Quantum error-correcting codes play an important role in quantum communications and quantum computations. After the pioneering work in [1-4], the theory of quantum codes has developed rapidly in recent years. As we know, the approach of constructing new quantum codes which have good parameters is an interesting research field. However, obtaining the parameters of the new quantum codes, especially the new good quantum codes, is a difficult problem. Recently, a lot of new quantum codes have been constructed by classical linear codes with Hermitian dual containing, which can be found in [5-10].

Cyclic codes over finite rings are an important class of codes from both a theoretical and a practical viewpoint. It has been shown that certain good quantum codes could be found as images of linear codes over some special rings under the Gray map (see[11]). In [12], Kai and Zhu established a construction for quantum codes from cyclic codes of odd length over finite chain ring $ \mathbb{F}_4+u\mathbb{F}_4 $, where $ u^2 = 0 $. Qian et al. in [13] gave a new method of constructing quantum codes from cyclic codes of odd length over finite ring $ \mathbb{F}_2+v\mathbb{F}_2 $, where $ v^2 = v $. Motivated by two papers above, we study quantum codes from cyclic codes over $ \mathbb{F}_q+u\mathbb{F}_q+v\mathbb{F}_q $ where $ u^2 = u, v^2 = v $, $ uv = vu = 0 $, and $ q = p^t $ for some prime $ p $ and positive integer $ t $.

In this paper, let $ R $ denote the finite ring $ \mathbb{F}_q+u\mathbb{F}_q+v\mathbb{F}_q $ with $ u^2 = u, v^2 = v $, and $ uv = vu = 0 $. In Section 2, we define the Gray map $ \varphi $ from $ R $ to $ \mathbb{F}_q^{3} $. Moreover, we investigate some results about linear codes over $ R $. In Section 3, we address the relation of Hermitian dual-containing codes between $ R $ and $ \mathbb{F}_q $. In light of the relation, we get quantum codes with new parameters over $ \mathbb{F}_q $.

The ring $ R $ is a finite commutative ring with characteristic $ p $ and it contains three maximal ideals which are

$ I_1 = \langle u, v\rangle, \; I_2 = \langle u-1, v\rangle, \; I_3 = \langle u, v-1\rangle. $

Obviously $ \frac{R}{I_1}, \frac{R}{I_2} $, and $ \frac{R}{I_3} $ are isomorphic to $ \mathbb{F}_q $. i.e., $ R\cong \mathbb{F}_q^{3} $. Therefore $ R $ is a principal ideal ring, i.e., $ R $ is a Frobenius ring.

Let $ R^n = \{{\bf x} = (x_1, \cdots, x_n)\, |\, x_j\in R\} $ be $ R $-module. A $ R $-submodule $ C $ of $ R^n $ is called a linear code of length $ n $ over $ R $. We assume throughout paper that all codes are linear.

Let $ {\bf x}, {\bf y}\in R^n $, the Euclidean inner product of $ {\bf x}, {\bf y} $ is defined as the following

$ {\bf x}\cdot{\bf y} = x_1y_1+\cdots+x_ny_n. $

We call

$ C^{\bot} = \{{\bf x}\in R^n\, |\, {\bf x}\cdot{\bf c} = 0, \, \forall \, {\bf c}\in C\} $

as the dual code of $ C $. Notice that $ C^{\bot} $ is linear if $ C $ is linear or not.

In [14], it is proved that for any linear code $ C $ over a finite Frobenius ring, $ |C|\cdot|C^{\bot}| = R^n. $ The following concepts and results can be found in [1].

The Gray map $ \varphi:\; R^{n}\rightarrow\mathbb{F}_{q}^{3n} $ is defined by $ \varphi({\bf{x}}) = (\beta(x_1), \cdots, \beta(x_n)) $ for $ {\bf{x}} = (x_1, \cdots, x_n) $, where $ \beta(a+ub+vc) = (a, a+b, a+c) $ for $ a+ub+vc\in R $ with $ a, b, c\in\mathbb{F} _{q} $. Using this map, we can define the Lee weight $ W_L $ and Lee distance $ d_L $ as follows.

