*(x)=b_{n-s}+\theta _t (b_{n-s-1})x+\cdots +\theta _t ^{n-s}(b_{0})x^{n-s}$.

5 Necessary and sufficient condition of self-dual skew cyclic and skew negacyclic codes over R

By Theorem 3.1, any $(1-2u_1-2u_2-\cdots -2u_m)$-skew constacyclic code over R is a direct product of skew cyclic and skew negacyclic codes over $F_q$. Here, in this section, we study the necessary and sufficient condition for skew cyclic codes to contain its dual.

In [30], it has shown that if $\hbox {gcd}(n,k)= 1$, then a skew cyclic (skew negacyclic) code C of length n over any finite field is equivalent to a cyclic (negacyclic) code, where k is the order of the automorphism of the finite field. Further, this result has been extended over a finite ring in [8]. Therefore, if $\hbox {gcd}(n,k)= 1$, then a skew cyclic (skew negacyclic) code C of length n over R is equivalent to a cyclic (negacyclic) code of length n over R.

Lemma 4.1

[14] Let C be a linear cyclic or negacyclic code with generator polynomial g(x) over $F_p$. Then, C contains its dual code if and only if

$$\begin{aligned} x^n - \lambda \equiv 0 (\hbox {mod} \, g(x) g^{*}(x)), \end{aligned}$$

where $g^{*}(x)$ is the reciprocal polynomial of g(x) and $\lambda = \pm 1.$

Theorem 4.1

Let $C=\oplus _{i} \eta _iC_i$ be a $(1-2u_1-2u_2-\cdots -2u_m)$-skew constacyclic code of length n over R, $\hbox {gcd}(n,k)= 1$ and $C_i=\langle g_i(x)\rangle$ for $i=0,1,\ldots ,2m$, where k is the order of the automorphism $\theta _t$. Then, $C^{\perp } \subseteq C$ if and only if $x^n- 1 \equiv 0 \, (\hbox {mod} \, f_j f_j^{*})$, $x^n+ 1 \equiv 0 \, (\hbox {mod} \, f_l f_l^{*})$, where $j=0,m+1,m+2,\ldots ,{2m}$ and $l=1,2,\ldots ,{m}$.

Proof

Let $x^n-1 \equiv 0 \, (\hbox {mod} \, f_j f_j^{*})$, $x^n+ 1 \equiv 0 \, (\hbox {mod} \, f_l f_l^{*})$, where $j=0,m+1,m+2,\ldots ,{2m}$ and $l=1,2,\ldots ,{m}$. Since $\hbox {gcd}(n, k)=1$, each $C_j$ are equivalent to cyclic code for $j=0,m+1,m+2,\ldots ,{2m}$ and $C_l$ are equivalent to negacyclic code $l=1,2,\ldots ,{m}$ Now, using Lemma 4.1, we have $C_{i}^\perp \subseteq C_{i},~i=0,1,\ldots ,2m.$ Therefore, $\eta _iC_{i}^\perp \subseteq \eta _iC_{i},~i=0,1,2,\ldots ,{2m}.$ Thus, $\oplus _{i} \eta _iC_i^\perp \subseteq \oplus _{i} \eta _iC_i$. Hence, $C^\perp \subseteq C$.

Conversely, as $\hbox {gcd}(n,k)= 1$, a $\lambda$-skew constacyclic code C of length n over R is equivalent to a $\lambda$-constacyclic code of length n over R. Suppose that $C^\perp \subseteq C$. Then, $\oplus _{i} \eta _iC_i^\perp \subseteq \oplus _{i} \eta _iC_i$. Since $C_{i}$ are linear codes over $F_q$ such that $\eta _iC_{i}\equiv C (\hbox {mod} \;\eta _i)$, for $i=0,1,2,\ldots ,{2m}$, $C_{i}^\perp \subseteq C_{i}$ for $i=0,1,2,\ldots ,{2m}.$ Therefore, $x^n-1 \equiv 0 \, (\hbox {mod} \, f_j f_j^{*})$, $x^n+ 1 \equiv 0 \, (\hbox {mod} \, f_l f_l^{*})$, where $j=0,m+1,m+2,\ldots ,{2m}$ and $l=1,2,\ldots ,{m}$. $\square$

Corollary 4.1

Let $C=\oplus _{i} \eta _iC_i$ be a $(1-2u_1-2u_2-\cdots -2u_m)$-skew constacyclic code of length n over R and $\hbox {gcd}(n,k)= 1$, where k is the order of the automorphism $\theta _t$. Then, $C^\perp \subseteq C$ if and only if $C_{i}^\perp \subseteq C_{i},~i=0,1,\ldots ,{2m}.$

