April 02, 2024
Let \(\mathbb{F}_q\) be the finite field of \(q=p^m\) elements where \(p\) is a prime and \(m\) is a positive integer. This paper considers \((\gamma,\Delta)\)-cyclic codes over a class of finite commutative non-chain rings \(\mathscr{R}_{q,s}=\mathbb{F}_q[v_1,v_2,\dots,v_s]/\langle v_i-v_i^2,v_iv_j=v_jv_i=0\rangle\) where \(\gamma\) is an automorphism of \(\mathscr{R}_{q,s}\), \(\Delta\) is a \(\gamma\)-derivation of \(\mathscr{R}_{q,s}\) and \(1\leq i\neq j\leq s\) for a positive integer \(s\). Here, we show that a \((\gamma,\Delta)\)-cyclic code of length \(n\) over \(\mathscr{R}_{q,s}\) is the direct sum of \((\theta,\Im)\)-cyclic codes of length \(n\) over \(\mathbb{F}_q\), where \(\theta\) is an automorphism of \(\mathbb{F}_q\) and \(\Im\) is a \(\theta\)-derivation of \(\mathbb{F}_q\). Further, necessary and sufficient conditions for both \((\gamma,\Delta)\)-cyclic and \((\theta,\Im)\)-cyclic codes to contain their Euclidean duals are established. Finally, we obtain many quantum codes by applying the dual containing criterion on the Gray images of these codes. The obtained codes have better parameters than those available in the literature.
example.eps
gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore
After the pioneering work of Hammons et al. [1] in 1994, codes over finite rings attracted many researchers for better error-correcting codes. Later,
several important research has been carried out over finite rings and explored plenty of suitable parameters; we refer [2]–[6]. Nevertheless, all of these works have been considered over finite commutative rings. Hence, it is natural to look at these works over the noncommutative ring to obtain codes with better parameters.
Towards this, in 2007, Boucher et al. [7] introduced skew cyclic codes as a generalized class of cyclic codes by using a non-trivial automorphism \(\theta\) on a finite field \(\mathbb{F}_q\). They proved that the noncommutative rings (skew polynomial rings) are worthy alphabets for producing new parameters. In addition, they have provided
a few codes with better parameters that are not known earlier over finite commutative rings. Since the factorization of the polynomial \(x^n-1\) plays an important role in the characterization of cyclic codes of length
\(n\), more factorization leads to the case of getting many new code parameters. Therefore, skew cyclic codes that generalize cyclic codes in a noncommutative setup attract many researchers. Later, Abualrub et al. [8] and Bhaintwal [9] introduced skew-quasi cyclic codes.
From an application point of view, recently, many authors have shown that skew cyclic codes are one of the important resources for producing new quantum codes along with classical codes (see [7], [10]–[14]).
However, all of the above works have been carried out on skew polynomial rings of automorphism type. Only a few works are available in the literature with both automorphisms and derivations. In [15]–[18], the authors generalized the notion of codes over skew polynomial rings with non-trivial automorphism \(\theta\) and
\(\theta\)- derivation \(\Im\) under the usual addition of polynomials and a specific polynomials multiplication involving \(\theta\) and \(\Im\). For the noncommutative ring \(\mathbb{F}_q[x,\theta;\Im]\) where \(\theta\) is the Frobenius automorphism \(a\mapsto
a^p\), \(p\) is the characteristic of \(\mathbb{F}_q\), the authors [15], [18] defined the inner \(\theta\)-derivation \(\Im\) induced by \(\beta\in \mathbb{F}_q^*\) of the form
\(a\mapsto \beta(\theta(a)-a)\). Further, Boulagouaz and Leroy [16] studied \((\sigma,\delta)\)-codes
with \(\sigma\)-derivation induced by the ring element. Recently, Sharma and Bhaintwal [19] have studied skew cyclic codes
over \(\mathbb{Z}_4+u\mathbb{Z}_4,\) \(u^2=1\) with both automorphism and inner derivation. In 2021, Ma et al. [20] studied \((\sigma,\delta)\)-skew quasi-cyclic codes over the ring \(\mathbb{Z}_4+u\mathbb{Z}_4,\) \(u^2=1\). Further, in
2021, Patel and Prakash [21] studied \((\theta,\delta_\theta)\)-cyclic codes over the ring \(\mathbb{F}_q[u,v]/\langle u^2-u, v^2-v, uv-vu \rangle\) via the decomposition method over \(\mathbb{F}_q\). Here, we extend our previous work [21] to a more general structure and propose a fruitful application of \((\gamma,\Delta)\)-cyclic codes in the context of quantum code construction. Per our survey, it is
worth mentioning that this is the first article proposing an application of \((\gamma,\Delta)\)-cyclic codes into quantum codes.
Quantum error-correcting codes play a significant role in protecting information against the disturbances such as decoherence occurring in the channel. Shor [22]
discovered the first quantum code. After that, Calderbank et al. [23] provided a method to obtain quantum codes from classical codes. This
technique became very popular among researchers and is known as the CSS (Calderbank-Shor-Steane) construction. Presently, quantum codes and their implementation from classical codes have gained significant attention. As a consequence, many quantum codes
with better parameters have been constructed from different families of linear codes such as cyclic, skew cyclic, skew constacyclic codes, etc., see [2]–[6], [24]–[27]. But the searching of new methods on different structures are still going on by which one can construct quantum codes efficiently with suitable parameters. Since getting new quantum codes
proportionally depend on the abundance of factors of \(x^n-1\), many authors have been exploring quantum codes in the setting of the skew polynomial ring with automorphism where \(x^n-1\)
indeed possesses more factorization than the commutative case. However, in this work, we extend all these works in a new direction by considering skew polynomial rings with non-trivial automorphisms and derivations. In this case, we have different
derivations for the same Frobenius automorphism having the form \(a\mapsto \beta(\theta(a)-a)\) for all \(\beta\in \mathbb{F}_q^*\).
The rest of the paper is structured as follows: In Section \(2\), we present some basic results and notations that will be useful for later sections. In Section \(3\), we discuss \((\theta,\Im)\)-cyclic codes over \(\mathbb{F}_q\) and derive a necessary and sufficient condition to contain their duals over \(\mathbb{F}_q\). Further, Section
\(4\) includes the results on \((\gamma,\Delta)\)-cyclic codes over \(\mathscr{R}_{q,s}\) and dual-containing property for these codes as well. Section \(5\) describes the applications of our obtained results by providing many new quantum codes with superior parameters. Finally, Section \(6\) concludes our work.
In this Section, we provide some preliminary results, definitions and notations which are used throughout this paper. We consider a finite non-chain ring \(\mathscr{R}_{q,s}:=\mathbb{F}_q[v_1,v_2,\dots,v_s]/\langle
v_i\\-v_i^2,v_iv_j=v_jv_i=0\rangle\) where \(1\leq i\neq j\leq s\) and \(s\) is a positive integer. This \(\mathscr{R}_{q,s}\) is a class of finite
commutative ring with unity for different values of \(q\) and \(s\). Further, \(\mathscr{R}_{q,s}\) can also be represented in the form of \(\mathscr{R}_{q,s}=\mathbb{F}_q+v_1\mathbb{F}_q+\cdots+v_s\mathbb{F}_q\) with \(v_i-v_i^2,v_iv_j=v_jv_i=0\). Moreover, \(\mathscr{R}_{q,s}\) is a non-chain
semi-local Frobenius ring having \(s + 1\) maximal ideals. For \(s=2\), there are three maximal ideals \(\langle v_1+v_2\rangle\), \(\langle 1-v_1\rangle\) and \(\langle 1-v_2\rangle\) in \(\mathscr{R}_{q,2}\), refer Islam20b?. Consider \[\zeta_0=\displaystyle\prod_{i=1}^{s}(1-v_i), ~~\text{and}~~ \zeta_j=v_j, ~~1\leq j\leq s.\] It is easy to verify that \(\sum_{i=0}^{s} \zeta_i=1\) and \[\zeta_i\zeta_j= \begin{cases} \zeta_i, &if~ i= j\\ 0, &if ~i\neq j \end{cases}.\] Therefore, by Chinese Remainder Theorem, \(\mathscr{R}_{q,s}=\zeta_0\mathscr{R}_{q,s}\oplus\zeta_1\mathscr{R}_{q,s}\oplus\cdots\oplus\zeta_s\mathscr{R}_{q,s}=\zeta_0\mathbb{F}_q\oplus\zeta_1\mathbb{F}_q\oplus\cdots\oplus\zeta_s\mathbb{F}_q\). From here we can conclude
that any element \(t\in\mathscr{R}_{q,s}\) can be uniquely written as \(t=\zeta_0t_0+\zeta_1t_1+\cdots+\zeta_st_s\), where \(t_i\in \mathbb{F}_q\). Also,
\(t\) is a unit in \(\mathscr{R}_{q,s}\) if and only if \(t_i\in \mathbb{F}_q^*\) for all \(i\).
Recall that a non-empty subset \(\mathcal{C}\) of \(\mathscr{R}_{q,s}^n\) is said to be a linear code of length \(n\) over \(\mathscr{R}_{q,s}\) if it is an \(\mathscr{R}_{q,s}\)-submodule of \(\mathscr{R}_{q,s}^n\) and the elements of \(\mathcal{C}\)
are called codewords. The Hamming weight \(w_H(c)\) of a codeword \(c=(c_0,c_1,\dots,c_{n-1})\in \mathcal{C}\) is the number of nonzero coordinates in \(c\).
The Hamming distance between any two codewords \(c\) and \(c'\) of \(\mathcal{C}\) is defined as \(d_H(c,c')=w_H(c-c')\) and the Hamming distance of a linear code \(\mathcal{C}\) is defined as \(d_H(\mathcal{C})= \min \{ d_H(x,y)~|~ x, ~y\in \mathcal{C}, x
\neq y \}.\) The Euclidean inner product of \(c\) and \(c'\) in \(\mathcal{R}^n\) is defined by \(c\cdot c' =
\sum_{i=0}^{n-1}c_ic'_i\) where \(c=(c_0,c_1,\dots,c_{n-1})\) and \(c'=(c'_0,c'_1,\dots,c'_{n-1})\) are codewords in \(\mathcal{C}\). The dual code of \(\mathcal{C}\) is defined by \(\mathcal{C}^\perp= \{c\in \mathscr{R}_{q,s}^n~|~ c\cdot c' = 0, ~\text{for all} ~c'\in
\mathcal{C} \}\). Also, a linear code \(\mathcal{C}\) is self-orthogonal if \(\mathcal{C}\subseteq \mathcal{C}^\perp\) and self-dual if \(\mathcal{C}=\mathcal{C}^\perp\). Further, let \(c=(c_0,c_1,\dots,c_{n-1})\in\mathcal{C}\subseteq \mathbb{F}_q^n\). If \(\mathcal{C}\) is an \([n,k,d]\) linear code, then from the Singleton bound, its minimum distance is bounded above by \(d \leq n - k + 1\), where \(d\) is the minimum distance, \(k\) is the dimension, and \(n\) is the length of the code. A code achieving the mentioned bound is called maximum-distance-separable (MDS). If the minimum distance of the code is one unit less
than the MDS, then the code is called almost MDS. A linear code is said to be optimal if it has the highest possible minimum distance for a given length and dimension.