For any element $ {{\bf{x}}} = (x_1, \cdots, x_n)\in R^n $, we define $ W_L({{\bf{x}}}) = W_H(\varphi({\bf{x}})) $, where $ W_H $ denotes the ordinary Hamming weight for codes over $ \mathbb{F} _{q} $. The Lee distance $ d_L({\bf{x}}, {\bf{y}}) $ between two codewords $ {\bf{x}} $ and $ {\bf{y}} $ is the Lee weight of $ {\bf{x}}-{\bf{y}} $.

Lemma 2.1 [1] The Gray map $ \varphi $ is a distance-preserving map from $ (R^n, \; \mathrm{Lee\; distance}) $ to $ (\mathbb{F}^{3n}, \; \mathrm{Hamming\; distance}) $ and also $ \mathbb{F}_{q} $-linear.

The following theorem is obvious.

Theorem 2.2 [1] If $ C $ is a linear code of length $ n $ over $ R $, size $ q^k $ and Lee distance $ d_L $, then $ \varphi(C) $ is a linear code over $ \mathbb{F}_{q} $ with parameters $ [3n, k, d_L] $.

Theorem 2.3 [1] If $ C $ is a linear code of length $ n $ over $ R $, then $ \varphi(C^{\perp}) = \varphi(C)^{\perp} $. Moreover, If $ C $ is a self-dual code, so is $ \varphi(C) $.

Let $ e_1 = 1-u-v, e_2 = u, e_3 = v $. It is easy to check that $ e_ie_j = \delta_{ij}e_i $ and $ \sum\limits_{k = 1}^{3}e_k = 1 $, where $ \delta_{ij} $ stands for Dirichlet function, i.e., $ \delta_{ij} = \left\{\begin{aligned} 1, \; \mathrm{if}\; i = j, \\ 0, \; \mathrm{if}\; i\neq j. \end{aligned}\right. $ According to [15], we have $ R = e_1R\oplus e_2R\oplus e_3R $.

Now, we mainly consider some familiar structural properties of linear code $ C $ over $ R $. The proof of following results can be found in [16], so we omit them here.

Let $ A_i\; (i = 1, 2, 3) $ be codes over $ R $. We denote

$ {A_1\oplus A_2\oplus A_3 = \{a_1+a_2+a_3|a_1\in A_1, a_2\in A_2, a_3\in A_3\}}. $

If $ C $ is a linear code of length $ n $ over $ R $, we define that

$ \begin{eqnarray*} &&C_1 = \{{\bf{a}}\in\mathbb{F}_{q}^{n}|\mathrm{there\; are}\; {\bf{b}}, {\bf{c}}\in\mathbb{F}_{q}^{n}\; \mathrm{such\; that}\; e_1{\bf{a}}+e_2{\bf{b}}+e_3{\bf{c}}\in C\}, \\ &&C_2 = \{{\bf{b}}\in\mathbb{F}_{q}^{n}|\mathrm{there\; are}\; {\bf{a}}, {\bf{c}}\in\mathbb{F}_{q}^{n}\; \mathrm{such\; that}\; e_1{\bf{a}}+e_2{\bf{b}}+e_3{\bf{c}}\in C\}, \\&&C_3 = \{{\bf{c}}\in\mathbb{F}_{q}^{n}|\mathrm{there\; are}\; {\bf{a}}, {\bf{b}}\in\mathbb{F}_{q}^{n}\; \mathrm{such\; that}\; e_1{\bf{a}}+e_2{\bf{b}}+e_3{\bf{c}}\in C\}. \end{eqnarray*} $

It is easy to verity that $ C_i(i = 1, 2, 3) $ are linear codes of length $ n $ over $ \mathbb{F}_{q} $. Furthermore, $ C = e_1C_1\oplus e_2C_2\oplus e_3C_3 $, and $ \mid C\mid = \mid C_1\mid\mid C_2\mid\mid C_3\mid $. Throughout the paper $ C_i(i = 1, 2, 3) $ will be reserved symbols referring to these special subcodes.

According to the above definitions and [17], we have the following theorem.