6 Quantum codes from $(1-2u_1-2u_2-\cdots -2u_m)$-skew constacyclic codes over R

Let H be a q-dimensional Hilbert space over the complex numbers $\mathbb {C}$. Define $H^{\otimes n}$ to be n-fold tensor product of the Hilbert space H, that is, $H^{\otimes n}=H\otimes H\otimes \cdots \otimes H$(n-times). Then, $H^{\otimes n}$ is a Hilbert space of $q^n$ dimension. A quantum code of length n and dimension k over $F_q$ is defined to be the Hilbert subspace of $H^{\otimes n}$. A quantum code with length n, dimension k and minimum distance d over $F_q$ is denoted by $[[n, k, d]]_q$.

Theorem 5.1

[14] (CSS Construction) Let $C_1$ and $C_2$ be $[n, k_1, d_1]_q$ and $[n, k_2, d_2]_q$ linear codes over $F_q$, respectively, with $C_{2}^{\perp }\subseteq C_1$. Furthermore, let $d=min\{d_1, d_2\}$. Then, there exists a quantum error-correcting code C with parameters $[[n, k_1+k_2-n, d]]_q$. In particular, if $C_{1}^{\perp }\subseteq C_1$, then there exists a quantum error-correcting code C with parameters $[[n, 2k_1-n, d_1]]$.

Theorem 5.2

Let $C=\oplus _{i} \eta _iC_i$ be a $(1-2u_1-2u_2-\cdots -2u_m)$-skew constacyclic code of length n over R and $\hbox {gcd}(n,k)= 1$. If $C_{i}^\perp \subseteq C_{i},~i=0,1,\ldots ,2m$, then $C^\perp \subseteq C$ and there exists a quantum error-correcting code with parameters $[[({2m}+1)n,~2k-({2m}+1)n,~d_{\mathrm{L}}]]$, where $d_{\mathrm{L}}$ denotes the minimum Lee weight of the code C and k denotes the dimension of the code $\Phi (C)$.

Proof

Let $C_{i}^\perp \subseteq C_{i}$ for $~i=0,1,\ldots ,2m$. Then, by the Corollary 4.1, we get $C^\perp \subseteq C$. Now let $x\in \Phi (C^\perp )=\Phi (C)^\perp$, then there exists $y \in C^\perp$ such that $x= \Phi (y)$, where $y\cdot y' =0$ for all $y' \in C$. Since $C^\perp \subseteq C$ and $y \in C^\perp$, we have $y \in C$. Hence, $x=\Phi (y) \in \Phi (C)$. Therefore, $\Phi (C)^\perp \subseteq \Phi (C)$. As $\Phi (C)$ is a $[(2m+1)n,k,d_{\mathrm{L}}]$ linear code over $F_q$. Then, by CSS Construction, there exists a quantum error-correcting code with parameters $[[(2m+1)n,~2k-(2m+1)n,~d_{\mathrm{L}}]]$. $\square$

Example 5.1

Let $R=F_{9}+u_1F_{9}+u_2F_{9}$ and $\lambda =(1-2u_1)$. Let $\theta (a)=a^3$ for $a\in F_{9}$.

$$\begin{aligned} x^{33}-1=(2 + 2x + x^2 + 2x^3 + x^5)^3(2 + x^2 + 2x^3 + x^4 + x^5)^3(2 + x)^3\in F_{9}[x,\theta ]. \end{aligned}$$

Let $f_i(x)=(2 + x^2 + 2x^3 + x^4 + x^5)^2$. Then $C_i=\langle f_i(x) \rangle$ are skew cyclic codes over $F_{9}$ with parameters [33, 23, 3]. As $gcd(33,2)=1$ so $C_i$ are equivalent to cyclic codes, also as $f_i(x)f_{i}^{*}(x)$ divide $x^{33}-1$, by Lemma 4.1, $C_i^\perp \subseteq C_i$, where $i=0,2$.

$$\begin{aligned} x^{33}+1=(1 + 2x + 2x^2 + 2x^3 + x^5)^3(1 + 2x^2 + 2x^3 + 2x^4 + x^5)^3(1 + x)^3\in F_{9}[x,\theta ]. \end{aligned}$$

Let $f_1(x)=(1 + 2x^2 + 2x^3 + 2x^4 + x^5)^2$. Then $C_1=\langle f_1(x) \rangle$ is a skew negacyclic code over $F_{9}$ with parameter [33, 23, 3]. As $gcd(33,2)=1$ so $C_1$ is equivalent to negacyclic code, also as $f_1(x)f_{1}^{*}(x)$ divides $x^{33}+1$, by Lemma 4.1, we get $C_1^\perp \subseteq C_1$.