Definition 1. Let \(\mathscr{R}_{q,s}\) be a finite ring and \(\gamma\) be an automorphism of \(\mathscr{R}_{q,s}\). Then a map \(\Delta : \mathscr{R}_{q,s} \rightarrow \mathscr{R}_{q,s}\) is said to be a \(\gamma\)-derivation of \(\mathscr{R}_{q,s}\) if
\(\Delta(x+y)=\Delta(x)+\Delta(y)\);
\(\Delta(xy)=\Delta(x)y+\gamma(x)\Delta(y)\)
for all \(x,y\in \mathscr{R}_{q,s}.\)
Let us consider an automorphism \(\theta:\mathbb{F}_q\rightarrow \mathbb{F}_q\) defined by \(\theta(a)=a^q\), for all \(a\in \mathbb{F}_q\) and a \(\theta\)-derivation \(\Im:\mathbb{F}_q\rightarrow \mathbb{F}_q\) defined by \(\Im(a)=\beta(\theta(a)-a)\), for all \(a\in \mathbb{F}_q\) and \(\beta\in \mathbb{F}_q^*\). Now, we extend the above maps over \(\mathscr{R}_{q,s}\) and define the skew polynomial ring with both automorphism and derivation over \(\mathscr{R}_{q,s}\). Let \(Aut(\mathscr{R}_{q,s})\) be the set of all automorphism of \(\mathscr{R}_{q,s}\) and \(\gamma\in Aut(\mathscr{R}_{q,s})\). We consider the set \[\mathscr{R}_{q,s}[x;\gamma, \Delta]=\{b_lx^l+\cdots+b_1x+b_0~|~ b_i\in \mathscr{R}~ \text{and}~ l\in \mathbb{N}\},\] where \(\Delta\) is a \(\gamma\)-derivation of \(\mathscr{R}_{q,s}\). Then \(\mathscr{R}_{q,s}[x;\gamma, \Delta]\) is a noncommutative ring unless \(\gamma\) is the identity under the usual addition of polynomials and multiplication is defined with respect to \(xb=\gamma(b)x+\Delta(b)\) for \(b\in \mathscr{R}_{q,s}\), known as a skew polynomial ring.
Definition 2. An element \(f(x)\in\mathscr{R}_{q,s}[x;\gamma, \Delta]\) is said to be a central element of \(\mathscr{R}_{q,s}[x;\gamma, \Delta]\) if \(f(x)b(x)=b(x)f(x),\) for all \(b(x)\in \mathscr{R}_{q,s}[x;\gamma, \Delta]\).
Definition 3. [28], [29] A pseudo-linear transformation \(T_{\gamma,\Delta}: \mathscr{R}_{q,s}^n \rightarrow \mathscr{R}_{q,s}^n\) is an additive map defined by \[\label{eq1} T_{\gamma,\Delta}(v)= \gamma(v)M + \Delta(v),\qquad{(1)}\] where \(v= (v_1,v_2,\dots,v_n)\in \mathscr{R}_{q,s}^n\), \(\gamma(v)= (\gamma(v_1),\gamma(v_2),\dots,\gamma(v_n))\in \mathscr{R}_{q,s}^n\), \(M\) is a matrix of order \(n\times n\) over \(\mathscr{R}_{q,s}\) and \(\Delta(v)= (\Delta(v_1),\Delta(v_2),\dots,\Delta(v_n))\in \mathscr{R}_{q,s}^n\). If \(\Delta=0\), then \(T_{\gamma}\) is known as semi-linear transformation.
Definition 4.
A code \(\mathcal{C}\) of length \(n\) over \(\mathscr{R}_{q,s}\) is said to be a \((\gamma, \Delta)\)-linear code if it is a left \(\mathscr{R}_{q,s}[x;\gamma, \Delta]\)-submodule of \(\frac{\mathscr{R}_{q,s}[x;\gamma, \Delta]}{\langle x^n-1 \rangle}\). Moreover, if \(x^n-1\) is a central element of \(\mathscr{R}_{q,s}[x;\gamma, \Delta]\), then \(\mathcal{C}\) is a central \((\gamma, \Delta)\)-linear code.
A code \(\mathcal{C}\) of length \(n\) over \(\mathscr{R}_{q,s}\) is said to be a \((\gamma, \Delta)\)-cyclic code if
\(\mathcal{C}\) is a \((\gamma, \Delta)\)-linear code;
\(T_{\gamma,\Delta}(\mathcal{C})\subseteq \mathcal{C}\), where \(T_{\gamma,\Delta}\) is as defined in Equation (?? ) and \(M\) is defined as \[M=\begin{pmatrix} 0 & 1 & \dots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \dots & 1\\ 1 & 0 & \dots & 0\\ \end{pmatrix}.\]
[[30]][]{#re1 label=“re1”} Let \(\mathscr{R}_{q,s}[x;\gamma, \Delta]\) be a skew polynomial ring, \(r\in \mathscr{R}_{q,s}\) and \(n\in \mathbb{N}\). Then \[x^nr=\gamma^n(r)x^n+a_{n-1}x^{n-1}+\cdots+a_1x+\Delta^n(r),\] for some \(a_{n-1},\dots, a_1\in\mathscr{R}_{q,s}\).
To find generator polynomials of \(( \gamma, \Delta)\)-cyclic codes over \(\mathscr{R}_{q,s},\) first we derive the right division algorithm in \(\mathscr{R}_{q,s}[x;\gamma, \Delta]\).
Theorem 1. ( The Right Division Algorithm)Let \(f(x),~g(x)\in \mathscr{R}_{q,s}[x;\gamma, \Delta]\) such that the leading coefficient of \(g(x)\) be a unit. Then there exist \(q(x),~r(x)\in \mathscr{R}_{q,s}[x;\gamma, \Delta]\) such that \[f(x)=q(x)g(x)+r(x),\] where \(r(x)=0\) or \(\deg~r(x)<\deg~g(x)\).
Proof. If \(f(x)=0\), then result follows by taking \(q(x),~r(x)=0\). If \(\deg~f(x)<\deg~g(x)\), then we take \(q(x)=0\) and \(r(x)=f(x)\). Further, for \(\deg f(x)\geq \deg g(x)\), we prove it by induction on \(\deg f(x)\).
It can be easily seen that the result is true for \(\deg f(x)=0\). Now, suppose the result is true for all polynomials of degree less than \(\deg~f(x)\). Let \(f(x)=f_0+f_1x+\cdots+f_sx^s\) and \(g(x)=g_0+g_1x+\cdots+g_tx^t\) be two polynomials in \(\mathscr{R}_{q,s}[x;\gamma, \Delta]\) such that \(f_s\neq0\) and \(g_t\) is a unit. Consider a polynomial \[h(x)=f(x)-f_s\gamma^{s-t}(g_t^{-1})x^{s-t}g(x).\] From Remark [re1], \(h(x)\) can be written as \[\begin{align} h(x)=&f(x)-f_s\gamma^{s-t}(g_t^{-1})x^{s-t}g(x) =f(x)-f_s\gamma^{s-t}(g_t^{-1})x^{s-t}(g_0+g_1x+\cdots+g_tx^t)\\
=&f(x)-f_s\gamma^{s-t}(g_t^{-1})x^{s-t}g_0-f_s\gamma^{s-t}(g_t^{-1})x^{s-t}g_1x-\cdots-f_s\gamma^{s-t}(g_t^{-1})x^{s-t}g_tx^t\\ =&f(x)-f_s\gamma^{s-t}(g_t^{-1})(\gamma^{s-t}(g_0)x^{s-t}+a_{s-t-1}x^{s-t-1}+\cdots+a_1x+\Delta^{s-t}(g_0))\\&
-\cdots-f_s\gamma^{s-t}(g_t^{-1})(\gamma^{s-t}(g_t)x^{s-t}+b_{s-t-1}x^{s-t-1}+\cdots+b_1x+\Delta^{s-t}(g_t))x^t\\ =&f(x)-f_s\gamma^{s-t}(g_t^{-1})(\gamma^{s-t}(g_0)x^{s-t}+a_{s-t-1}x^{s-t-1}+\cdots+a_1x+\Delta^{s-t}(g_0))\\&
-\cdots-f_s\gamma^{s-t}(g_t^{-1})\gamma^{s-t}(g_t)x^{s-t}x^t-f_s\gamma^{s-t}(g_t^{-1})b_{s-t-1}x^{s-t-1}x^t-\cdots-\\&f_s\gamma^{s-t}(g_t^{-1})\Delta^{s-t}(g_t)x^t\\
=&f(x)-f_s\gamma^{s-t}(g_t^{-1})(\gamma^{s-t}(g_0)x^{s-t}+a_{n-1}x^{s-t-1}+\cdots+a_1x+\Delta^{s-t}(g_0))\\& -\cdots-f_sx^{s}-f_s\gamma^{s-t}(g_t^{-1})b_{s-t-1}x^{s-1}-\cdots-f_s\gamma^{s-t}(g_t^{-1})\Delta^{s-t}(g_t)x^t\\
=&f_0+f_1x+\cdots+f_sx^s-f_s\gamma^{s-t}(g_t^{-1})(\gamma^{s-t}(g_0)x^{s-t}+a_{s-t-1}x^{s-t-1}+\cdots\\&+a_1x+\Delta^{s-t}(g_0))-\cdots-f_sx^{s}-f_s\gamma^{s-t}(g_t^{-1})b_{s-t-1}x^{s-1}-\cdots-f_s\gamma^{s-t}(g_t^{-1})\\&\Delta^{s-t}(g_t)x^t\\
=&f_0+f_1x+\cdots+f_{s-1}x^{s-1}-f_s\gamma^{s-t}(g_t^{-1})(\gamma^{s-t}(g_0)x^{s-t}+a_{s-t-1}x^{s-t-1}\\&+\cdots+a_1x+\Delta^{s-t}(g_0))-\cdots-f_s\gamma^{s-t}(g_t^{-1})b_{s-t-1}x^{s-1}-\cdots-f_s\gamma^{s-t}(g_t^{-1})\\&\Delta^{s-t}(g_t)x^t
\end{align}\] where \(a_1,a_2,\dots,a_{s-t-1},b_1,b_2,\dots,b_{s-t-1}\in \mathscr{R}_{q,s}\). Now, we can conclude that \(\deg~h(x)<\deg~f(x)\). Hence, by induction on \(\deg~h(x),\) there exist \(b(x), r(x)\in \mathscr{R}_{q,s}[x;\gamma, \Delta]\) such that \[h(x)=b(x)g(x)+r(x),\] where \(r(x)=0\) or \(\deg~r(x)<\deg~g(x)\). Thus, \[\begin{align} f(x)&=h(x)+f_s\gamma^{s-t}(g_t^{-1})x^{s-t}g(x) \\
&=b(x)g(x)+r(x)+f_s\gamma^{s-t}(g_t^{-1})x^{s-t}g(x)\\ &=(b(x)+f_s\gamma^{s-t}(g_t^{-1})x^{s-t})g(x)+r(x)\\ &=q(x)g(x)+r(x),
\end{align}\] where \(q(x)=b(x)+f_s\gamma^{s-t}(g_t^{-1})x^{s-t}\in \mathscr{R}_{q,s}[x;\gamma, \Delta]\) and \(r(x)=0\) or \(\deg~r(x)<\deg~g(x)\). This gives the required result. ◻
Similarly, one can define the left division algorithm. In above Theorem 1, if \(r(x)=0,\) then \(g(x)\) is called a right divisor of \(f(x)\) or \(f(x)\) is a left multiple of \(g(x)\) in \(\mathscr{R}_{q,s}[x;\gamma, \Delta]\). Throughout this paper, we consider the right division.
In the following, we discuss the main algebraic properties of \((\theta,\Im)\)-cyclic codes in \(R=\mathbb{F}_q[x;\theta,\Im]\) and provide a necessary and sufficient condition for these codes to contain their Euclidean duals. In [16], Boulagouaz and Leroy introduced the notion of \((f,\gamma,\Delta)\)-cyclic codes. Moreover, a \((\theta,\Im)\)-cyclic code \(\mathcal{C}\) is the subset of \(\mathbb{F}_q^n\) consisting of the coordinates of the elements of \(Rg(x)/\langle x^n-1\rangle\) in the basis \(\{1,x,\dots,x^{n-1}\}\) for some right monic factors \(g(x)\) of \(x^n-1\).