Theorem 2.4. If $ C = e_1C_1\oplus e_2C_2\oplus e_3C_3 $ is a linear code of length $ n $ over $ R $, then $ C^{\perp} = e_1C_1^{\perp} \oplus e_2C_2^{\perp}\oplus e_3C_3^{\perp} $.

The next theorem gives a computation for minimum Lee distance $ d_L $ of a linear code of length $ n $ over $ R $.

Theorem 2.5. If $ C = e_1C_1\oplus e_2C_2\oplus e_3C_3 $ is a linear code of length $ n $ over $ R $, then $ d_L(C) = \min\{d_H(C_1), d_H(C_2), d_H(C_3)\} $.

Proof By Theorem 2.3, we have, $ d_L(C) = d_H(\varphi(C)) $.

For any codeword $ {\bf{x}} $, it can be written as $ {\bf{x}} = e_1{\bf{a}}+e_2{\bf{b}}+e_3{\bf{c}} $, where $ {\bf{a}}\in C_1, {\bf{b}}\in C_2, {\bf{c}}\in C_3 $. Thus,

$ \varphi({\bf{x}}) = ({\bf{a}}, {\bf{b}}, {\bf{c}}) = ({\bf{a}}, {\bf{0}}, {\bf{0}})+({\bf{0}}, {\bf{b}}, {\bf{0}})+({\bf{0}}, {\bf{0}}, {\bf{c}}). $

This means that $ d_L(C) = \min\{d_H(C_1), d_H(C_2), d_H(C_3)\} $.

In this section, we assume that $ q = l^2 $, where $ l $ is a power of the prime $ p $. Consider the involution $ ^{-} $: $ \; a\rightarrow a^{l} $ defined on $ \mathbb{F}_{l^2} $. For any $ r = e_1a+e_2b+e_3c\in R $, we denote the involution on $ R $ by $ ^{-} $ defined by $ \overline{r} = e_1a^l+e_2b^l+e_3c^l. $

For a given linear code $ C $ of length $ n $ over $ R $, denoted by $ C^{\perp_{H}} $ the Hermitian dual of $ C $ defined with respect to the form $ [{\bf{x}}, {\bf{y}}]_{H}: = \sum\limits_{i = 1}^{n}x_{i}\overline{y_{i}} $, where $ {\bf{x}} = (x_1\cdots, x_n), {\bf{y}} = (y_1\cdots, y_n)\in R^n $. The code $ C $ is said to be Hermitian dual-containing if $ C^{\perp_{H}}\subset C $, and Hermitian self-dual if $ C^{\perp_{H}} = C $.

We first recall the definition of reciprocal polynomial in $ \mathbb{F}_{l^2}[x] $. For any polynomial $ f(x) = \sum\limits_{i = 0}^{k}a_{i}x^{i} $ of degree $ k $ ($ a_{k}\neq0) $ over $ \mathbb{F}_{l^2} $, let $ f^{\ast}(x) $ denote the reciprocal polynomial of $ f(x) $ given by

$ f^{\ast}(x) = x^{k}f(\frac{1}{x}) = \sum\limits_{i = 0}^{k}a_{k-i}x^{i}. $

Next, we extend the involution map to polynomials in $ \mathbb{F}_{l^{2}}[x] $. For $ f(x) = a_0+a_1x+\cdots+a_{n-1}x^{n-1} $ in $ \mathbb{F}_{l^{2}}[x] $, we set $ \overline{f(x)} = \overline{a_0}+\overline{a_1}x+\cdots+\overline{a_{n-1}}x^{n-1} $. The conjugate reciprocal polynomial of $ f(x) $ is denoted as $ f^{†}(x) $ and is equal to $ \overline{f^{*}(x)} $. $ f(x) $ is said to be self-conjugate reciprocal if $ f(x) = f^{†}(x) $. Otherwise, $ f(x) $ and $ f^{†}(x) $ form a conjugate reciprocal pair.

The following lemma is going to play an important role in constructing quantum codes.

