New Optimal Asymmetric Quantum Codes Derived from Negacyclic Codes

Abstract

The construction of quantum maximum-distance-separable (MDS) codes have been studied by many researchers for many years. Here, by using negacyclic codes, we construct two families of asymmetric quantum codes. The first family is the asymmetric quantum codes with parameters $[[q^{2}+1,q^{2}+1-2(t+k+1),(2k+2)/(2t+2)]]_{q^{2}}$, where 0≤t≤k≤(q−1)/2, $q \equiv1(\operatorname{mod} 4)$, and k, t are positive integers. The second one is the asymmetric quantum codes with parameters $[[(q^{2}+1)/2,(q^{2}+1)/2-2(t+k),(2k+1)/(2t+1)]]_{q^{2}}$, where 1≤t≤k≤(q−1)/2, and k, t are positive integers. Moreover, the constructed asymmetric quantum codes are optimal and different from the codes available in the literature.

1 Introduction

Nowadays, the theory of quantum error-correcting codes have been studied by many authors. Some researchers studied the construction of efficient quantum error-correcting codes over unbiased quantum channels (see [1–9] for more details). We can see that quantum error-correcting code, realizing all-optical communications in the quantum communication, plays an important role in the optical switching in [10–13]. Quantum codes defined over quantum channels where qudit-flip errors and phase-shift errors may have different probabilities are called asymmetric quantum codes. In many quantum mechanical systems, the occurrence of bit flip and phase flip errors is quite different. Steane firstly introduced the concept of asymmetric quantum errors in 1996 [14]. Since then, the constructions of quantum codes have been extended to asymmetric quantum channels. Loffe et al. used BCH codes and LDPC codes to correct qubit-flip errors and phase-shift errors respectively in [15]. Asymmetric stabilizer codes were constructed derived from LDPC codes and several families of asymmetric quantum codes were constructed by using classical BCH and RS codes in [16, 17]. In [18], there are some families of asymmetric quantum codes derived from BCH codes were constructed. S.A. Aly et al. studied asymmetric quantum codes derived from cyclic codes and subsystem codes in [19]. Leng et al. presented some families of good asymmetric quantum BCH codes in [20]. L. Wang et al. presented the construction of nonadditive asymmetric quantum codes and asymptotically good asymmetric quantum codes derived from algebraic-geometry codes in [21].

In [22], authors studied the general construction of asymmetric quantum codes under the trace Hermitian inner product. M.F. Ezerman et al. studied two systematic construction of asymmetric quantum codes in [23]. Y. Chee et al. constructed pure q-ary asymmetric quantum codes and these codes can attain the quantum Sigleton bound in [24]. Recently, constructions of new families of asymmetric quantum codes were presented in [25–28]. J. Qian et al. studied the asymmetric quantum codes by using the q ²-ary cyclotomic cosets in [29]. Recently, M.F. Ezerman et al. also studied the pure asymmetric quantum codes and obtained some good codes in [32]. P. Dong et al. proposed and efficient quantum circuit for encoding and decoding of the [[8, 3, 5]] stabilizer code in [31].

Cyclic codes are important classes of linear codes that have been applied to the construction of quantum codes. However, few people constructed asymmetric quantum codes by using negacyclic codes. In this paper, we will use negacyclic codes to construct new asymmetric quantum codes. Based on the classical negacyclic codes, we propose the construction of new families of asymmetric quantum codes. Some of these codes aren’t covered by the codes available in the literatures. Moreover, the proposed asymmetric quantum codes are optimal.

The organization of this paper is as follows: In Sect. 2, we present some definitions and basic results of negacyclic codes. In Sect. 3, we state some basic definitions of asymmetric quantum codes. In Sect. 4, two classes of asymmetric quantum codes are constructed.

2 Negacyclic Codes

In this section, we recall some basic results about negacyclic codes in [32–35].

