New asymmetric quantum codes over $\textbf{F}_{q^2}$

Abstract

Two families of asymmetric quantum codes of length $n=q^{2m}-1$ over $\textbf{F}_{q^{2}}$ are constructed in this paper. By a detailed analysis of properties about $q^{2}$-ary cyclotomic cosets modulo n, Hermitian dual-containing conditions for a family of primitive narrow-sense BCH codes are presented. Consequently, a series of asymmetric quantum BCH codes are constructed via the CSS-like construction and pairs of nested BCH codes. The parameters of new asymmetric quantum codes presented here are better than those available in the literatures before, and the real Z-distance are much larger than $\delta _{max}+1$.

1 Introduction

Asymmetric quantum codes are an efficient coding scheme against the qubit-flip errors $\sigma _{x}$, phase-flip errors $\sigma _{z}$ and the combined qubit-phase flip errors $\sigma _{y}$ in quantum communication, which occurs with different probability. In most cases, the phase-flip errors occur more frequently than qubit-flip errors [1,2,3,4]. Therefore, putting asymmetric quantum codes to use in the asymmetric quantum channels is an issue worth considering. In the past 15 years, researchers have focused on the construction of asymmetric quantum codes, and obtained many codes with good parameters. Aly [5,6,7] derived some families of asymmetric quantum codes from imprimitive non-narrow-sense BCH codes and RS codes over finite fields. Sarvepalli et al. [8] utilized the construction of asymmetric quantum codes from dual-containing BCH codes and LDPC codes. L. Wang et al. [9] extend the characterization of nonadditive symmetric quantum codes to the asymmetric case, established a relationship of asymmetric quantum codes with classical error-correcting codes and obtained an asymptotic bound on asymmetric quantum codes from algebraic geometry codes. Ezerman et al. [10, 11] proposed CSS-like constructions and exploited pairs of nested linear codes under one of the Euclidean, trace Euclidean, Hermitian and trace Hermitian inner products constructed good parameters CSS-like asymmetric quantum codes. La Guardia [12,13,14] constructed many good asymmetric quantum codes which derived from Euclidean (Hermitian) dual-containing BCH codes by CSS construction. Recently, the constructions of asymmetric quantum codes have been studied by many researchers, and some families of new asymmetric quantum codes are constructed by utilizing constacyclic codes, negacyclic codes and MDS codes et al. [15,16,17,18,19,20]. Especially, Grassl [21] present a new propagation rule for CSS codes, and this construction applies to asymmetric quantum codes from the CSS construction as well.

In [22], the authors have utilized Euclidean dual-containing BCH code to construct asymmetric quantum codes of length $n=q^{m}-1$ over $\textbf{F}_{q}$ where $q\ge 5$. Reference [23] presented a class of special code length asymmetric quantum codes of length $n=\frac{q^{2m}-1}{q^{2}-1}$ over $\textbf{F}_{q^{2}}$. Inspired by the previous work mentioned [22, 23], in this paper, the construction of $q^{2}$-ary asymmetric quantum codes with code length $n=q^{2m}-1$($q\ge 3$) from Hermitian dual-containing primitive narrow-sense BCH codes were studied. And the lower bound $\delta$ on Z-distance and X-distance of these codes were provided here. Furthermore, we exactly calculate the parameters of two families of asymmetric quantum codes with special Z-distance, where our Z-distance can be much larger than $\delta _{max}+1$ given in Theorem 2.3 Ref. [24].

This paper is organized as follows. In Sect. 2, basic concepts on $q^{2}$-cyclotomic cosets, BCH codes and asymmetric quantum codes are reviewed. In Sect. 3, the conditions regarding Hermitian dual-containing BCH codes were discussed. In Sect. 4, a families of asymmetric quantum codes were constructed from Hermitian dual-containing BCH codes. In Sect. 5, we compared the parameters of the new codes with the ones available in the literature. Finally, the paper is summarized with a discussion in Sect. 6.

2 Preliminary

In this section, we will review the basic concepts on $q^{2}$-cyclotomic cosets of modulo n and BCH codes. Let q be a prime power, $\textbf{F}_{q^{2}}$ be the finite field with $q^{2}$ elements. For $n=q^{2m}-1$ denotes the code length, ${\mathcal {B}}^{\perp _{h}}$ denotes the Hermitian dual of BCH code ${\mathcal {B}}$, and an asymmetric quantum BCH code Q is denoted by $[[n,k,d_{z}/d_{x}]]$. For more details, refer to [6,7,8,9].