Lemma 3.1 Let $ C = e_1C_1\oplus e_2C_2\oplus e_3C_3 $ be a cyclic codes of length $ n $ over $ R $. Then $ C = \langle e_1g_1(x), e_2g_2(x), e_3g_3(x)\rangle $ and $ \mid C\mid = q^{3n-\sum\limits_{i = 1}^{3}\mathrm{deg}g_i(x)} $, where $ g_i(x) $ is a generator polynomial of cyclic codes $ C_i $ of length $ n $ over $ \mathbb{F}_{l^2} $ for i = 1, 2, 3.

Proof Since $ C_i = \langle g_i(x)\rangle\subset\frac{\mathbb{F}_{l^2}[x]}{\langle x^n-1\rangle} $ for $ i = 1, 2, 3 $, and $ C = e_1C_1\oplus e_2C_2\oplus e_3C_3 $, $ C = \{ c(x) |c(x) = e_1f_1(x)+e_2f_2(x)+e_3f_3(x), f_i(x)\in C_i, i = 1, 2, 3\} $. Thus

$ C\subset\langle e_1g_1(x), e_2g_2(x), e_3g_3(x)\rangle. $

On the other hand, for any

$ e_1g_1(x)r_1(x)+e_2g_2(x)r_2(x)+e_3g_3(x)r_3(x)\in \langle e_1g_1(x), e_2g_2(x), e_3g_3(x)\rangle\subset \frac{R[x]}{\langle x^n-1\rangle}, $

where $ r_1(x), r_2(x) $ and $ r_3(x)\in \frac{R[x]}{\langle x^n-1\rangle} $, there exist $ s_1(x), s_2(x) $ and $ s_3(x)\in \mathbb{F}_{l^2}[x] $ such that $ e_1r_1(x) = e_1s_1(x), e_2r_2(x) = e_2s_2(x) $, and $ e_3r_3(x) = e_3s_3(x) $. Hence,

$ e_1g_1(x)r_1(x)+e_2g_2(x)r_2(x)+e_3g_3(x)r_3(x) = e_1g_1(x)s_1(x)+e_2g_2(x)s_2(x)+e_3g_3(x)s_3(x), $

which implies that $ \langle e_1g_1(x), e_2g_2(x), e_3g_3(x)\rangle\subset C $. Therefore, $ C = \langle e_1g_1(x), e_2g_2(x), e_3g_3(x)\rangle $.

Similar to the proof of Theorems 2.4 and 2.5, we can prove the following two theorems.

Theorem 3.2 Let $ C $ be a linear code of length $ n $ over $ R $. Then

$ \mathrm{(1)} $ $ \varphi(C^{\perp_{H}}) = \varphi(C)^{\perp_{H}} $;

$ \mathrm{(2)} $ if $ C $ is a Hermitian self-dual code, then $ \varphi(C) $ is a Hermitian self-dual code of length $ 3n $ over $ \mathbb{F}_{l^{2}} $;

$ \mathrm{(3)} $ if $ C $ is a Hermitian dual-containing code, then $ \varphi(C) $ is a Hermitian dual-containing code of length $ 3n $ over $ \mathbb{F}_{l^{2}} $.

Theorem 3.3 Let $ C = e_1C_1\oplus e_2C_2\oplus e_3C_3 $ be a linear code of length $ n $ over $ R $. Then

$ \mathrm{(1)} $ $ C^{\perp_{H}} = e_1C_1^{\perp_{H}} \oplus e_2C_2^{\perp_{H}}\oplus e_3C_3^{\perp_{H}} $. Furthermore, $ C $ is a Hermitian self-dual code if and only if $ C_1, C_2, C_3 $ are Hermitian self-dual codes over $ \mathbb{F}_{l^{2}} $, and $ C $ is a Hermitian dual-containing code if and only if $ C_1, C_2, C_3 $ are Hermitian dual-containing codes over $ \mathbb{F}_{l^{2}} $;

$ \mathrm{(2)} $ if $ C = \langle e_1g_1(x), e_2g_2(x), e_3g_3(x)\rangle $ is a cyclic codes of length $ n $ over $ R $, where