Throughout this paper, let p be an odd prime number, q be a prime power of p, and $F_{q^{2}}$ be a finite field with q ² elements. Let $a^{q}=(a_{0}^{q},a_{1}^{q},\ldots,a_{n-1}^{q})$ denote the conjugation of the vector a=(a ₀,a ₁,…,a _n−1). For u=(u ₀,u ₁,…,u _n−1), $v=(v_{0},v_{1},\ldots,v_{n-1})\in F_{q^{2}}^{n}$, we can define the Hermitian inner product as follows:

$$\langle u,v\rangle_h=u_0^qv_0+u_1^qv_1+\cdots+u_{n-1}^qv_{n-1}. $$

If C is a k-free submodule with length n, then C is said to be [n,k]-linear code. The number of nonzero components of c∈C is said to be the weight wt(c) of the codeword c. The minimum nonzero weight d of all codewords in C is said to be the minimum weight of C. For a q ²-ary linear code C of length n, the Hermitian dual code of C can be defined as follows:

$$C^{\bot_h}=\{u\in F_{q^2}^n\mid\langle u,v\rangle_h=0\ \mbox{for all}\ v\in C \}. $$

From the above definition, we can see that a q ²-ary linear code C of length n is called Hermitian self-orthogonal if $C\subseteq C^{\perp_{h}}$.

If a q ²-ary linear code C of length n is invariant under the permeation of $F_{q^{2}}$, i.e.,

$$(c_0,c_1,\ldots,c_{n-1})\rightarrow(ac_{n-1},c_0,c_1, \ldots,c_{n-2}), $$

then C is constacyclic code. If a=1, then C is said to be a cyclic code. If a=−1, then C is called negacyclic code.

We can see that xc(x) corresponds to a negacyclic shift of c(x) in the quotient ring $F_{q^{2}}[x]/\langle x^{n}+1\rangle$. Then, a q ²-ary negacyclic code C of length n is an ideal of $F_{q^{2}}[x]/\langle x^{n}+1\rangle$ and C can be generated by a monic polynomial g(x) of x ⁿ+1.

Let gcd(n,q)=1. Then x ⁿ+1 doesn’t have multiple roots. Let m be the multiplicative order of q ² modulo 2n. Let β be primitive 2nth root of unity in $F_{q^{2m}}$ and $\alpha=\beta^{2}\in F_{q^{2m}}$. Then, α is a primitive nth root of unity. Hence,

$$x^n+1=\varPi_{i=0}^{n-1}\bigl(x-\beta \alpha^i\bigr)=\varPi_{i=0}^{n-1}\bigl(x- \beta^{2i+1}\bigr). $$

The q ²-cyclotomic coset module 2n containing i are defined by C _i , $C_{i}=\{i,iq^{2},iq^{4},\ldots, iq^{2(m_{i}-1)}\}$, where m _i is the smallest positive integer such that $iq^{2m_{i}}\equiv i\operatorname{mod} 2n$.

Let $\mathcal{O} _{2n}$ be the set of all odd integers from 1 to 2n. The defining set of a negacyclic code C=〈g(x)〉 of length n is the set $Z=\{i\in\mathcal{O}_{2n}\mid\beta^{i}\ \mathrm{is\ a\ root\ of}\ g(x)\}$. We can see that the defining set Z is a union of some q ²-cyclotomic cosets module 2n and $\operatorname{dim}(C)=n-|Z|$.

The following propositions in [33, 34] play an important role in constructing asymmetric quantum codes.

Proposition 1

Let C be an [n,k] negacyclic code over $F_{q^{2}}$ with defining set Z. Then the Hermitian dual $C^{\perp_{h}}$ is also negacyclic and has defining set $Z^{\perp_{h}}=\{z\in\mathcal {O}_{2n}|-qz(\operatorname{mod} 2n)\notin Z\}$.

Proposition 2

(The BCH bound for negacyclic codes)

Let C be a q ²-ary negacyclic code of length n. If the generator polynomial g(x) of C has the elements {β ¹⁺²ⁱ∣0≤i≤d−2} as the roots where β is a primitive 2nth root of unity, then the minimum distance of C is at least d.

Proposition 3

(Singleton boud)

If an [n,k,d] linear code over F _q exists, then k≤n−d+1.

Now, let us recall the following result in [29].

Proposition 4

Let C _i be cyclic codes of length n over F _q with defining set T _i for i=1,2. Then C ₁⊆C ₂ if and only if T ₂⊆T ₁.

We can obtain the following similar result from the defining set of negacyclic codes.

Proposition 5

Let C _i be q ²-ary negacyclic codes of length n with defining set Z _i for i=1,2. Then C ₁⊆C ₂ if and only if Z ₂⊆Z ₁.

3 Error group and asymmetric quantum codes

In this section, we state some definitions and some basic results in [17–19] and [22–29].