Then

$$\begin{aligned} C=\langle \gamma _0f_0(x)\rangle ,\gamma _1f_1(x) ,\gamma _{2}f_{2}(x)\rangle \end{aligned}$$

is a $\lambda$-skew constacyclic code of length 33 over R. Thus, $\Phi (C)$ is a code over $F_{9}$ with parameters [99, 69, 3]. As $C_i^\perp \subseteq C_i$ for $i=0,1,2$, we get $C^\perp \subseteq C$. Now using Theorem 5.2, we get a quantum code with parameter [[99, 39, 3]].

Example 5.2

Let $R=F_{81}+u_1F_{81}+u_2F_{81}$ and $\lambda =(1-2u_1)$. Let $\theta (a)=a^3$ for $a\in F_{81}$.

$$\begin{aligned} x^{13}-1= & {} (2 + x)(2 + x^2 + x^3)(2 + 2x + x^3)(2 + 2x + 2x^2 + x^3)\\&(2+ x + x^2 + x^3)\in F_{81}[x,\theta ]. \end{aligned}$$

Let $f_0(x)=(2 + 2x + x^3)(2 + 2x + 2x^2 + x^3)$ and $f_2(x)=(2 + x^2 + x^3)( 2 + x + x^2 + x^3).$ Then $C_i=\langle f_i(x) \rangle$ are skew cyclic codes of length 13 over $F_{81}$ with parameters [13, 7, 4]. As $\hbox {gcd}(13,4)= 1$, so $C_i$ are equivalent to cyclic codes of length 13 and $f_i(x)f_{i}^{*}(x)$ divide $x^{13}-1$, by Lemma 4.1, we get $C_i^\perp \subseteq C_i$, for $i=0,2$.

$$\begin{aligned} x^{13}+1= & {} (1 + x)(1 + 2x^2 + x^3)(1 + 2x + x^3)(1 + 2x + x^2 + x^3)\\&(1 + x + 2x^2 + x^3)\in F_{81}[x,\theta ]. \end{aligned}$$

Let $f_1(x)=(1 + 2x^2 + x^3)( 1 + 2x + x^2 + x^3)$. Then $C_1=\langle f_1(x) \rangle$ is skew negacyclic code over $F_{81}$ with parameter [13, 7, 4]. As $gcd(13,4)= 1$, so $C_1$ is equivalent to negacyclic code of length 13 and as $f_1(x)f_{1}^{*}(x)$ divides $x^{13}+1$. Therefore, by Lemma 4.1, we get $C_1^\perp \subseteq C_1$. Thus,

Then

$$\begin{aligned} C=\langle \gamma _0f_0(x), \gamma _1f_1(x), \gamma _2f_2(x)\rangle \end{aligned}$$

is a $\lambda$-skew constacyclic code of length 13 over R. Thus, $\Phi (C)$ is a code over $F_{25}$ with parameters [39, 21, 4]. As $C_i^\perp \subseteq C_i$ for $i=0,1,2$, we get $C^\perp \subseteq C$. Now, using Theorem 5.2, we get a quantum code with parameter [[39, 3, 4]]. This quantum code is new in the literature.

Example 5.3

Let $R=F_{27}+u_1F_{27}+u_{2}F_{27}+u_3F_{27}+u_4F_{27}$ and $\lambda =(1-2u_1-2u_2)$. Let $\theta (a)=a^3$ for $a\in F_{27}$.

$$\begin{aligned} x^{11}-1=(2+x)(2+x^2+2x^3+x^4+x^5)(2+2x+x^2+2x^3+x^5)\in F_{27}[x,\theta ]. \end{aligned}$$

Let $f_i(x)=(2 + x^2 + 2x^3 + x^4 + x^5)$. Then $C_i=\langle f_i(x) \rangle$ are skew cyclic codes over $F_{27}$ with parameters [11, 6, 5]. As $gcd(11,3)=1$ so $C_i$ are equivalent to cyclic codes, also as $f_i(x)f_{i}^{*}(x)$ divide $x^{11}-1$, by Lemma 4.1, we get $C_i^\perp \subseteq C_i$, where $i=0,3,4$.

$$\begin{aligned} x^{11}+1= & {} (1+x)(1+2x^2+2x^3+2x^4+x^5)(1+2x+2x^2+2x^3+x^5)\\&\in F_{27}[x,\theta ]. \end{aligned}$$

Let $f_j(x)=(1 + 2x + 2x^2 + 2x^3 + x^5)$. Then $C_j=\langle f_j(x) \rangle$ are skew negacyclic codes over $F_{27}$ with parameters [11, 6, 5]. As $gcd(11,3)=1$ so $C_j$ are equivalent to negacyclic codes, also as $f_j(x)f_{j}^{*}(x)$ divide $x^{11}+1$, by Lemma 4.1, we get $C_j^\perp \subseteq C_j$, where $j=1,2$.