$ C_1 = \langle g_1(x)\rangle, C_2 = \langle g_2(x)\rangle, C_3 = \langle g_3(x)\rangle $

and

$ x^n-1 = g_1(x)h_1(x) = g_2(x)h_2(x) = g_3(x)h_3(x) $

in $ \mathbb{F}_{l^{2}}[x] $, then

$ C^{\perp_{H}} = \langle e_1h_1^{†}(x), e_2h_2^{†}(x), e_3h_3^{†}(x)\rangle = \langle e_1h_1^{†}(x)+e_2h_2^{†}(x)+e_3h_3^{†}(x)\rangle. $

In particular, if $ C_1, C_2 $, and $ C_3 $ are Hermitian dual-containing code over $ \mathbb{F}_{l^{2}} $ with parameters $ [n, k_1, d_1], [n, k_2, d_2] $, and $ [n, k_3, d_3] $, respectively, then $ \varphi(C) $ is a Hermitian dual-containing code over $ \mathbb{F}_{l^{2}} $ with parameters $ [3n, k_1+k_2+k_3, \min\{d_1, d_2, d_3\}] $.

It is easy to prove the following lemma.

Lemma 3.4 Let $ C $ be a cyclic code with a generator polynomial $ g(x) $ over $ \mathbb{F}_{l^{2}} $, where $ x^n-1 = g(x)h(x) $. Then

$ C^{\perp_{H}}\subset C \Leftrightarrow h(x)h^{\dagger}(x)\equiv0(\mathrm{mod}\; x^n-1) $

or equivalently

$ C^{\perp_{H}}\subset C \Leftrightarrow x^n-1\equiv0(\mathrm{mod}\; g(x)g^{\dagger}(x)). $

Combining Lemma 3.4 with Theorem 3.3, we have the following corollary.

Corollary 3.5 Let $ C = \langle g(x)\rangle $ be a cyclic codes of length $ n $ over $ R $, where

$ g(x) = e_1g_1(x)+e_2g_2(x)+e_3g_3(x), C_1 = \langle g_1(x)\rangle, C_2 = \langle g_2(x)\rangle, C_3 = \langle g_3(x)\rangle $

and

$ x^n-1 = g_1(x)h_1(x) = g_2(x)h_2(x) = g_3(x)h_3(x) $

in $ \mathbb{F}_{l^{2}}[x] $. Then

$ C^{\perp_{H}}\subset C \Leftrightarrow x^n-1\equiv0(\mathrm{mod}\; g_i(x)g_i^{\dagger}(x)), i = 1, 2, 3. $

A $ l $-ary quantum code $ Q $ of length $ n $ and size $ K $ is a $ K $-dimensional subspace of the $ q^n $-dimensional Hilbert space $ \mathbb{H} = (C^{q})^{\otimes n} = C^{q}\otimes\cdots\otimes C^{q} $. Let $ k = \mathrm{log}_{l}(K) $. We use $ [[n, k, d]]_{l} $ to denote a $ l $-ary quantum code of length $ n $ with size $ q^{k} $ and minimum distance $ d $. If a quantum code has minimum distance $ d $, then it can detect any $ d-1 $ and correct any $ \lfloor\frac{d-1}{2}\rfloor $ errors. One of the principal problems in quantum coding theory is to construct quantum codes with the best possible minimum distance. Recently, some classes of $ l $-ary good quantum code have been found by employing the following Hermitian construction (see[3, 6, 8-11]).

Lemma 3.6 [10] If $ C $ is a Hermitian dual-containing code over $ \mathbb{F}_{l^{2}} $ with parameters $ [n, k, d] $, then there exists a a $ l $-ary $ [[n, 2k-n, \geq d]]_{l} $ quantum code.

Combining Theorem 3.3 with Lemma 3.6, we give the parameters of quantum codes obtained from the cyclic codes over $ R $ containing their Hermitian duals.