Let H be the Hilbert space $H=\mathcal{C}^{q^{n}}=\mathcal {C}^{q}\bigotimes\cdots\bigotimes\mathcal{C}^{q}$. Let |x〉 be the vectors of an orthonormal basis of $\mathcal{C}^{q}$, where the notions x are elements of F _q .

Consider a,b∈F _q , the unitary operators X(a) and Z(b) in $\mathcal{C}^{q}$ are defined by X(a)∣x〉=∣x+a〉, Z(b)∣x〉=ω ^tr(bx)∣x〉, where ω=exp(2πi/p) is a pth root of unity and tr is the trace map from F _q to F _p .

Consider that $a=(a_{1},a_{2},\ldots,a_{n})\in F_{q}^{n}$ and $b=(b_{1},b_{2},\ldots ,b_{n})\in F_{q}^{n}$. Denote by

$$X(a)=X(a_1)\otimes X(a_2)\otimes \cdots\otimes X(a_n), $$

$$Z(a)=Z(b_1)\otimes Z(b_2)\otimes \cdots\otimes Z(b_n), $$

tensor products of n error operators. The set $E_{n}=\{X(a)Z(b)\mid a,b \in F_{q}^{n}\}$ is an error basis on the complex vector space $\mathcal{C}^{q^{n}}$ and the set $G_{n}=\{\omega^{c}X(a)Z(b)\mid a,b\in F_{q}^{n},c\in F_{p}\}$ is the error group associated with E _n . For a quantum error e=ω ^c X(a)Z(b)∈G _n the quantum weight ω _Q (e), the X-weight ω _X (e) and the Z-weight ω _Z (e) of e, are defined respectively by

$$\omega_Q(e)=\sharp\bigl\{i: 1\leq i\leq n,(a_i,b_i) \neq(0,0)\bigr\}, $$

$$\omega_X(e)=\sharp\{i: 1\leq i\leq n, a_i \neq0 \}, $$

$$\omega_Z(e)=\sharp\{i: 1\leq i\leq n, b_i \neq0 \}. $$

In [26], a q-ary asymmetric quantum code Q, denoted by [[n,k,d _z /d _x ]], is a q ^k-dimensional subspace of the Hilbert space $\mathcal{C}^{q^{n}}$ and can control all qubit-flip errors up to ⌊(d _x −1)/2⌋ and all phase-flip errors up to ⌊(d _z −1)/2⌋.

The following basic definitions and results in [16, 29] will be applied to construct optimal asymmetric quantum codes.

Theorem 1

(CSS construction)

Let C _i be a classical code with parameters [n,k _i ,d _i ] for i=1,2, with $C^{\perp}_{1}\subseteq C_{2}$. Then there exists an asymmetric quantum code Q with parameters [[n,k ₁+k ₂−n,d _z /d _x ]], where $d_{x}=wt(C_{1}\backslash C^{\perp}_{2})$ and $d_{z}=wt(C_{2}\backslash C^{\perp}_{1})$.

Proposition 6

If the asymmetric quantum code C with parameters [[n,k ₁+k ₂−n,d _z /d _x ]] exists, then C satisfies the asymmetric quantum Singleton bound

$$k\leq n-d_x-d_z+2. $$

If C satisfy the equality k=n−d _x −d _z +2, then it is called an optimal code.

4 Code Constructions

In this section, we will use negacyclic codes of length n=q ²+1 and n=(q ²+1)/2 to construct asymmetric quantum MDS codes respectively. In [29], authors have constructed new asymmetric quantum MDS codes with parameters $[[q^{2}+1,q^{2}+1-2(k+i-2),(2k+3)/(2i+3)]]_{q^{2}}$, where k and i are positive integers, and 0≤i≤k≤q/2−1.

Remark 1

By computing, we can obtain the asymmetric quantum codes with parameters $[[q^{2}+1,q^{2}+1-2(k+i+2),(2k+3)/(2i+3)]]_{q^{2}}$, not $[[q^{2}+1,q^{2}+1-2(k+i-2),(2k+3)/(2i+3)]]_{q^{2}}$ in [29].

We have the following lemma from [33] that play an important role in the quantum construction.

Lemma 1

Let n=q ²+1 with $q \equiv1 (\operatorname{mod} 4)$, and s=n/2. If C is a q ²-ary negacyclic code of length n with defining set $Z=\bigcup _{i=0}^{\delta}C_{s-2i}$, where 0≤δ≤(q−1)/2, then $C^{\perp_{h}}\subseteq C$.