Then

$$\begin{aligned} C=\langle \gamma _0f_0(x)\rangle ,\gamma _1f_1(x),\cdots ,\gamma _{4}f_{4}(x)\rangle \end{aligned}$$

is a $\lambda$-skew constacyclic code of length 11 over R. Thus, $\Phi (C)$ is a code over $F_{27}$ with parameters [55, 30, 5]. As $C_i^\perp \subseteq C_i$ for $i=0,1,\dots ,4$, we get $C^\perp \subseteq C$. Now using Theorem 5.2, we get a quantum code with parameter [[55, 5, 5]]. This quantum code is new in the literature.

Example 5.4

Let $R=F_{49}+u_1F_{49}+\cdots +u_6F_{49}$ and $\lambda =(1-2u_1-2u_2-2u_3)$. Let $\theta (a)=a^7$ for $a\in F_{49}$.

$$\begin{aligned} \begin{array}{ccc}x^{15}-1&{}=&{}(3+x)(5+x)(6+x)(1 + x + x^2 + x^3 + x^4)(4 + x + 2x^2 + 4x^3 + x^4)\\ &{}&{}\times (2 + x + 4x^2 + 2x^3 + x^4)\in F_{49}[x,\theta ]. \end{array} \end{aligned}$$

Let $f_i(x)=(4 + x + 2x^2 + 4x^3 + x^4)(5 + x)$. Then $C_i=\langle f_i(x) \rangle$ are skew cyclic codes over $F_{49}$ with parameters [15, 10, 3]. As $gcd(15,2)=1$ so $C_i$ are equivalent to cyclic codes, also as $f_i(x)f_{i}^{*}(x)$ divide $x^{15}-1$, by Lemma 4.1, we get $C_i^\perp \subseteq C_i$, where $i=0,4,5,6$.

$$\begin{aligned} \begin{array}{ccl}x^{15}+1&{}=&{}(1+x)(2+x)(4+x)(1 + 6x + x^2 + 6x^3 + x^4)\\ &{}&{}\times (4 + 6x + 2x^2 + 3x^3 + x^4)(2 + 6x + 4x^2 + 5x^3 + x^4)\in F_{49}[x,\theta ].\end{array} \end{aligned}$$

Let $f_j(x)=(2 + 6x + 4x^2 + 5x^3 + x^4)(4 + x)$. Then $C_j=\langle f_j(x) \rangle$ are skew negacyclic codes over $F_{49}$ with parameters [15, 10, 3]. As $gcd(15,2)=1$ so $C_j$ are equivalent to negacyclic codes, also as $f_j(x)f_{j}^{*}(x)$ divide $x^{15}+1$, by Lemma 4.1, we get $C_j^\perp \subseteq C_j$, where $j=1,2,3$.

Then

$$\begin{aligned} C=\langle \gamma _0f_0(x)\rangle ,\gamma _1f_1(x),\cdots ,\gamma _{6}f_{6}(x)\rangle \end{aligned}$$

is a $\lambda$-skew constacyclic code of length 15 over R. Thus, $\Phi (C)$ is a code over $F_{49}$ with parameters [105, 70, 3]. As $C_i^\perp \subseteq C_i$ for $i=0,1,\dots ,6$, we get $C^\perp \subseteq C$. Now using Theorem 5.2, we get a quantum code with parameter [[105, 35, 3]].