Theorem 3.7 Let $ C = e_1C_1\oplus e_2C_2\oplus e_3C_3 $ be a cyclic code of length $ n $ over $ R $, where

$ C_1 = \langle g_1(x)\rangle, C_2 = \langle g_2(x)\rangle, C_3 = \langle g_3(x)\rangle $

and

$ x^n-1 = g_1(x)h_1(x) = g_2(x)h_2(x) = g_3(x)h_3(x) $

in $ \mathbb{F}_{l^{2}}[x] $. If $ x^n-1\equiv0(\mathrm{mod}\; g_i(x)g_i^{\dagger}(x)), i = 1, 2, 3 $, then there exists a quantum code with the parameters $ [[3n, 3n-2s, \geq d_L]]_{l} $, where $ s = \sum\limits_{i = 1}^{3}\mathrm{deg}g_i(x) $ and $ d_L $ is the minimum Lee distance of the code $ C $.

Using Theorem 3.7, we give some new quantum codes.

Example 1 In $ \mathbb{F}_4 $,

$ x^7-1 = (x + 1)(x^3+x+1)(x^3+x^2+1). $

Let $ g_1(x) = g_2(x) = g_3(x) = x^3+x+1 $. Then, by Theorem 3.7 and a computer programme, we get a $ [[21, 3, \geq3]]_2 $ quantum code.

Example 2 In $ \mathbb{F}_9 $,

$ x^{13}-1 = (x + 2)(x^3+2x+2)(x^3+x^2+2)(x^3+x^2+x+2)(x^3+2x^2+2x+2). $

Let $ g_1(x) = g_2(x) = g_3(x) = x^3+x^2+x+2. $ Then, by Theorem 3.7 and a computer programme, we get a $ [[39, 15, \geq3]]_3 $ quantum code.

The following result can be found in [12].

Lemma 3.8 Let $ l $ be an odd prime power. Then

$ \mathrm{(1)} $ If $ l\equiv1\; \mathrm{mod}\; 4 $, there exists a Hermitian dual-containing code over $ \mathbb{F}_{l^{2}} $ with parameters $ [l^2+1, l^2-d+2, d] $, where $ 2\leq d\leq l+1 $ is even;

$ \mathrm{(2)} $ There exists a Hermitian dual-containing code over $ \mathbb{F}_{l^{2}} $ with parameters

$ [\frac{l^2+1}{2}, \frac{l^2+1}{2}-d+1, d], $

where $ 3\leq d\leq l $ is odd.

By Theorem 3.3, we can immediately get the following lemma:

Lemma 3.9 Let $ l $ be an odd prime power. Then

$ \mathrm{(1)} $ If $ l\equiv1\; \mathrm{mod}\; 4 $, then there exists a Hermitian dual-containing code over $ \mathbb{F}_{l^{2}} $ with parameters $ [3l^2+3, 3l^2-3d+6, d] $, where $ 2\leq d\leq l+1 $ is even;

$ \mathrm{(2)} $ There exists a Hermitian dual-containing code over $ \mathbb{F}_{l^{2}} $ with parameters

$ [\frac{3(l^2+1)}{2}, \frac{3(l^2+1)}{2}-3d+3, d], $

where $ 3\leq d\leq l $ is odd.

Then by Lemmas 3.6 and 3.9, we have the following theorem.

Theorem 3.10. Let $ l $ be an odd prime power. Then

$ \mathrm{(1)} $ If $ l\equiv1\; \mathrm{mod}\; 4 $, then there exists a $ l $-ary $ [[3l^2+3, 3l^2-6d+9, \geq d]]_{l} $ quantum code, where $ 2\leq d\leq l+1 $ is even;

$ \mathrm{(2)} $ There exists a $ l $-ary

$ [[\frac{3(l^2+1)}{2}, \frac{3(l^2+1)}{2}-6d+6, \geq d]]_{l} $

quantum code, where $ 3\leq d\leq l $ is odd.

In Table 1, we list some quantum codes obtained from Theorem 3.10. The table shows that our quantum codes have new parameters compared with the previous quantum codes available (see [18]).

We have developed a new method of constructing quantum codes from cyclic codes over finite ring $ R $. Using this method, we have constructed new quantum codes. We believe that cyclic codes over finite ring $ R $ will be a good source for constructing new quantum codes. In a future work, we will use the computer algebra system MAGMA to find more new quantum codes.