Theorem 2. Let \(g(x)=g_0+g_1x+\cdots+ g_rx^r\in R\) be a monic polynomial.
A \((\theta,\Im)\)-cyclic code of length \(n\) corresponding to \(Rg(x)/\langle x^n-1\rangle\) is a free left \(\mathbb{F}_q\)-module of dimension \(n-r\) where \(r=\deg~g(x)\).
If \(v=(v_0,v_1,\dots,v_{n-1})\in \mathcal{C},\) then \(T_{\theta,\Im}(v)\in \mathcal{C}\).
The rows of the matrix which generates the code \(\mathcal{C}\) are given by \[T_{\theta,\Im}^k(g_0,g_1,\dots,g_r,0,0,\dots,0),~~~~\text{for}~0\leq k\leq n-r-1.\]
Proof.
We have \(x^n-1=h(x)g(x)\) for some monic polynomials \(h(x)\in R\). Hence, as left \(R\)-module, we have \(Rg(x)/\langle x^n-1\rangle\cong R/\langle h(x)\rangle\). Since \(h\) is monic, \(R/\langle h(x)\rangle\) is a free left \(\mathbb{F}_q\)-module of rank \(deg~h(x)=n-r\).
\(v=(v_0,v_1,\dots,v_{n-1})\in \mathcal{C}\) if and only if \(v(x):=\sum_{i=0}^{n-1} v_{i}x^{i}+\langle x^n-1\rangle\in Rg(x)/\langle x^n-1\rangle\). Since \(xv(x)\in Rg(x)/\langle x^n-1\rangle\) and left multiplication by \(x\) on \(R/\langle x^n-1\rangle\) corresponds to the action of \(T_{\theta,\Im}\) on \(\mathbb{F}_q^n\), we have \(T_{\theta,\Im}(v)\in \mathcal{C}\).
We have \(T_{\theta,\Im}^k(v_0,v_1,\dots,v_{n-1})\in \mathcal{C}\) for any \(k\geq 0\). On the other hand, it is clear that \(g, xg, x^2g,\dots,\\ x^{n-r-1}g\) are left linearly independent over \(\mathbb{F}_q\), all are taken modulo \(x^n-1\) and hence form a basis of \(Rg(x)/\langle x^n-1\rangle\). In codewords representation, this implies that the vectors \(T_{\theta,\Im}^k(g_0,g_1,\dots,g_r,0,\dots,0)\) form a left \(\mathbb{F}_q\)-basis for \(\mathcal{C}\), \(0\leq k\leq n-r-1.\)
◻
Theorem 3. Let \(\mathcal{C}\) be a left \(R\)-submodule of \(R/\langle x^n-1\rangle\). Then \(\mathcal{C}\) is a \((\theta,\Im)\)-cyclic submodule generated by a monic polynomial of the smallest degree in \(\mathcal{C}\).
Proof. Let \(g(x)\in \mathcal{C}\) be a monic smallest degree polynomial among nonzero polynomials in \(\mathcal{C}\) and \(c(x)\in \mathcal{C}\). Then by Theorem 1, there exist unique polynomials \(q(x)\) and \(r(x)\) in \(R\) such that \(c(x) = q(x)g(x) + r(x)\) where \(r(x)=0~ \text{or}~ \deg r(x)<\deg g(x).\) As \(\mathcal{C}\) is a left \(R\)-submodule, we have \(r(x)=c(x)-q(x)g(x)\in \mathcal{C}\). This is a contradiction to the assumption that \(g(x)\) is of the smallest degree in \(\mathcal{C}\) unless \(r(x)=0\). This implies \(c(x) = q(x)g(x)\) and hence \(\mathcal{C}\) is a \((\theta,\Im)\)-cyclic submodule generated by \(g(x)\). ◻
Theorem 4. Let \(\mathcal{C}=\langle g(x) \rangle\) be a left \(R\)-submodule of \(R/\langle x^n-1\rangle\), where \(g(x)\) is a monic polynomial of smallest degree in \(\mathcal{C}\). Then \(g(x)\) is a right divisor of \(x^n-1\).
Proof. Consider a monic smallest degree polynomial \(g(x)\) in \(\mathcal{C}\). From Theorem 1, there exist polynomials \(q(x)\) and \(r(x)\) in \(R\) such that \(x^n-1 = q(x)g(x) + r(x)\), where \(\deg r(x)<\deg g(x).\) Since \(g(x)\) and \(x^n-1=0\) are in \(\mathcal{C}\), this implies \(r(x)=(x^n-1)-q(x)g(x)\in \mathcal{C}\). But, \(g(x)\) is smallest in \(\mathcal{C}\). Therefore, \(r(x)=0\) and hence \(g(x)\) is a right divisor of \(x^n-1\). ◻
Let \(\mathcal{C}\) be a \((\theta,\Im)\)-cyclic code of length \(n\) over \(\mathbb{F}_q\) generated by the right divisor \(g(x)\) of \(x^n-1\), where \(g(x)=g_0+g_1x+\cdots+g_rx^r\in R\) and \(g_r=1\). Then from the above discussion, we can conclude that \(\mathcal{C}\) is a free left \(\mathbb{F}_q\)-module of dimension \(k=n-\deg g(x)\). Now, by using [29], the generator matrix of \(\mathcal{C}\) is given by \[\label{eq3} G= \begin{pmatrix} g\\ T_{\theta,\Im}(g)\\ \vdots\\ T_{\theta,\Im}^{k-1}(g) \end{pmatrix}\tag{1}\] where \(g=(g_0,g_1,g_2,\dots,g_{r})\) is the codeword corresponding to \(g(x)\). Moreover, it is well known that \(\dim(\mathcal{C})+\dim(\mathcal{C}^\perp)=n\). Therefore, \(\dim(\mathcal{C}^\perp)=n-k=r\). Further, for our convenience, we define a one-to-one correspondence between the algebraic structures and combinatorial structures of \((\theta,\Im)\)-cyclic codes as follows: \[\tau:~~~~\mathbb{F}_q^n~~~~~\longrightarrow~~~~ \mathbb{F}_q[x;\theta,\Im]/\langle x^n-1\rangle\] \[(c_0,c_1,c_2,\dots,c_{n-1})\longmapsto c_0+c_1x+c_2x^2+\dots+c_{n-1}x^{n-1}.\]
Theorem 5. Let \(\mathcal{C}=\langle g(x) \rangle\) be a \((\theta,\Im)\)-cyclic code of length \(n\) over \(\mathbb{F}_q\), for some right divisor \(g(x)\) of \(x^n-1\). Let \(x^n-1=h(x)g(x)=g(x)h'(x)\) for some monic skew polynomials \(g(x),h(x),h'(x)\in R\). Then \(c(x)\in \mathbb{F}_q[x;\theta,\Im]/\langle x^n-1\rangle\) is contained in \(\mathcal{C}\) if and only if \(c(x)h'(x)=0\) in \(\mathbb{F}_q[x;\theta,\Im]/\langle x^n-1\rangle\).
Proof. Let \(c(x)\in \mathbb{F}_q[x;\theta,\Im]/\langle x^n-1\rangle\) be contained in \(\mathcal{C}\). Then \(c(x)=a(x)g(x)\) for some \(a(x)\in R\). Now, \[\begin{align} c(x)&=a(x)g(x) ~\text{for some} ~a(x)\in R \\ c(x)h'(x)&=a(x)g(x)h'(x)=a(x)h(x)g(x)\\ &=a(x)(x^n-1)=0 ~\text{in}~ \mathbb{F}_q[x;\theta,\Im]/\langle x^n-1\rangle. \end{align}\] Conversely, let \(c(x)h'(x)=0\) for some \(c(x)\) in \(\mathbb{F}_q[x;\theta,\Im]/\langle x^n-1\rangle\). Then \(c(x)h'(x)=q(x)(x^n-1)\) for some \(q(x)\in \mathbb{F}_q[x;\theta,\Im]/\langle x^n-1\rangle\). Also, \[c(x)h'(x)=q(x)(x^n-1)=q(x)h(x)g(x)=q(x)g(x)h'(x).\] This implies that \(c(x)=q(x)g(x)\in \langle g(x) \rangle=\mathcal{C}\) as \(h'(x)\) is a nonzero polynomial. ◻
Now, with the help of the above-defined correspondence, the following theorem provides the generator matrix of the dual code \(\mathcal{C}^\perp\) of \((\theta,\Im)\)-cyclic code \(\mathcal{C}\) of length \(n\) over \(\mathbb{F}_q\).
Theorem 6. Let \(\mathcal{C}=\langle g(x) \rangle\) be a \((\theta,\Im)\)-cyclic code of length \(n\) over \(\mathbb{F}_q\) for some right divisor \(g(x)\) of \(x^n-1\) and \(x^n-1=h(x)g(x)=g(x)h'(x)\) for some monic skew polynomials \(g(x),h(x),h'(x)\in R\). Then \(\deg g(x)\) linearly independent columns of the matrix \[H= \begin{pmatrix} h'\\ T_{\theta,\Im}(h')\\ \vdots\\ T_{\theta,\Im}^{n-1}(h') \end{pmatrix}\] form a basis of \(\mathcal{C}^\perp\).
Proof. Consider a \((\theta,\Im)\)-cyclic code \(\mathcal{C}\) of length \(n\) over \(\mathbb{F}_q\). Let \(\mathcal{C}=\langle g(x) \rangle\) where \(g(x)\) is a right divisor of \(x^n-1\), and its leading coefficient is a unit. Then there exists \(h(x)=h_0+h_1x+\cdots+h_kx^k\in \mathbb{F}_q[x;\theta,\Im]/\langle x^n-1\rangle\) such that \(x^n-1=h(x)g(x)=g(x)h'(x)\). Now, for \(c(x)=c_0+c_1x+\cdots+c_{n-1}x^{n-1}\in \mathcal{C}\), we have \[\tau(c(T_{\theta,\Im})(h'))=c(x)h'(x)=a(x)g(x)h'(x)=a(x)h(x)g(x)=a(x)(x^n-1)=0.\] for some \(a(x)\) in \(\mathbb{F}_q[x;\theta,\Im]/\langle x^n-1\rangle\) and \(c(x)h'(x)\) is taken modulo \(x^n-1\). This implies \(c(T_{\theta,\Im})(h')=0\). Thus, \(0=c(T_{\theta,\Im})(h')=c_0+ c_1T_{\theta,\Im}(h')+c_2T_{\theta,\Im}^2(h')+\cdots+c_{n-1}T_{\theta,\Im}^{n-1}(h').\) This shows that \((c_0,c_1,c_2,\dots,c_{n-1}).H=0\) for any \(c=(c_0,c_1,c_2,\dots,c_{n-1})\in \mathcal{C}\). Also, \(\tau(T_{\theta,\Im}^k(h'))=x^kh'(x)\) for \(k=0,1,\dots, n-deg~ h'(x)-1=deg ~g(x)-1\) and hence \(\{h',T_{\theta,\Im}(h'),T_{\theta,\Im}^2(h'),\dots,T_{\theta,\Im}^{r-1}(h')\}\) are linearly independent. ◻
We now derive a necessary and sufficient condition for \((\theta,\Im)\)-cyclic codes to contain their duals codes over \(\mathbb{F}_q\).
Theorem 7. Let \(\mathcal{C}=\langle g(x) \rangle\) be a \((\theta,\Im)\)-cyclic code of length \(n\) over \(\mathbb{F}_q\), for some right divisor \(g(x)\) of \(x^n-1\) and \(x^n-1=h(x)g(x)=g(x)h'(x)\) for some monic skew polynomials \(g(x),h(x),h'(x)\in R\). Then \(\mathcal{C}^{\perp}\subseteq \mathcal{C}\) if and only if \(h'(x)h'(x)\) is divisible by \(x^n-1\) from the right.