Theorem 2

Let n=q ²+1 with $q \equiv1 (\operatorname{mod} 4)$, and s=n/2. There exist asymmetric quantum codes with parameters $[[q^{2}+1,q^{2}+1-2(t+k+1),(2k+2)/(2t+2)]]_{q^{2}}$, where k and t are positive integers, and 0≤t≤k≤(q−1)/2.

Proof

Without losing generality, we can assume that the defining set of negacyclic code C ₁ is given by $Z_{1}=\bigcup _{i=0}^{t}C_{s-2i}$, where 0≤t≤(q−1)/2. The generator polynomial of the negacyclic code C ₁ is

$$g_1(x)=\bigl(x-\alpha^{s-2t}\bigr)\cdots\bigl(x- \alpha^{s-2}\bigr) \bigl(x-\alpha^s\bigr) \bigl(x-\alpha ^{s+2}\bigr)\cdots\bigl(x-\alpha^{s+2t}\bigr). $$

It is clearly that the minimum distance of C ₁ is at least 2(t+1) from Proposition 2. From Proposition 3, we can see that the minimum distance of C ₁ is 2(t+1). Hence C ₁ is a q ²-ary [n,n−2t−1,2t+2]-negacyclic code and C ₁ is a MDS negacyclic code. Moreover, the Hermitian dual $C_{1}^{\perp_{h}}$ is also a MDS negacyclic code with parameters $[n,2t+1,n-2t]_{q^{2}}$ from Proposition 1. Now, we suppose that the defining set of negacyclic code C ₂ is given by $Z_{2}=\bigcup_{i=0}^{k}C_{s-2i}$, where 0≤t≤k≤(q−1)/2. The generator polynomial of negacyclic code C ₂ is

$$g_2(x)=\bigl(x-\alpha^{s-2k}\bigr)\cdots\bigl(x- \alpha^{s-2}\bigr) \bigl(x-\alpha^s\bigr) \bigl(x-\alpha ^{s+2}\bigr)\cdots\bigl(x-\alpha^{s+2k}\bigr). $$

It is clearly that the minimum distance of C ₂ is at least 2k+2 from Proposition 2. From Proposition 3, we can see that the minimum distance of C ₂ is 2(k+1). Hence C ₂ is a MDS negacyclic code with parameters $[n,n-2k-1,2(k +1)]_{q^{2}}$. From Lemma 1 and Proposition 5, we know that $C_{1}^{\perp_{h}} \subseteq C_{2}$. Then, from Theorem 1, we know that there exist asymmetric quantum codes with parameters $[[q^{2}+1,q^{2}+1-2(t+k+1),(2k+2)/(2t+2)]]_{q^{2}}$. □

Remark 2

From Theorem 2, we can see that d _z +d _x =2t+2+2k+2=q ²+1−(q ²+1)+2(t+k+1)+2. Then, from Proposition 6, we know that the constructed asymmetric quantum codes with parameters $[[q^{2}+1,q^{2}+1-2(t+k+1),(2k+2)/(2t+2)]]_{q^{2}}$ attain asymmetric quantum Singleton bound. Hence, these asymmetric quantum codes are optimal.

Example 1

Let q=5, then n=26. We suppose that the defining set of negacyclic code C ₁ is given by Z ₁=C ₁₃={13}. Then, C ₁ is a MDS negacyclic code with parameters [26,25,2]₂₅. Moreover, $C_{1}^{\perp}$ is also a MDS negacyclic code with parameters [26,1,26]₂₅. We can also suppose that the defining set of negacyclic code C ₂ is given by Z ₂=C ₁₃∪C ₁₁∪C ₉={13,11,15,17,9}. Then, C ₂ is a MDS negacyclic code with parameters [26,21,6]₂₅. From Theorem 2, there exists optimal asymmetric quantum code with parameters [[26,20,6/2]]₂₅.

If the defining set of negacyclic code C ₁ takes as Z ₁=C ₁₃∪C ₁₁={13,11,15}. Then, C ₁ is a MDS negacyclic code with parameters [26,23,4]₂₅. Moreover, $C_{1}^{\perp}$ is also a MDS negacyclic code with parameters [26,3,24]₂₅. From Theorem 2, there exists optimal asymmetric quantum code with parameters [[26,18,6/4]]₂₅.