Definition 2.1

If $gcd(q,n)=1$, the $q^{2}$-cyclotomic coset of modulo n containing x is defined by

$$\begin{aligned} C_{x}=\left\{ {x,xq^{2},x(q^{2})^{2},...,x(q^{2})^{(k-1)}}\right\} (\bmod n) \end{aligned}$$

where k is the smallest positive integer such that $q^{2k}x\equiv x(\bmod n)$.

A cyclic code of length $n=q^{2m}-1$ over $\textbf{F}_{q^{2}}$ is called a BCH code with designed distance $\delta$ if its generator polynomial

$$\begin{aligned} g(x)=\prod _{z\in T}(x-\xi ^{z}), T=C_{b}\cup C_{b+1}\cup \cdots \cup C_{b+\delta -2}, \end{aligned}$$

where $C_{x}$ denotes the $q^{2}$-cyclotomic coset of modulo n containing x, $\xi$ is a primitive element of $\textbf{F}_{q^{2}}$ and $m=ord_{n}(q^{2})$ is the multiplicative order of q modulo n. According to the concept of defining set, such a BCH code can also be defined, see following Definition 2.2.

Definition 2.2

Let $gcd(q,n)=1$. If $\xi$ is a primitive n-th root of unity in some field containing $\textbf{F}_{q^{2}}$, $T=C_{b}\cup C_{b+1}\cup \cdots \cup C_{b+\delta -2}=T_{[b,b+\delta -2]}$, the cyclic code of length n with defining set T is called a BCH code of designed distance $\delta$. If $b=1$, ${\mathcal {C}}$ is called a narrow-sense BCH code, if $n=q^{2m}-1$, ${\mathcal {C}}$ is called primitive.

Lemma 2.1

If $gcd(q,n)=1$, ${\mathcal {C}}$ is a cyclic code over $\textbf{F}_{q^{2}}$ with defining set T, ${\mathcal {C}}^{\perp _{h}}\subseteq {\mathcal {C}}$ if and only if $T\cap T^{-q}=\emptyset$, where $T^{-q}=\left\{ n-qt(\bmod n)\mid t\in T\right\}$.

Let ${\mathcal {B}}_{1}$ and ${\mathcal {B}}_{2}$ be $q^{2}$-ary BCH codes of length n, and with defining set $T_{1}$ and $T_{2}$, respectively. From above Lemma 3.1, we know ${\mathcal {B}}^{\perp _{h}}_{1}\subseteq {\mathcal {B}}_{2}$ if and only if $T_{1}^{-q}\cap T_{2}=\emptyset$. Thus, we have

Lemma 2.2

Let ${\mathcal {B}}_{1}$ and ${\mathcal {B}}_{2}$ be $q^{2}$-ary BCH code with defining set $T_{1}$ and $T_{2}$, then ${\mathcal {B}}_{1}^{\perp _{h}}\subseteq {\mathcal {B}}_{2}$ if and only if $T_{1}^{\perp _{h}}\supseteq T_{2}$.

According to [2, 5, 6, 8], an asymmetric quantum code $[[n,k,d_{z}/d_{x}]]$ can control all $\lfloor \frac{d_{x}-1}{2}\rfloor$ qubit-flip errors and all $\lfloor \frac{d_{z}-1}{2}\rfloor$ phase-flip errors. At the same time, which can detect $d_{x}-1$ qubit-flip errors and $d_{z}-1$ phase-flip errors. Based on CSS construction, in 2013, Ezerman et al. proposed constructions are called CSS-like construction and utilized pairs of nested subfield linear codes under Hermitian inner products. The following Theorem 2.3 is CSS-like construction for asymmetric quantum codes.

Theorem 2.3

(CSS-Like Construction) For $i=1,2$, let ${\mathcal {C}}_{i}$ be a classical linear code with parameters $[n,k_{i},d_{i}]_{q^{2}}$. If ${\mathcal {C}}_{1}^{\perp _{h}}\subseteq {\mathcal {C}}_{2}$, then there exists an asymmetric quantum code with parameters $[[n,k(\delta _{1})+k(\delta _{2})-n, d_{z}/d_{x}]]_{q^{2}}$, where $\left\{ d_{x}, d_{z}\right\} =\left\{ d_{1}, d_{2}\right\}$.

3 Construction of Hermitian dual-containing BCH codes

In this section, the construction of asymmetric quantum codes derived from pairs of nested nonprimitive Hermitian dual-containing BCH codes were discussed as follow. If fixed n, denote $T=\bigcup ^{r}_{i=1}C_{i}$, define $u=\min \left\{ x \mid x \in T^{-q}\right\}$ and $v=\max \left\{ y \mid y\in T^{-q}\right\}$, Similar to [22], we have following Lemma 2.2.