Example 5.5

Let $R=F_{121}+u_1F_{121}+\cdots +u_{10}F_{121}$ and $\lambda =(1-2u_1-\cdots -2u_{5})$. Let $\theta (a)=a^{11}$ for $a\in F_{121}$.

$$\begin{aligned} \begin{array}{ccl}x^{15}-1&{}=&{}(2+x)(6+x)(7+x)(8+x)(10+x)(9 + 3x + x^2)(5 + 4x + x^2)\\ &{}&{}\times (4 + 9x + x^2)(3 + 5x + x^2)(1 + x + x^2)\in F_{121}[x,\theta ].\end{array} \end{aligned}$$

Let $f_i(x)=(4 + 9x + x^2)(5 + 4x + x^2)(6 + x)(8 + x)$. Then $C_i=\langle f_i(x) \rangle$ are skew cyclic codes over $F_{49}$ with parameters [15, 9, 5]. As $gcd(15,2)=1$ so $C_i$ are equivalent to cyclic codes, also as $f_i(x)f_{i}^{*}(x)$ divide $x^{15}-1$, by Lemma 4.1, we get $C_i^\perp \subseteq C_i$, where $i=0,6,7,8,9,10$.

$$\begin{aligned} \begin{array}{ccl}x^{15}+1&{}=&{}(1+x)(3+x)(4+x)(5+x)(9+x)(9 + 8x + x^2)(5 + 7x + x^2)\\ &{}&{}\times (4 + 2x + x^2)(3 + 6x + x^2)(1 + 10x + x^2)\in F_{121}[x,\theta ].\end{array} \end{aligned}$$

Let $f_j(x)=(5 + 7x + x^2)(3 + x)(4 + 2x + x^2)(5 + x)$. Then $C_j=\langle f_j(x) \rangle$ are skew negacyclic codes over $F_{121}$ with parameters [15, 9, 5]. As $gcd(15,2)=1$ so $C_j$ are equivalent to negacyclic codes, also as $f_j(x)f_{j}^{*}(x)$ divide $x^{15}+1$, by Lemma 4.1, we get $C_j^\perp \subseteq C_j$, where $j=1,2,3,4,5$.

Then

$$\begin{aligned} C=\langle \gamma _0f_0(x)\rangle ,\gamma _1f_1(x),\cdots ,\gamma _{10}f_{10}(x)\rangle \end{aligned}$$

is a $\lambda$-skew constacyclic code of length 15 over R. Thus, $\Phi (C)$ is a code over $F_{121}$ with parameters [165, 99, 5]. As $C_i^\perp \subseteq C_i$ for $i=0,1,\ldots ,10$, we get $C^\perp \subseteq C$. Now, using Theorem 5.2, we get a quantum code with parameter [[165, 33, 5]]. This quantum code is new in the literature.

The following table contains some new quantum error-correcting codes over $F_9$. Let $R=F_9+u_1F_9+u_2F_9$, first column of the table denotes the length of $(1-2u_1)$-skew constacyclic codes over R, second column denotes the generator polynomials of skew cyclic codes $C_i$ for $i=0, 2$, column third denotes the generator polynomial of skew negacyclic code $C_1$, column four denotes the parameters of the gray images of $(1-2u_1)$-skew constacyclic codes over R, and the last column denotes the parameters of the associated quantum codes. We write coefficients of generator polynomials in descending order, for example, the polynomial $x^8+\alpha ^2 x^6+x^5+\alpha x^4+\alpha x^3+ x^2+\alpha ^2 x+\alpha ^5$ is represented by $10\alpha ^21\alpha \alpha 1\alpha ^2\alpha ^5$.

n	$f_0(x)=f_2(x)$	$f_1(x)$	$\Phi (C)$	[[n, k, d]]
27	111	121	[81, 75, 3]	[[81, 69, 3]]
75	11111	12121	[225, 213, 5]	[[225, 201, 5]]
91	1112	1211	[273, 264, 4]	[[273, 255, 4]]
99	102122	102221	[297, 282, 5]	[[297, 267, 5]]
117	1112	1121	[351, 342, 4]	[[351, 333, 4]]
135	111	121	[405, 399, 3]	[[405, 393, 3]]
135	11111	12121	[405, 393, 5]	[[405, 381, 5]]

Comparison: Compared to previously known quantum error-correcting codes in the references [10, 19, 20], some of our quantum error-correcting codes $[[({2m}+1)n,~2k-({2m}+1)n,~d_{\mathrm{L}}]]$ are new. In the above table, our quantum codes [[225, 201, 5]], [[351, 333, 4]], [[405, 393, 3]] and [[405, 381, 5]] have better parameters than the known quantum codes [[224, 196, 5]], [[352, 329, 4]], [[401, 389, 3]] and [[401, 369, 5]], respectively, in [10].

7 Conclusion

In this paper, we have given the structural properties of $(1-2u_1-2u_2-\cdots -2u_m)$-skew constacyclic codes over the ring $R=F_q+u_1F_{q}+\cdots +u_{2m}F_{q}$. As an application of this class of codes over R, we have obtained some new quantum codes over the field $F_q$. For future work, it would be interesting to find quantum codes over $F_q$ by taking another gray map over R.