Example 2

Let q=13, then n=170. We suppose that the defining set of negacyclic code C ₁ is given by Z ₁=C ₈₅={85}. Then, C ₁ is a MDS negacyclic code with parameters [170,169,2]₁₆₉. Moreover, $C_{1}^{\perp}$ is also a MDS negacyclic code with parameters [170,1,170]₁₆₉.

We can also suppose that the defining set of negacyclic code C ₂ is given by Z ₂=C ₈₅∪C ₈₃∪C ₈₁∪C ₇₉∪C ₇₇∪C ₇₅∪C ₇₃={85,83,87,81,89,79,91,77,93,75,95,73,97}. Then, C ₂ is a MDS negacyclic code with parameters [170,157,14]₁₆₉. From Theorem 2, there exist some optimal asymmetric quantum codes with parameters [[170,156,14/2]]₁₆₉.

If the defining set of negacyclic code C ₁ is given by Z ₁=C ₈₅∪C ₈₃={85,83,87}. Then, C ₁ is a MDS negacyclic code with parameters [170,167,4]₁₆₉. Moreover, $C_{1}^{\perp}$ is also a MDS negacyclic code with parameters [170,3,168]₁₆₉. From Theorem 2, there exist some optimal asymmetric quantum codes with parameters [[170,154,14/4]]₁₆₉.

If the defining set of negacyclic code C ₁ is given by Z ₁=C ₈₅∪C ₈₃∪C ₈₁={85,83,87,81,89}. Then, C ₁ is a MDS negacyclic code with parameters [170,165,6]₁₆₉. Moreover, $C_{1}^{\perp}$ is also a MDS negacyclic code with parameters [170,5,166]₂₅. From Theorem 2, there exist some optimal asymmetric quantum codes with parameters [[170,152,14/6]]₁₆₉.

If the defining set of negacyclic code C ₁ is given by Z ₁=C ₈₅∪C ₈₃∪C ₈₁∪C ₇₉={85,83,87,81,89,79,91}. Then, C ₁ is a MDS negacyclic code with parameters [170,163,8]₁₆₉. Moreover, $C_{1}^{\perp}$ is also a MDS negacyclic code with parameters [170,7,164]₂₅. From Theorem 2, there exist some optimal asymmetric quantum codes with parameters [[170,150,14/8]]₁₆₉.

If the defining set of negacyclic code C ₁ is given by Z ₁=C ₈₅∪C ₈₃∪C ₈₁∪C ₇₉∪C ₇₇={85,83,87,81,89,79,91,77,93}. Then, C ₁ is a MDS negacyclic code with parameters [170,161,10]₁₆₉. Moreover, $C_{1}^{\perp}$ is also a MDS negacyclic code with parameters [170,9,162]₂₅. From Theorem 2, there exist some optimal asymmetric quantum codes with parameters [[170,148,14/10]]₁₆₉.

If the defining set of negacyclic code C ₁ is given by Z ₁=C ₈₅∪C ₈₃∪C ₈₁∪C ₇₉∪C ₇₇∪C ₇₅={85,83,87,81,89,79,91,77,93,75,95}. Then, C ₁ is a MDS negacyclic code with parameters [170,169,12]₁₆₉. Moreover, $C_{1}^{\perp}$ is also a MDS negacyclic code with parameters [170,11,160]₂₅. From Theorem 2, there exist some optimal asymmetric quantum codes with parameters [[170,146,14/12]]₁₆₉.

Now, we can construct asymmetric quantum code with length n=(q ²+1)/2 by using the following lemma in [33].

Lemma 2

Let n=(q ²+1)/2. If C is a q ²-ary negacyclic code of length n with defining set $Z=\bigcup_{i=1}^{\delta}C_{2i-1}$, where 1≤δ≤(q−1)/2, then $C^{\perp_{h}}\subseteq C$.

Theorem 3

Let n=(q ²+1)/2. There exist asymmetric quantum codes with parameters $[[(q^{2}+1)/2,(q^{2}+1)/2-2(t+k),(2k+1)/(2t+1)]]_{q^{2}}$, where k and i are positive integers, and 1≤t≤k≤(q−1)/2.