Proof. Let \(\mathcal{C}=\langle g(x) \rangle\) be a \((\theta,\Im)\)-cyclic code over \(\mathbb{F}_q\) such that \(\mathcal{C}^{\perp}\subseteq \mathcal{C}\). Note that \(h'(x)\in \mathcal{C}^\perp\) and \(\mathcal{C}^\perp\subseteq \mathcal{C}=\langle g(x) \rangle\). Thus, \(h'(x)=p(x)g(x)\) for some \(p(x)\in R\). Now, multiplying both sides by \(h'(x)\) from right, we get \[h'(x)h'(x)=p(x)g(x)h'(x)=p(x)(x^n-1).\] Hence, \(h'(x)h'(x)\) is divisible by \(x^n-1\) from the right.
Conversely, let \(h'(x)h'(x)\) be divisible by \(x^n-1\) from the right. Then \(h'(x)h'(x)=b(x)(x^n-1)\) for some \(b(x)\in R\). Now, consider \(a(x)\in\mathcal{C}^\perp=\langle h'(x)\rangle\), then \(a(x)=c(x)h'(x)\) for some \(c(x)\in R\). Multiplying both sides by \(h'(x)\) from right and using \(h'(x)h'(x)=b(x)(x^n-1)\), we get \[\begin{align} a(x)h'(x)=c(x)h'(x)h'(x)&=c(x)b(x)(x^n-1)\\ &=c(x)b(x)h(x)g(x) =c(x)b(x)g(x)h'(x),\\ &\Big(a(x)-c(x)b(x)g(x)\Big)h'(x)=0. \end{align}\] As \(h'(x)\) is a nonzero polynomial, we have \(a(x)-c(x)b(x)g(x)=0\), which gives \(a(x)=c(x)b(x)g(x)\). Therefore, \(a(x)\in \mathcal{C}=\langle g(x) \rangle\). Thus, \(\mathcal{C}^\perp\subseteq \mathcal{C}\). ◻
Here, we present an example to show the construction of \((\theta,\Im)\)-cyclic codes over \(\mathbb{F}_q\) with the help of our derived results.
Example 1. Let \(q=49,n=14\). In \(\mathbb{F}_{49}\), the Frobenius automorphism \(\theta:\mathbb{F}_{49}\longrightarrow \mathbb{F}_{49}\) is defined by \(\theta(a)=a^7\) whereas the \(\theta\)-derivation \(\Im\) is defined by \(\Im(a)=w^2(\theta(a)-a)\) for all \(a\in \mathbb{F}_{49}\). Therefore, \(R=\mathbb{F}_{49}[x;\theta,\Im]\) is a skew polynomial ring. In \(\mathbb{F}_{49}[x;\theta,\Im],\) we have \[\begin{align} x^{14}-1=&(w^9x^{12} + 3x^{11} + w^{41}x^{10} + w^{13}x^9 + w^{37}x^8 + w^{47}x^7 + w^{18}x^5 +6x^4 + w^{38}x^3 \\&+ w^{18}x^2 + w^{28}x + w^{12})(w^{39}x^2 + w^3x + w^{17})=h(x)g(x)\\ =&(w^{39}x^2 + w^3x + w^{17})(w^9x^{12} + 3x^{11} + w^{41}x^{10} + w^{13}x^9 + w^{37}x^8 + w^{47}x^7 \\&+ w^{33}x^5 +4x^4 + w^{17}x^3 + w^{37}x^2 + w^{13}x + w^{23})=g(x)h'(x). \end{align}\] Consider \(g(x)=w^{39}x^2 + w^3x + w^{17}\), \(h(x)=w^9x^{12} + 3x^{11} + w^{41}x^{10} + w^{13}x^9 + w^{37}x^8 + w^{47}x^7 + w^{18}x^5 +6x^4 + w^{38}x^3 + w^{18}x^2 + w^{28}x+ w^{12}\) and \(h'(x)=w^9x^{12} + 3x^{11} + w^{41}x^{10} + w^{13}x^9 + w^{37}x^8 + w^{47}x^7 + w^{33}x^5+4x^4 + w^{17}x^3 + w^{37}x^2 + w^{13}x + w^{23}\). Then, by Theorem 6 and Equation 1 , \(\mathcal{C}\) is a \((\theta,\Im)\)-cyclic codes over \(\mathbb{F}_{49}\) of length \(14\) which is generated by \(g(x)\). The generator and parity check matrices of \(\mathcal{C}\) are given by Equation 1 and Theorem 6 respectively. Since, \(h'(x)h'(x)\) is divisible by \(x^{14}-1\) from the right and hence the code \(\mathcal{C}\) is also a dual-containing code, i.e., \(\mathcal{C}^{\perp}\subseteq \mathcal{C}\).
In this section, our main focus is on the algebraic properties of \((\gamma,\Delta)\)-cyclic codes over \(\mathscr{R}_{q,s}\) via decomposition over \(\mathbb{F}_q\). To do so, we consider a linear code \(\mathcal{C}\) of length \(n\) over \(\mathscr{R}_{q,s}\). Now, we define \[\mathcal{C}_i=\left\{t_i\in \mathbb{F}_q^n~|~ \sum_{i=0}^{s}\zeta_it_i\in \mathcal{C}, ~\text{for some}~ t_0,t_1,\dots,t_{i-1},t_{i+1},\dots,t_s\in \mathbb{F}_q^n \right\}\] for \(0\leq i\leq s\). Then \(\mathcal{C}_i\) is linear code of length \(n\) over \(\mathbb{F}_q\) and \(\mathcal{C}\) can be decomposed as follows \[\mathcal{C}=\displaystyle\bigoplus_{i=0}^{s}\zeta_i\mathcal{C}_i.\] Further, we consider a map \(\gamma:\mathscr{R}_{q,s}\rightarrow \mathscr{R}_{q,s}\) defined by \[\gamma(r)=\sum_{i=0}^{s}\zeta_i\theta(r_i)\] where \(r=\sum_{i=0}^{s}\zeta_ir_i\) and \(\theta\in \text{Aut}(\mathbb{F}_q)\) defined by \(\theta(r_i)=r_i^{p^t}\) for all \(r_i\in \mathbb{F}_q\). Then \(\gamma\) is an automorphism on \(\mathscr{R}_{q,s}\). Next, we define a map \(\Delta: \mathscr{R}_{q,s}\rightarrow \mathscr{R}_{q,s}\) such that \[\Delta(r)=(1+v_1+v_2+\cdots+v_{r})(\gamma(r)-r)\] where \(r=\sum_{i=0}^{r}r_iv_i\) and \(r_i\in \mathbb{F}_q\).
Theorem 8. The above defined map \(\Delta\) is a \(\gamma\)-derivation of \(\mathscr{R}_{q,s}\).
Proof. Let \(r,t\in\mathscr{R}_{q,s}\), we have \[\begin{align} \Delta(r+t)&=(1+v_1+v_2+\cdots+v_{s})(\gamma(r+t)-(r+t)) \\ &=(1+v_1+v_2+\cdots+v_{s})(\gamma(r)-r)+(1+v_1+v_2+\cdots+v_{s})(\gamma(t)-t)\\ &=\Delta(r)+\Delta(t) \end{align}\] and \[\begin{align} \Delta(rt)=&(1+v_1+v_2+\cdots+v_{s})(\gamma(rs)-rt) \\ =&(1+v_1+v_2+\cdots+v_{s})(\gamma(r)\gamma(t))-(1+v_1+v_2+\cdots+v_{s})rt\\ =&(1+v_1+v_2+\cdots+v_{s})(\gamma(r)\gamma(t))-(1+v_1+v_2+\cdots+v_{s})rt\\&+(1+v_1+v_2+\cdots+v_{s})\gamma(r)t-(1+v_1+v_2+\cdots+v_{s})\gamma(r)t\\ =&(1+v_1+v_2+\cdots+v_{s})\gamma(r)(\gamma(t)-t)-(1+v_1+v_2+\cdots+v_{s})(r-\gamma(r))t\\ =&(1+v_1+v_2+\cdots+v_{s})\gamma(r)(\gamma(t)-t)+(1+v_1+v_2+\cdots+v_{s})(\gamma(r)-r)t\\ =&\Delta(r)t+\gamma(r)\Delta(t). \end{align}\] Hence, \(\Delta\) is a \(\gamma\)-derivation of \(\mathscr{R}_{q,s}\). ◻
Further, with the help of the defined decomposition of \(\mathcal{C}\), we discuss the algebraic properties of \((\gamma,\Delta)\)-cyclic codes over \(\mathscr{R}_{q,s}\).
Theorem 9. Let \(\mathcal{C}=\displaystyle\bigoplus_{i=0}^{s}\zeta_i\mathcal{C}_i\) be a linear code of length \(n\) over \(\mathscr{R}_{q,s}\) where \(\mathcal{C}_i\) is a linear code of length \(n\) over \(\mathbb{F}_q\) for \(i=0,1,2,\dots,s\). Then \(\mathcal{C}\) is a \((\gamma,\Delta)\)-cyclic code of length \(n\) over \(\mathscr{R}_{q,s}\) if and only if \(\mathcal{C}_i\) is a \((\theta,\Im)\)-cyclic code of length \(n\) over \(\mathbb{F}_q\) for \(i=0,1,2,\dots,s\).
Proof. Let \(\mathcal{C}=\displaystyle\bigoplus_{i=0}^{s}\zeta_i\mathcal{C}_i\) be a \((\gamma,\Delta)\)-cyclic code of length \(n\) over \(\mathscr{R}_{q,s}\) and \(a^i=(a_0^i,a_1^i,\dots,a_{n-1}^i)\\\in \mathcal{C}_i\), for \(0\leq i \leq s\). Consider \(r_j=\sum_{i=0}^{s}\zeta_ia_j^i\) for \(0\leq j\leq n-1\). Then \(r=(r_0,r_1,\dots,r_{n-1})\in \mathcal{C}\) and \(T_{\gamma,\Delta}(r)\in \mathcal{C}\). Again, we have \(\gamma(r_j)=\sum_{i=0}^{s}\zeta_i\theta(a_j^i)\) and \(\Delta(r_j)=
\Delta(\sum_{i=0}^{s}\zeta_ia_j^i)=\Delta(\zeta_0a_j^0)+\Delta(\zeta_1a_j^1)+\cdots+\Delta(\zeta_0a_j^s)\) for \(0\leq j\leq n-1\). Also, \[\begin{align} \Delta(\zeta_0a_j^0)&=
\Delta(\zeta_0)a_j^0+\gamma(\zeta_0)\Im(a_j^0)\\ &=\Big((1+v_1+\cdots+v_s)(\gamma(\zeta_0)-\zeta_0)\Big)a_j^0+\zeta_0\Im(a_j^0)\\ &=\zeta_0\Im(a_j^0).
\end{align}\] Similarly, \(\Delta(\zeta_ia_j^i)=\zeta_i\Im(a_j^i)\) for \(i=1,2,\dots,s\) and \(0\leq j\leq n-1\). Hence, \(T_{\gamma,\Delta}(r)=\sum_{i=0}^{s}\zeta_iT_{\theta,\Im}(a^i)\). This implies that \(T_{\theta,\Im}(a^i)\in \mathcal{C}_i\) for \(i=0,1,2,\dots,s\). Thus, \(\mathcal{C}_i\) is a \((\theta,\Im)\)-cyclic code of length \(n\) over \(\mathbb{F}_q\) for \(i=0,1,2,\dots,s\).