Proof

We can assume that the defining set of negacyclic code C ₁ is given by $Z_{1}=\bigcup_{i=1}^{t} C_{2i-1}$, where 1≤t≤(q−1)/2. The generator polynomial of negacyclic code C ₁ is

$$g_1(x)=\bigl(x-\alpha^{1-2t}\bigr)\cdots\bigl(x- \alpha^{-1}\bigr) (x-\alpha)\cdots\bigl(x-\alpha^{2t-1}\bigr). $$

We know the minimum distance of C ₁ is at least 2t+1 from Proposition 2. Moreover, we can see that the minimum distance of C ₁ is 2t+1 from Proposition 3. Therefore, C ₁ is a negacyclic code with parameters $[n,n-2t,2t+1]_{q^{2}}$ and C ₁ is a MDS negacyclic code. Moreover, $C_{1}^{\perp}$ is also a MDS negacyclic code with parameters $[n,2t,n-2t+1]_{q^{2}}$ from Proposition 1.

Now, we suppose that the defining set of negacyclic code C ₂ is given by $Z_{2}=\bigcup_{i=1}^{k}C_{2i-1}$, where 1≤t≤k≤(q−1)/2. The generator polynomial of negacyclic code C ₂ is

$$g_2(x)=\bigl(x-\alpha^{1-2k}\bigr)\cdots\bigl(x- \alpha^{-1}\bigr) (x-\alpha)\cdots\bigl(x-\alpha^{2k-1}\bigr). $$

Similar to the above discussion, C ₂ is a MDS negacyclic code with parameters $[[n,n-2k,2k +1]]_{q^{2}}$. From Lemma 2 and Proposition 5, we can obtain that $C_{1}^{\perp_{h}} \subseteq C_{2}$. Then, from Theorem 1, there exist asymmetric quantum codes with parameters $[[(q^{2}+1)/2,(q^{2}+1)/2-2(t+k), (2k+1)/(2t+1)]]_{q^{2}}$. □

Remark 3

From Theorem 3, we can see that d _z +d _x =2t+1+2k+1=(q ²+1)/2−(q ²+1)/2+2(t+k)+2. Then, the constructed asymmetric quantum codes $[[(q^{2}+1)/2,(q^{2}+1)/2-2(t+k),(2k+1)/(2t+1)]]_{q^{2}}$ attain asymmetric Singleton bound. Hence, these asymmetric quantum codes are optimal.

Example 3

Let q=5, then n=13. We suppose that the defining set of negacyclic code C ₁ is given by Z ₁=C ₁={1,25}. Then, C ₁ is a MDS negacyclic code with parameters [13,11,3]₂₅. Moreover, $C_{1}^{\perp}$ is also a MDS negacyclic code with parameters [13,2,12]₂₅.

We can also suppose that the defining set of negacyclic code C ₂ is given by Z ₂=C ₁∪C ₃={1,25,3,23}. Then, C ₂ is a MDS negacyclic code with parameters [13,9,5]₂₅. From Theorem 3, there exists optimal asymmetric quantum code with parameters [[13,7,5/3]]₂₅.

Remark 4

The constructed optimal asymmetric quantum codes in this paper are different from optimal asymmetric quantum codes constructed by using the q ²-ary cyclotomic cosets in [29]. In [26], Guardia used Reed-Solomon codes to construct optimal asymmetric quantum codes with parameters [[p−1,p−2d+2,d/(d−1)]] _p , where p is a prime number. In [24], authors used generalized Reed-Solomon code to construct optimal asymmetric quantum codes with parameters $[[2^{m}+2,2^{m}-4,4/4]]_{2^{m}}$, where m is a positive integer. We can compare the new constructed optimal asymmetric quantum codes with the optimal asymmetric quantum codes in the [24] and [26]. We can find that the lower bound d _z of new constructed asymmetric quantum codes is greater than the lower bound for d _x . These quantum codes are able to correct quantum errors with great asymmetry.

5 Conclusions

In this paper, we construct two class of asymmetric quantum code by using negacyclic codes. The first class is the asymmetric quantum codes with parameters $[[q^{2}+1,q^{2}+1-2(t+k+1),(2k+2)/(2t+2)]]_{q^{2}}$, where 0≤t≤k≤(q−1)/2, $q \equiv1(\operatorname{mod}4)$, and k, t are positive integers. The second one is the asymmetric quantum codes with parameters $[[(q^{2}+1)/2,(q^{2}+1)/2-2(t+k),(2k+1)/(2t+1)]]_{q^{2}}$, where 1≤t≤k≤(q−1)/2, and k, t are positive integers. Here, the constructed asymmetric quantum codes are optimal. There is an interesting topic to consider encoding asymmetric quantum codes and encoding circuit.