Conversely, suppose \(\mathcal{C}_i\) is a \((\theta,\Im)\)-cyclic code of length \(n\) over \(\mathbb{F}_q\). Let \(r=(r_0,r_1,\dots,\\r_{n-1})\in \mathcal{C}\) where \(r_j=\sum_{i=0}^{s}\zeta_ia_j^i\) for \(0\leq j\leq n-1\). Consider, \(a^i=(a_0^i,a_1^i,\dots,a_{n-1}^i)\), for \(0\leq i \leq s\). Then \(a^i\in \mathcal{C}_i\) and also \(T_{\theta,\Im}(a^i)\in
\mathcal{C}_i\). Similar to the first part of the proof, we have \[\gamma(r_j)=\sum_{i=0}^{s}\zeta_i\theta(a_j^i)\] and \[\Delta(r_j)=
\Delta\Big(\sum_{i=0}^{s}\zeta_ia_j^i\Big)=\sum_{i=0}^{s}\zeta_i\Im(a_j^i)\] for \(i=0,1,2,\dots,s\) and \(0\leq j\leq n-1\). Then \[\begin{align}
T_{\gamma,\Delta}(r)=\gamma(r)M + \Delta(r)=&\Big(\gamma(r_{n-1})+ \Delta(r_0),\gamma(r_{o})+ \Delta(r_1),\gamma(r_{1})+ \Delta(r_2),\dots,\gamma(r_{n-2})+\\& \Delta(r_{n-1})\Big)\\ =&\sum_{i=0}^{s}\zeta_iT_{\theta,\Im}(a^i) \in
\displaystyle\bigoplus_{i=0}^{s}\zeta_i\mathcal{C}_i=\mathcal{C}.
\end{align}\] Therefore, \(\mathcal{C}\) is a \((\gamma,\Delta)\)-cyclic code of length \(n\) over \(\mathscr{R}_{q,s}\). ◻
Theorem 10. Let \(\mathcal{C}=\displaystyle\bigoplus_{i=0}^{s}\zeta_i\mathcal{C}_i\) be a \((\gamma,\Delta)\)-cyclic code of length \(n\) over \(\mathscr{R}_{q,s}\). Then \(\mathcal{C}=\langle \zeta_0g_0(x),\\\zeta_1g_1(x),\dots,\zeta_sg_s(x)\rangle\) and \(|\mathcal{C}|= q^{(s+1)n-\sum_{i=0}^{s} \deg( g_i(x))}\), where \(g_i(x)\) is a generator polynomial of \(\mathcal{C}_i\) for \(i=0,1,2,\dots,s\).
Proof. Let \(\mathcal{C}=\displaystyle\bigoplus_{i=0}^{s}\zeta_i\mathcal{C}_i\) be a \(({\gamma,\Delta})\)-cyclic code of length \(n\) over \(\mathscr{R}_{q,s}\). Then, by Theorem 9, \(\mathcal{C}_i\) is a \(({\theta,\Im})\)-cyclic
code over \(\mathbb{F}_q,\) for \(i=0,1,2,\dots,s\). This implies that \(\mathcal{C}_i=\langle g_i(x)\rangle \subseteq \mathbb{F}_q[x;\theta,\Im]/\langle
x^n-1\rangle\) for \(i=0,1,2,\dots,s\). Thus, \[\mathcal{C}=\left\{r(x)|r(x)=\sum_{i=0}^{s}\zeta_ig_i(x), g_i(x)\in \mathcal{C}_i\right\}.\] Hence, \(\mathcal{C}\subseteq \langle \zeta_0g_0(x),\zeta_1g_1(x),\dots,\zeta_sg_s(x)\rangle\).
On the other hand, we consider \(\zeta_0f_0(x)g_0(x)+\zeta_1f_1(x)g_1(x)+\cdots+\zeta_sf_s(x)g_s(x)\in \langle \zeta_0g_0(x),\\\zeta_1g_1(x),\dots,\zeta_sg_s(x)\rangle\subseteq \mathscr{R}_{q,s}[x;\gamma,\Delta]/\langle
x^n-1\rangle\) where \(f_i(x)\in \mathscr{R}_{q,s}[x;\gamma,\Delta]/\langle x^n-1\rangle\) for \(i=0,1,2,\dots,s\). Then there exists \(s_i(x)\in
\mathbb{F}_q[x;\theta,\Im]/\langle x^n-1\rangle\) such that \(\zeta_if_i(x)=\zeta_is_i(x)\) for \(i=0,1,2,\dots,s\). This implies that \(\langle
\zeta_0g_0(x),\zeta_1g_1(x),\dots,\zeta_sg_s(x)\rangle\subseteq \mathcal{C}\). Thus, \(\mathcal{C}=\langle \zeta_0g_0(x),\zeta_1g_1(x),\dots,\zeta_sg_s(x)\rangle\).
Moreover, \(|\mathcal{C}|=|\mathcal{C}_0||\mathcal{C}_1|\cdots|\mathcal{C}_s|=q^{n-\deg (g_0(x))}q^{n-\deg(g_1(x))}\cdots q^{n-\deg(g_s(x))}\\=q^{(s+1)n-\sum_{i=0}^{s} \deg( g_i(x))}\). ◻
Theorem 11. Let \(\mathcal{C}=\displaystyle\bigoplus_{i=0}^{s}\zeta_i\mathcal{C}_i\) be a \((\gamma,\Delta)\)-cyclic code of length \(n\) over \(\mathscr{R}_{q,s}\) and \(x^n-1=h_i(x)g_i(x)=g_i(x)h'_i(x)\) for some monic skew polynomials \(g_i(x),h_i(x),h'_i(x)\in\mathbb{F}_q[x;\theta,\Im]\) for \(i=0,1,2,\dots,s\). Then \(\mathcal{C}^\perp\subseteq \mathcal{C}\) if and only if \(h_i'(x)h_i'(x)\) is divisible by \(x^n-1\) from the right.
Proof. Let \(h_i'(x)h_i'(x)\) be divisible by \(x^n-1\) from the right for \(i=0,1,2,\dots,s\). Then, by Theorem 7, we have \(\mathcal{C}_i^\perp\subseteq \mathcal{C}_i\), \(i=0,1,2,\dots,s\). This implies that \(\displaystyle\bigoplus_{i=0}^{s}\zeta_i\mathcal{C}_i^\perp\subseteq \displaystyle\bigoplus_{i=0}^{s}\zeta_i\mathcal{C}_i\). Hence, \(\mathcal{C}^\perp\subseteq \mathcal{C}\).
Conversely, let \(\mathcal{C}^\perp\subseteq \mathcal{C}\) , then \(\displaystyle\bigoplus_{i=0}^{s}\zeta_i\mathcal{C}_i^\perp\subseteq
\displaystyle\bigoplus_{i=0}^{s}\zeta_i\mathcal{C}_i\). Now, considering modulo \(\zeta_i\), we get \(\mathcal{C}_i^\perp\subseteq \mathcal{C}_i\) for \(i=0,1,2,\dots,s\). Thus, \(h_i'(x)h_i'(x)\) is divisible by \(x^n-1\) on the right for \(i=0,1,2,\dots,s\). ◻
The next corollary is a direct consequence of the Theorem 11.
Corollary 1. Let \(\mathcal{C}=\langle g(x)\rangle\) be a \((\gamma,\Delta)\)-cyclic code of length \(n\) over \(\mathscr{R}_{q,s}\) and \(x^n-1=h_i(x)g_i(x)=g_i(x)h'_i(x)\) for some monic skew polynomials \(g_i(x),h_i(x),h'_i(x)\in\mathbb{F}_q[x;\theta,\Im]\). Then \(\mathcal{C}^\perp\subseteq \mathcal{C}\) if and only if \(\mathcal{C}_i^\perp\subseteq \mathcal{C}_i\) for \(i=0,1,2,\dots,s\).
The theory of quantum error-correcting codes plays a key role in quantum information theory. For a long time, it has been difficult to provide a satisfactory solution to the problem of protecting information from quantum noises. However, after the introduction of the first quantum error-correcting codes by Shor et al. [22], a stream of great developments has emerged in information theory. Let \(H_q(\mathbb{C})\) be a \(q\)-dimensional Hilbert vector space. Then the set of \(n\)-fold tensor product \(H_q^n(\mathbb{C})=\underbrace{H_q(\mathbb{C})\otimes H_q(\mathbb{C})\otimes\cdots \otimes H_q(\mathbb{C})}_{n\rm\;times}\) is a \(q^n\)-dimensional Hilbert space. Here, a \(q^k\) dimensional subspace of \(H_q^n(\mathbb{C})\) is called a quantum code with parameters \([[n,k,d]]_q\) where \(d\) is the minimum distance, and \(k\) is the dimension of the quantum code. Also, \(\mathcal{C}\) is dual-containing if \(\mathcal{C}^\perp\subseteq \mathcal{C}\). Moreover, in \(1997\), the quantum Singleton bound for binary codes was introduced by Knill and Laflamme [31]. In \(1998\), Calderbank et al. [23] provided the quantum Singleton bound for all codes over finite fields as \(k+2d\leq n+2.\) A quantum code is said to be a quantum MDS code if it attains the Singleton bound. In this section, we study quantum error correcting codes from \((\gamma,\Delta)\)-cyclic code of length \(n\) over \(\mathscr{R}_{q,s}\). In order to construct quantum error-correcting codes, we first derive a necessary and sufficient condition for \((\gamma,\Delta)\)-cyclic codes to be dual containing. Note that a quantum code \([[n,k,d]]_q\) is said to be better than \([[n',k',d']]_q\) if any one of the following or both hold:
\(d>d'\) when the code rate \(\frac{k}{n}=\frac{k'}{n'}\) (Larger distance with same code rate).
\(\frac{k}{n}>\frac{k'}{n'}\) when the distance \(d=d'\) (Larger code rate with same distance).
Now, we define a Gray map and study \(\mathbb{F}_q\)-images of \((\gamma,\Delta)\)-cyclic codes. Let \(GL_{s+1}(\mathbb{F}_q)\) be the set of all \((s+1)\times (s+1)\) invertible matrices over \(\mathbb{F}_q\). Now, \(\varphi:\mathscr{R}_{q,s}\longrightarrow \mathbb{F}_q^{s+1}\) define by \[\varphi(r)=(r_0,r_1,\dots,r_s)G,\] where \(r=\sum_{i=0}^{s}\zeta_ir_i\in \mathscr{R}_{q,s}\), \(G\in GL_{s+1}(\mathbb{F}_q)\) such that \(GG^T=kI_{s+1}\), \(G^T\) is the transpose matrix of \(G\), \(k\in \mathbb{F}_q^*\) and \(I_{s+1}\) is the identity matrix of order \(s+1\). It is easy to check that \(\varphi\) is a bijection and can be extended over \(\mathscr{R}_{q,s}^n\) componentwise. If we define Gray distance for a linear code \(\mathcal{C}\) by \(d_G(\mathcal{C})=d_H(\varphi(\mathcal{C}))\), then \(\varphi\) is a linear distance preserving map from \((\mathscr{R}_{q,s}^n,d_G)\) to \((\mathbb{F}_q^{n(s+1)},d_H)\), where \(d_H\) is the Hamming distance in \(\mathbb{F}_q\).
The Gray map \(\varphi\) is an \(\mathbb{F}_q\)-linear and distance preserving map from \(\mathscr{R}_{q,s}^n\) (Gray distance) to \(\mathbb{F}_q^{(s+1)n}\) (Hamming distance).
Proof. Let \(a=(a_0,a_1,\dots,a_{n-1}),~b=(b_0,b_1,\dots,b_{n-1})\in \mathscr{R}_{q,s}^n\), where \(a_j= \sum_{i=0}^{s} \zeta_ia_j^i\), \(b_j= \sum_{i=0}^{s} \zeta_ib_j^i\) for \(j=0,1,\dots,n-1\) and \(a_j^i,~b_j^i\in \mathbb{F}_q\). Then \[\begin{align} \varphi(a+b)=&\varphi(a_0+b_0,a_1+b_1,\dots,a_{n-1}+b_{n-1})\\ =&\varphi (\zeta_0(a_0^0+b_0^0)+\zeta_1(a_0^1+b_0^1)+\cdots+\zeta_{s}(a_0^{s}+b_0^{s}),\dots, \zeta_0(a_{n-1}^0+b_{n-1}^0)\\&+\zeta_1(a_{n-1}^1+b_{n-1}^1)+\cdots+\zeta_{s}(a_{n-1}^{s}+b_{n-1}^{s}))\\ =&[(a_0^0+b_0^0,a_0^1+b_0^1,\dots,a_{0}^{s}+b_0^{s})G,\dots, (a_{n-1}^0+b_{n-1}^0,a_{n-1}^1+b_{n-1}^1,\dots,\\&a_{n-1}^{s}+b_{n-1}^{s})G]\\ =&[(a_0^0,a_0^1,\dots,a_{0}^{s})G,\dots, (a_{n-1}^0,a_{n-1}^1,\dots,a_{n-1}^{s})G]+[(b_0^0,b_0^1,\dots,b_{0}^{s+1})G,\dots,\\& (b_{n-1}^0,b_{n-1}^1,\dots,b_{n-1}^{s})G]\\ =&\varphi(a)+\varphi(b). \end{align}\] Now, for any \(\lambda\in \mathbb{F}_q\), we have \[\begin{align} \varphi(\lambda a)&=\varphi(\lambda a_0,\lambda a_1,\dots,\lambda a_{n-1})\\ &=\varphi (\lambda \zeta_0a_0^0+\lambda \zeta_1a_0^1+\cdots+\lambda \zeta_sa_0^s,\dots, \lambda \zeta_0a_{n-1}^0+\lambda \zeta_1a_{n-1}^1+\cdots+\lambda \zeta_sa_{n-1}^s)\\ &=[(\lambda a_0^0,\lambda a_0^1,\dots,\lambda a_{0}^s)G,\dots, (\lambda a_{n-1}^0,\lambda a_{n-1}^1,\dots,\lambda a_{n-1}^s)G]\\ &=[\lambda(a_0^0,a_0^1,\dots,a_{0}^s)G,\dots, \lambda(a_{n-1}^0,a_{n-1}^1,\dots,a_{n-1}^s)G]\\ &=\lambda[(a_0^0,a_0^1,\dots,a_{0}^s)G,\dots, (a_{n-1}^0,a_{n-1}^1,\dots,a_{n-1}^s)G]\\ &=\lambda\varphi(a). \end{align}\] Moreover, \(d_G(a,b)=\omega_G(a-b)=\omega_H(\varphi(a-b))=\omega_H(\varphi(a)-\varphi(b))=d_H(\varphi(a),\varphi(b))\). Hence, \(\varphi\) is a distance preserving map. ◻
Theorem 12. If \(\mathcal{C}\) is an \([n,k,d_G]\) linear code over \(\mathscr{R}_{q,s}\), then \(\varphi(\mathcal{C})\) is a \([(s+1)n,k,d_H]\) linear code over \(\mathbb{F}_q\).
Proof. Follows directly from Proposition [prop1] and the definition of the Gray map. ◻
The Gray map \(\varphi\) preserves the orthogonality as shown in the next result.
Lemma 1. Let \(\mathcal{C}\) be a \((\gamma,\Delta)\)-cyclic code of length \(n\) over \(\mathscr{R}_{q,s}\). Then \(\varphi(\mathcal{C})^\perp=\varphi(\mathcal{C}^\perp)\). Further, \(\mathcal{C}\) is self-dual if and only if \(\varphi(\mathcal{C})\) is self-dual.
Proof. Let \(c=(c_0,c_1,\dots,c_{n-1})\in \mathcal{C}\) and \(d=(d_0,d_1,\dots,d_{n-1})\in \mathcal{C}^{\perp}\) where \(a_j= \sum_{i=0}^{s} \zeta_ic_j^i\), \(b_j= \sum_{i=0}^{s} \zeta_id_j^i\) for \(j=0,1,\dots,n-1\) and \(a_j^i,~b_j^i\in \mathbb{F}_q\). Now, \(c\cdot d=\sum_{j=0}^{n-1}c_jd_j=0\) gives \(\sum_{j=0}^{n-1}(c^0_jd^0_j+c^1_jd^1_j+\dots+c^s_jd^s_j)=0\). Again, \[\begin{align} &\varphi(c)=[(c_0^0,c_0^1,\dots,c_0^s)G,\dots, (c_{n-1}^0,c_{n-1}^1,\dots,c_{n-1}^s)G]=(\alpha_0G,\dots,\alpha_{n-1}G)\\ &\text{and}\\ &\varphi(d)=[(d_0^0,d_0^1,\dots,d_0^s)G,\dots, (d_{n-1}^0,d_{n-1}^1,\dots,d_{n-1}^s)G]=(\beta_0G,\dots,\beta_{n-1}G), \end{align}\] where \(\alpha_j=(c_j^0,c_j^1,\dots,c_j^s)\) and \(\beta_j=(d_j^0,d_j^1,\dots,d_j^s)\) for \(0\leq j\leq n-1\) and \(GG^T=k I_{s+1}\). Also, \[\begin{align} \varphi(c)\cdot \varphi(d)=\varphi(c) \varphi(d)^T&=\sum_{j=0}^{n-1}\alpha_jGG^T\beta_j^T\\ &=k\sum_{j=0}^{n-1}\alpha_j\beta_j^T\\ &=k\sum_{j=0}^{n-1}(c^0_jd^0_j+c^1_jd^1_j+\dots+c^s_jd^s_j)=0. \end{align}\] Since \(c\in \mathcal{C}\) and \(d\in \mathcal{C}^{\perp}\) are arbitrary, \(\varphi(\mathcal{C}^{\perp})\subseteq (\varphi(\mathcal{C}))^{\perp}\). On the other hand, as \(\varphi\) is a bijective linear map, \(\mid \varphi(\mathcal{C}^{\perp})\mid= \mid (\varphi(\mathcal{C}))^{\perp}\mid\). Therefore, \(\varphi(\mathcal{C}^{\perp})=(\varphi(\mathcal{C}))^{\perp}\). ◻
Lemma 2 ([32], Theorem 3). Let \(\mathcal{C}\) be an \([n,k,d]\) linear code over \(\mathbb{F}_q\) such that \(\mathcal{C}^{\perp}\subseteq \mathcal{C}\). Then there exists a quantum code with parameters \([[n,2k-n,d]]_q\).
Theorem 13. Let \(\mathcal{C}=\displaystyle\bigoplus_{i=0}^{s}\zeta_i\mathcal{C}_i\) be a \((\gamma,\Delta)\)-cyclic code of length \(n\) over \(\mathscr{R}_{q,s}\). Also, let \(\mathcal{C}_i=\langle g_i(x) \rangle\) be a \((\theta,\Im)\)-cyclic code over \(\mathbb{F}_q\) where \(x^n-1=h_i(x)g_i(x)=g_i(x)h'_i(x)\) for some monic skew polynomials \(g_i(x),h_i(x),h'_i(x)\in\mathbb{F}_q[x;\theta,\Im],\) for \(i=0,1,\dots,s\). Further, let \(h_i'(x)h_i'(x)\) be divisible by \(x^n-1\) from the right for \(i=0,1,\dots,s\). Then there exists a quantum code with parameters \([[(s+1)n,2k-(s+1)n,d_H]]_q\).
Proof. Let \(h_i'(x)h_i'(x)\) be divisible by \(x^n-1\) from right for \(i=0,1,\dots,s\). Then from Theorem 11, we have \(\mathcal{C}^{\perp}\subseteq \mathcal{C}\). Also, by Lemma 1, we have \(\varphi(\mathcal{C}^{\perp})=\varphi(\mathcal{C})^{\perp}\), and hence \(\varphi(\mathcal{C})^{\perp}\subseteq \varphi(\mathcal{C})\). Thus, \(\varphi(\mathcal{C})\) is a dual containing linear code with parameters \([(s+1)n,k,d_H]\) over \(\mathbb{F}_q\). Further, by Lemma 2, there exists a quantum code with parameters \([[(s+1)n,2k-(s+1)n,d_H]]_q\). ◻
Next, with the help of our established results, we construct many new quantum codes possessing better parameters than the existing codes, which are appeared in [33], [34]. In the following examples, \(\mathbb{F}_q^*=\langle w \rangle\) denotes the cyclic group of non-zero elements of \(\mathbb{F}_q\) generated by \(w\in \mathbb{F}_q\). All examples’ computations are carried out using the Magma computation system [35].
Example 2. Let \(q=8\), \(s=3\) and \(\mathscr{R}_{8,3}=\mathbb{F}_8[v_1,v_2,v_3]/\langle v_1^2-v_1, v_2^2-v_2,v_3^2-v_3,v_1v_2=v_2v_1 =v_2v_3 = v_3v_2 = v_3v_1 =v_1v_3 = 0\rangle\), where \(\mathbb{F}_8=\mathbb{F}_2(w)\) and \(w^3+w+1=0\). Let \(n=30\), \(\theta:\mathbb{F}_8\longrightarrow \mathbb{F}_8\) be the Frobenius automorphism defined by \(\theta(a)=a^2\), and the \(\theta\)-derivation \(\Im:\mathbb{F}_8\longrightarrow \mathbb{F}_8\) is defined by \(\Im(a)=w(\theta(a)-a)\) for all \(a\in \mathbb{F}_8\). Therefore, \(\mathbb{F}_8[x;\theta,\Im]\) is a skew polynomial ring. In \(\mathbb{F}_8[x;\theta,\Im],\) we have \[\begin{align} x^{30}-1=&(w^6x^{29} + w^4x^{28} + w^6x^{27} + w^4x^{26} + w^3x^{25} + x^{24} + w^6x^{23} + w^4x^{22} + w^6x^{21}\\& + w^4x^{20}+ w^6x^{19} + w^4x^{18} + w^6x^{17} + w^4x^{16} + w^6x^{15} + w^4x^{14} + w^3x^{13} \\&+ x^{12} + w^6x^{11} + w^4x^{10} + w^3x^9 + x^8 + w^6x^7 + w^4x^6 + w^3x^5 + x^4 + w^6x^3\\& + w^4x^2+ w^3x + 1)(w^2x + 1)=h_0(x)g_0(x)\\ =&(w^2x + 1)(w^6x^{29} + w^4x^{28} + w^6x^{27} + w^4x^{26} + w^6x^{25} + w^4x^{24} + w^6x^{23} \\&+ w^4x^{22} + w^6x^{21} + w^4x^{20} + w^6x^{19} + w^4x^{18} + w^6x^{17} + w^4x^{16} + w^6x^{15} \\&+ w^4x^{14} + w^6x^{13} + w^4x^{12} + w^6x^{11} + w^4x^{10} + w^6x^9 + w^4x^8 + w^6x^7 \\&+ w^4x^6 + w^6x^5 + w^4x^4 + w^6x^3 + w^4x^2 + w^6x + w^4)=g_0(x)h_0'(x)\\ x^{30}-1=&(w^5x^{28} + w^3x^{27} + w^2x^{26} + w^3x^{25} + w^3x^{24} + w^4x^{23} + w^6x^{22} + w^5x^{21} \\&+ w^6x^{20} + wx^{19} + w^3x^{18} + x^{17} + w^6x^{16} + w^3x^{15} + w^6x^{14} + w^6x^{13} + w^4x^{12} \\&+ w^2x^{11} + w^6x^9 + w^5x^8 + w^6x^6 + w^4x^5 + w^5x^4 + w^4x^2 + w^2x + 1)(wx^2 \\&+ w^4x + w^6)=h_1(x)g_1(x)\\ =&(wx^2 + w^4x + w^6)(w^5x^{28} + w^3x^{27} + w^2x^{26} + w^3x^{25} + x^{24} + w^3x^{22} + w^6x^{21} \\&+ w^4x^{20} + x^{19} + w^5x^{18} + x^{16} + w^6x^{15} + w^5x^{13} + w^3x^{12} + w^2x^{11} + w^3x^{10} \\&+ x^9 + w^3x^7 + w^6x^6 + w^4x^5 + x^4 + w^5x^3 + x + w^6)=g_1(x)h_1'(x)\\ x^{30}-1=&(w^6x^{28} + w^6x^{27} + wx^{26} + wx^{24} + x^{23} + w^5x^{22} + x^{20} + w^6x^{19} + w^5x^{17} \\&+ w^3x^{16} + w^2x^{15} + w^3x^{14} + x^{13} + w^4x^{12} + w^6x^9 + wx^8 + w^6x^7 + wx^6\\& + wx^5 + w^2x^4 + wx^3 + w^4x^2 + w^2x)(w^4x^2 + w^3x + w)=h_2(x)g_2(x)\\ =&(w^4x^2 + w^3x + w)(w^6x^{28} + w^6x^{27} + wx^{26} + w^2x^{24} + wx^{23} + wx^{22} + w^3x^{21}\\& + w^2x^{20} + w^2x^{19} + w^5x^{18} + w^5x^{17} + x^{16} + w^2x^{15} + w^6x^{13} + w^6x^{12} + wx^{11} \\&+ w^2x^9 + wx^8 + wx^7 + w^3x^6 + w^2x^5 + w^2x^4 + w^5x^3 +w^5x^2 + x + w^2)\\&=g_2(x)h_2'(x)\\ x^{30}-1=&(x^{28} + w^4x^{27} + w^3x^{26} + w^3x^{25} + wx^{24} + w^4x^{23} + x^{22} + w^4x^{20} + x^{18} + x^{17} \\&+ w^5x^{16} + w^5x^{15} + w^4x^{14} + w^5x^{13} + w^6x^{11} + w^5x^{10} + wx^9 + w^4x^8 + x^7\\& + wx^6 + wx^5 + w^3x^3 + w^6x^2 + w^6x + 1)(x^2 + w^2x + w^4)=h_3(x)g_3(x)\\ =&(x^2 + w^2x + w^4)(x^{28} + w^4x^{27} + w^3x^{26} + w^3x^{25} + wx^{24} + w^2x^{23} + w^5x^{22}\\& + w^2x^{21} + w^4x^{20} + wx^{19} + w^6x^{18} + x^{17} + w^5x^{16} + w^6x^{15} + x^{13} + w^4x^{12} \\&+ w^3x^{11} + w^3x^{10} + wx^9 + w^2x^8 + w^5x^7 + w^2x^6 + w^4x^5 + wx^4 + w^6x^3\\& + x^2 + w^5x + w^6)=g_3(x)h_3'(x) \end{align}\] Now, let \(g_0=w^2x + 1, g_1=wx^2 + w^4x + w^6\), \(g_2=w^4x^2 + w^3x + w\) and \(g_3=x^2 + w^2x + w^4\). Then \(\mathcal{C}_i=\langle g_i(x)\rangle\) is a \((\theta,\Im)\)-cyclic code of length \(30\) over \(\mathbb{F}_8\) for \(i=0,1,2,3\). Then by Theorem 9, \(\mathcal{C}=\displaystyle\bigoplus_{i=0}^{s}\zeta_i\mathcal{C}_i\) is a \((\gamma,\Delta)\)-cyclic code of length \(30\) over \(\mathscr{R}_{8,3}\). Let \[\label{eq31} G= \begin{pmatrix} 1&w&w^3&1\\ w&1&1&w^3\\ w^3&1&1&w\\ 1&w^3&w&1\\ \end{pmatrix}\in GL_4(\mathbb{F}_8)\qquad{(2)}\] such that \(GG^T=I_4\). Then \(\varphi(\mathcal{C})\) is a \([120,114,4]\) linear code over \(\mathbb{F}_8\). Again, \[\begin{align} h_0'(x)h_0'(x)=&(w^2x^{28} + x^{27} + x^{26} + w^5x^{25} + w^4x^{24} + w^2x^{22} + x^{21} + x^{20} + w^5x^{19}\\& + w^2x^{18} + x^{17}+ x^{16} + w^5x^{15} + w^2x^{14} + x^{13} + w^3x^{12} + w^4x^{11} + x^{10 }\\&+ w^5x^9 + w^4x^8 + w^2x^6 + x^5 + w^3x^4 + w^4x^3 + x^2 + w^5x + w^4)(x^{30}-1)\\ h_1'(x)h_1'(x)=&(wx^{26} + w^4x^{25} + w^2x^{24} + wx^{23} + wx^{22} + w^3x^{21} + w^5x^{19} + w^4x^{16}\\& + x^{15}+ w^2x^{14} + w^3x^{13} + wx^{12} + w^4x^{11} + w^6x^{10} + w^6x^9 + w^2x^8 + wx^7 \\&+ w^4x^6 + x^4 + x^2 + w^6x + w^6)(x^{30}-1)\\ h_2'(x)h_2'(x)=&(w^4x^{26} + w^3x^{25 }+ w^3x^{24} + w^4x^{23} + x^{22} + w^3x^{21} + w^4x^{20} + w^3x^{19} \\&+ w^2x^{18}+ wx^{17} + wx^{16} + w^5x^{15} + x^{14} + w^5x^{13} + w^5x^{12} + w^2x^{11} \\&+ w^6x^9 + w^6x^8 + w^2x^7 + w^5x^6 + x^5 + w^4x^4 + w^6x^3 + w^5x^2 + w^5x)\\&(x^{30}-1)\\ h_3'(x)h_3'(x)=&(x^{26} + w^2x^{25} + w^5x^{24} + x^{23} + w^2x^{22} + wx^{20} + x^{19} + w^3x^{18} + w^6x^{17} \\&+ wx^{16} + x^{15} + wx^{14} + w^6x^{12} + w^6x^{11} + w^5x^{10} + x^9 + w^3x^8 + x^7 \\&+ x^6 + w^6x^5 + w^6x^3 + w^4x^2 + w^6x + w^6)(x^{30}-1). \end{align}\] From above, we see that \(h_i'(x)h_i'(x)\) is divisible by \((x^{30}-1)\) on the right for \(i=0,1,2,3\). Hence, by Theorem 13, there exists a quantum code with parameters \([[120,108,4]]_8\) which has the same length and distance but better code rate than the best-known code \([[120,104,4]]_8\) given by [33].
Example 3. Let \(q=25\), \(s=3\) and \(\mathscr{R}_{25,3}=\mathbb{F}_{25}[v_1,v_2,v_3]/\langle v_1^2-v_1, v_2^2-v_2,v_3^2-v_3,v_1v_2=v_2v_1 =v_2v_3 = v_3v_2 = v_3v_1 =v_1v_3 = 0\rangle\). Let \(n=30\), \(\theta:\mathbb{F}_{25}\longrightarrow \mathbb{F}_{25}\) be the Frobenius automorphism defined by \(\theta(a)=a^5\), and the \(\theta\)-derivation \(\Im:\mathbb{F}_{25}\longrightarrow \mathbb{F}_{25}\) is defined by \(\Im(a)=w(\theta(a)-a)\) for all \(a\in \mathbb{F}_{25}\). Therefore, \(\mathbb{F}_{25}[x;\theta,\Im]\) is a skew polynomial ring. In \(\mathbb{F}_{25}[x;\theta,\Im],\) we have \[\begin{align} x^{20}-1=&(w^{19}x^{19} + x^18 + w^{20}x^{17} + w^4x^{16} + w^{15}x^{15} + w^{20}x^{14} + x^{13} + w^{19}x^{12} + w^7x^{11} \\&+ w^2x^{10} + w^{10}x^9 + 3x^8 + w^3x^7 + w^{17}x^6 +w^{11}x^5 + 2x^4 + 3x^2 + 4x + 3)(wx \\&+ w^{17})=h_0(x)g_0(x)\\ =&(wx + w^{17})(w^{19}x^{19} + x^{18} + w^{20}x^{17} + w^4x^{16} + w^{15}x^{15} + wx^{14} + 2x^{13} + w^2x^{12}\\& + w^{10}x^{11} + w^{21}x^{10} + w^7x^9 + 4x^8 + w^8x^7 + w^{16}x^6 + w^3x^5 + w^{13}x^4 + 3x^3\\& + w^{14}x^2+ w^{22}x + w^9)=g_0(x)h_0'(x)\\ x^{20}-1=&(w^{14}x^{18} + w^8x^{17} + w^{17}x^{16} + 3x^{15} + 2x^{14} + w^{21}x^{13} + w^8x^{12} + w^{10}x^{11} + wx^{10}\\& + 4x^9 + w^{19}x^7 + w^{19}x^6 + w^9x^5 + 2x^4 + 4x^3 + x +2)(w^{10}x^2+ 2x+w^{11})\\& =h_1(x)g_1(x)\\ =&(w^{10}x^2+ 2x+w^{11})(w^{14}x^{18 }+ w^8x^{17} + w^{17}x^{16} + 3x^{15} + 2x^{14} + w^{15}x^{13} \\&+ w^3x^{12} + x^{11} + w^{15}x^{10} + 3x^9 + w^4x^8 + 4x^7 + w^4x^6 + w^{10}x^5 + x^4 + 2x^3 \\&+w^{11}x^2 + w^{19}x+ w^{19}) =g_1(x)h_1'(x)\\ x^{20}-1=&(w^{23}x^{19} + w^{19}x^{18} + w^3x^{17} + w^{14}x^{16} + x^{15} + w^4x^{14} + w^{19}x^{13} + w^{10}x^{12} \\&+ w^{13}x^{11} + 2x^{10} + w^2x^9 + w^{13}x^8 + w^7x^7 + w^{21}x^6 + 4x^5 + 2x^4 + 3x^2 + 4x \\&+ 3)(w^5x + 3)=h_2(x)g_2(x)\\ =&(w^5x + 3)(w^{23}x^{19 }+ w^{19}x^{18} + w^3x^{17} + w^{14}x^{16} + x^{15} + w^5x^{14} + wx^{13} +\\& w^9x^{12} + w^{20}x^{11} + 2x^{10} + w^{11}x^9 + w^7x^8 + w^{15}x^7 + w^2x^6 + 4x^5 + w^{17}x^4 \\&+ w^{13}x^3 + w^{21}x^2 + w^8x + 3)=g_2(x)h_2'(x)\\ x^{20}-1=&(w^{10}x^{18} + 4x^{17} + w^{11}x^{16} + 4x^{15} + w^{16}x^{14} + w^7x^{13} + 3x^{12} + w^8x^{11} \\&+ w^{21}x^{10} + w^{13}x^9 + 3x^8 + 2x^7 + w^{14}x^6 + w^2x^5 + 4x^4 +4x^3 + x^2 + 4x\\& + 2)(w^{14}x^2 + w^{19}x + w^{15})=h_3(x)g_3(x)\\ =&(w^{14}x^2 + w^{19}x + w^{15})(w^{10}x^{18} + 4x^{17} + w^{11}x^{16} + 4x^{15} + w^{16}x^{14 }+ w^2x^{13}\\& + x^{12} + 3x^{11} + w^4x^{10} + w^4x^9 + w^7x^8 + 2x^7 + w^{22}x^6 + w^9x^5 + w^{10}x^4 \\&+ x^3 +w^{13}x + w^2 )=g_3(x)h_3'(x) \end{align}\] Now, let \(g_0(x)=wx + w^{17}, g_1(x)=w^{10}x^2 + 2x + w^{11}\), \(g_2(x)= w^5x + 3\) and \(g_3(x)=w^{14}x^2 + w^{19}x + w^{15}\). Then \(\mathcal{C}_i=\langle g_i(x)\rangle\) is a \((\theta,\Im)\)-cyclic code of length \(20\) over \(\mathbb{F}_{25}\) for \(i=0,1,2,3\). Then by Theorem 9, \(\mathcal{C}=\displaystyle\bigoplus_{i=0}^{s}\zeta_i\mathcal{C}_i\) is a \((\gamma,\Delta)\)-cyclic code of length \(20\) over \(\mathscr{R}_{{25},3}\). Let \[\label{mowezqfl} G= \begin{pmatrix} -1&1&1&1\\ 1&1&1&-1\\ 1&-1&1&1\\ 1&1&-1&1\\ \end{pmatrix}\in GL_4(\mathbb{F}_{25})\qquad{(3)}\] such that \(GG^T=4I_2\). Then \(\varphi(\mathcal{C})\) is a \([80,74,4]\) linear code over \(\mathbb{F}_{25}\). Again, \[\begin{align} h_0'(x)h_0'(x)=&(3x^{18} + w^4x^{17} + w^{19}x^{16} + w^2x^{15} + w^{23}x^{13} + w^8x^{12} + w^9x^{11} + w^{11}x^{10}\\& + w^{21}x^8 + wx^7 + w^5x^6 + 3x^5 + wx^3 + 2x^2 + 3x + w^{15})(x^{20}-1)\\ h_1'(x)h_1'(x)=&(w^4x^{16} + w^{13}x^{15} + w^{14}x^{14} + w^2x^{13} + w^{17}x^{12} + w^9x^{11} + w^{20}x^{10} + w^7x^9 \\&+ x^8 + w^{14}x^7 + w^{22}x^6 + w^{21}x^5 + w^8x^4 + 4x^3 + w^{22}x^2 +wx + w^{13})\\&(x^{20}-1)\\ h_2'(x)h_2'(x)=&(3x^{18} + w^{20}x^{17} + w^{10}x^{16} + 2x^{15} + w^{19}x^{13} + 4x^{12} + w^{21}x^{11} + 4x^{10}\\& + w^9x^8 + w^{15}x^7 + 2x^6 + 3x^5 + w^5x^3 + wx^2 + w^{13}x + 1)(x^{20}-1)\\ h_3'(x)h_3'(x)=&(w^{20}x^{16} + w^{23}x^{15} + 2x^{14} + w^{14}x^{13} + w^{21}x^{12} + w^{13}x^{11} + w^{16}x^{10} \\&+w^{11}x^9 + w^5x^8 + w^{11}x^7 + w^{11}x^6 + w^{17}x^5 + w^8x^4 + w^8x^3 + w^{11}x^2 \\&+ w^{16}x + w^{20})(x^{20}-1). \end{align}\] From above we see that \(h_i'(x)h_i'(x)\) is divisible by \(x^{20}-1\) on the right for \(i=0,1,2,3\). Hence, by Theorem 13, there exists a quantum code with parameters \([[80,68,4]]_{25}\) which has the same length and distance, but better code rate than the best-known code \([[80,64,4]]_{25}\) given by [34].
Let \(\mathcal{C}\) be a \((\theta,\Im)\)-cyclic code of length \(n\) over \(\mathbb{F}_q\) where \(\mathcal{C}=\langle g(x)\rangle\) and \(x^n-1=h(x)g(x)=g(x)h'(x)\) for some monic skew polynomials \(g(x),h(x),h'(x)\in\mathbb{F}_q[x;\theta,\Im]\). Further, let \(h'(x)h'(x)\) be divisible by \(x^n-1\) from the right. Therefore, by Theorem 11, we get the dual containing codes with \(\mathbb{F}_q\)-parameters \([n,k,d]_q\) (enlisted in the fourth column of Table 1). Also, by Lemma 2, we construct quantum codes \([[n,k,d]]_q\) (in the fifth column), in which some codes satisfy the equality \(n-k+2-2d=2\) (Near to MDS), and some are MDS (maximum-distance-separable). Let \(\mathcal{C}=\displaystyle\bigoplus_{i=0}^{s}\zeta_i\mathcal{C}_i\) be a \((\gamma,\Delta)\)-cyclic code of length \(n\) over \(\mathscr{R}_{q,s}\) where \(\mathcal{C}_i=\langle g_i(x)\rangle\) is a \((\theta,\Im)\)-cyclic code of length \(n\) over \(\mathbb{F}_q\) and \(x^n-1=h_i(x)g_i(x)=g_i(x)h'_i(x)\) for some monic skew polynomials \(g_i(x),h_i(x),h'_i(x)\in\mathbb{F}_q[x;\theta,\Im]\) for \(i=0,1,2,\dots,s\). Further, let \(h_i'(x)h_i'(x)\) is divisible by \(x^n-1\) from the right for \(i=0,1,2,\dots,s\). Therefore, by Theorem 11, we get the dual containing codes with \(\mathbb{F}_q\)-parameters \([n,k,d]_q\) (enlisted in the fifth column of Table 2). Also, by Theorem 13, we construct quantum codes \([[n,k,d]]_q\) (in the sixth column), which beat the parameters of best-known codes (in the seventh column) given by the online database [33], [34]. Also, the first and second columns represent \(s\) and \((n,q)\), respectively. Moreover, in third column we present \(\theta\)-derivations \(\Im(a)\) for \(a\in\mathbb{F}_q\). Note that in fourth column we give generator polynomials \(g_i\) for \(\mathcal{C}_i\) (\(i=0,1,2,\dots,s\)) which is a right factor of \(x^n-1\) in \(\mathbb{F}_q[x;\theta,\Im]\). In order to make Table 2 precise, we enlist the coefficients of polynomials in decreasing powers of \(x\). For example, we write \(w^702w\) to represent the polynomial \(w^7x^3+2x+w\).
| \((n,q)\) | \(\Im(a)\), \(a\in \mathbb{F}_q\) | \(g(x)\) | \(\mathcal{C}\) | Obtained | Remark | ||||||||||
| Codes | |||||||||||||||
| \((30,8)\) | \(w(\theta(a)-a)\) | \((w^3w^5)\) | \([30,29,2]_8\) | \([[30,28,2]]_8\) | MDS | ||||||||||
| \((30,8)\) | \(w(\theta(a)-a)\) | \((ww^4w^6)\) | \([30,28,2]_8\) | \([[30,26,2]]_8\) | Near to MDS | ||||||||||
| \((14,49)\) | \(w^2(\theta(a)-a)\) | \((w^{39}w^3w^{17})\) | \([14,12,2]_{49}\) | \([[14,10,2]]_{49}\) | Near to MDS | ||||||||||
| \((24,49)\) | \(w^2(\theta(a)-a)\) | \((w^41)\) | \([24,23,2]_{49}\) | \([[24,22,2]]_{49}\) | MDS | ||||||||||
| \((24,49)\) | \(\theta(a)-a\) | \((w^2w^{17})\) | \([24,23,2]_{49}\) | \([[24,22,2]]_{49}\) | MDS | ||||||||||
| \((20,25)\) | \(w(\theta(a)-a)\) | \((w^{14}w^{19}w^{15})\) | \([20,18,2]_{25}\) | \([[20,16,2]]_{25}\) | Near to MDS | ||||||||||
| \((24,25)\) | \(\theta(a)-a\) | \((w^{3}w^4w^{20})\) | \([24,22,2]_{25}\) | \([[24,20,2]]_{25}\) | Near to MDS | ||||||||||
| \((20,25)\) | \(w(\theta(a)-a)\) | \((ww^{17})\) | \([20,19,2]_{25}\) | \([[20,18,2]]_{25}\) | MDS |
| \(s\) | \((n,q)\) | \(\Im(a)\), \(a\in \mathbb{F}_q\) | \([g_0(x),g_1(x),\dots,g_s(x)]\) | \(\varphi(\mathcal{C})\) | Obtained | Existing | |||||||||
| Codes | Codes | ||||||||||||||
| \(2\) | \((48,9)\) | \(w^2(\theta(a)-a)\) | \((w^71w^3,w^5w^2,w^512)\) | \([144,139,3]_9\) | \([[144,134,3]]_9\) | \([[146,134,3]]_9\) [33] | |||||||||
| \(3\) | \((36,9)\) | \(w^2(\theta(a)-a)\) | \((w^21w^5,1w^3, w^7w^2,2ww^3)\) | \([144,138,4]_9\) | \([[144,132,4]]_9\) | \([[146,128,4]]_9\) [33] | |||||||||
| \(3\) | \((32,9)\) | \(w^2(\theta(a)-a)\) | \((w^2ww, w^6w^22, 2w^2ww^3,w^31)\) | \([128,120,4]_9\) | \([[128,112,4]]_9\) | \([[129,103,4]]_9\) [33] | |||||||||
| \(2\) | \((42,9)\) | \(w^2(\theta(a)-a)\) | \((ww^6, w^6ww^3,w^7w^2)\) | \([126,122,3]_{9}\) | \([[126,118,3]]_{9}\) | \([[130,118,3]]_{9}\) [34] | |||||||||
| \(2\) | \((60,4)\) | \(w(\theta(a)-a)\) | \((www^2,1w^2w^2, 11w^2)\) | \([180,174,3]_{4}\) | \([[180,168,3]]_{4}\) | \([[185,167,3]]_{4}\) [34] | |||||||||
| \(3\) | \((20,25)\) | \(w(\theta(a)-a)\) | \((ww^{17}, w^{10}2w^{11}, w^53,w^{14} w^{19}w^{15})\) | \([80,74,4]_{25}\) | \([[80,68,4]]_{25}\) | \([[80,64,4]]_{25}\) [34] | |||||||||
| \(3\) | \((40,25)\) | \(w(\theta(a)-a)\) | \((w^{19}w^{10},w^{10}w^{14},w^{11} w^{17},w^{14}w^4)\) | \([120,116,3]_{25}\) | \([[120,112,3]]_{25}\) | \([[120,106,3]]_{25}\) [34] | |||||||||
| \(3\) | \((30,8)\) | \(w(\theta(a)-a)\) | \((w^21, ww^4w^6, w^4w^3w,1w^2w^4)\) | \([120,114,4]_8\) | \([[120,108,4]]_8\) | \([[120,104,4]]_8\) [33] | |||||||||
| \(3\) | \((32,8)\) | \(w(\theta(a)-a)\) | \((w^6w^3w^4, w^2w^5w^5, ww^6,w^6w^5w^2)\) | \([128,121,4]_8\) | \([[128,114,4]]_8\) | \([[128,112,4]]_8\) [33] |
In this paper, we have constructed many quantum codes over a class of finite commutative non-chain rings \(\mathscr{R}_{q,s}\), with better parameters than the codes available in recent literature. Particularly, we have obtained \((\gamma,\Delta)\)-cyclic codes using a set of idempotents over \(\mathscr{R}_{q,s}\) and established results on their algebraic structure. Towards construction of quantum codes, a necessary and sufficient condition to contain their dual codes has established. Finally, we have obtained many better quantum codes. On the other hand, exploring applications in the quantum computation of these codes is still open as future research work.
The first and second authors are thankful to the Department of Science and Technology (DST), Govt. of India, for financial support under CRG/2020/005927, vide Diary No. SERB/F/6780/ 2020-2021 dated 31 December 2020 and Ref No. DST/INSPIRE/03/2016/
001445, respectively.
Data Availability Statement: The authors declare that [the/all other] data supporting the findings of this study are available within the article. Any clarification may be requested from corresponding author provided it is
essential.
Competing interests: The authors declare that there is no conflict of interest regarding the publication of this manuscript.
Use of AI tools declaration The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this manuscript.