Lemma 3.1

If ${\mathcal {B}}$ is a $q^{2}$-ary narrow-sense BCH code of length n with defining set $T=\cup _{i=1}^{r} C_{i}$ where $r<\delta _{max}$, and $T^{\perp _{h}}=Z_{n}\backslash T^{-q}$. Then ${\mathcal {B}}$ and ${\mathcal {B}}^{\perp _{h}}$ has design distance $\delta ({\mathcal {B}})=r+1$ and $\delta ({\mathcal {B}}^{\perp _{h}})\le \max \left\{ u,n-v-1\right\}$, respectively.

Proof

The defining set of BCH code ${\mathcal {B}}$ is $T=\cup _{i=1}^{r} C_{i}=T_{[1,r]}$, so we have the maximal design distance of narrow sense BCH code ${\mathcal {B}}$ is $r+1$, then we have $\delta ({\mathcal {B}})=r+1$.

Since $T^{\perp _{h}}=Z_{n}\setminus (T^{-q})$=$Z_{n}-\left\{ n-qx \mid x\in T\right\}$=$\left\{ 0,1,2,\cdots ,n-1\right\} -\left\{ u,u+s,\cdots ,v-t,v\right\} =\left\{ 0,1,2,\cdots ,u-1,\cdots ,v+1,\cdots ,n-1\right\}$, then $T^{\perp _{h}}$ contain u or $n-v-1$ integer. From Definition 2.2, thus we have $\delta ({\mathcal {B}}^{\perp _{h}})\le \max \left\{ u,n-v-1\right\}$.

For constructing asymmetric quantum codes via CSS-Like construction, let $q\ge 3$, the conditions regarding Hermitian dual-containing BCH codes of code length $n=q^{2m}-1$ were discussed as follow Theorem 3.2. $\square$

Theorem 3.2

Let $n=q^{2m}-1$, where $q\ge 3$ is prime power and $m\ge 4$.

• (I) If $1\le i\le q-2$, $2\le j\le q$. For $\delta _{1}=(i\cdot q+j)\cdot q-1$, $\delta _{1}< \delta _{2}\le q^{(2\,m-1)}-(i\cdot q+j)$, then there exist narrow sense BCH codes satisfying ${\mathcal {B}}_{1}^{\perp _{h} }(n,\delta _{1})\subseteq {\mathcal {B}}_{2}(n,\delta _{2})$.

• (II) If $2\le i\le q^{2}-q$. For $\delta _{1}=q^{3}-i$, $\delta _{1}< \delta _{2}\le i\cdot q^{(2m-3)}-1$, then there exist narrow sense BCH codes satisfying ${\mathcal {B}}_{1}^{\perp _{h} }(n,\delta _{1})\subseteq {\mathcal {B}}_{2}(n,\delta _{2})$.

• (III) If $1\le j\le q^{2}-1$

• (1) For $\delta _{1}=j\cdot q^{2\,s+1}-1$, $\delta _{1}<\delta _{2}\le q^{2\,m-2\,s-1}-j$, where $1\le s\le [\frac{m}{2}]-1$, then there exist narrow sense BCH codes satisfying ${\mathcal {B}}_{1}^{\perp _{h} }(n,\delta _{1})\subseteq {\mathcal {B}}_{2}(n,\delta _{2})$;

• (2) For $\delta _{1}= q^{2\,s+1}-j$, $\delta _{1}<\delta _{2}\le j\cdot q^{2\,m-2\,s-1}-1$, where $2\le s\le [\frac{m+1}{2}]-1$, then there exist narrow sense BCH codes satisfying ${\mathcal {B}}_{1}^{\perp _{h} }(n,\delta _{1})\subseteq {\mathcal {B}}_{2}(n,\delta _{2})$.

Proof

Here, we only show item (II) since the other cases are similar.

Let $n=q^{2m}-1$, $q\ge 3$. Since $\delta _{1}=q^{3}-i$ where $2\le i\le q^{2}-q$, then narrow sense BCH code ${\mathcal {B}}_{1}(n,\delta _{1})$ with defining set $T_{1}=\cup _{t=1}^{\delta _{1}-1} C_{t}=\cup _{t=1}^{q^{3}-i-1} C_{t}$. Let $T^{-q}_{1}=\left\{ n-qx_{i} \mid x_{i}\in T_{1}\right\}$, since ${\mathcal {B}}_{1}^{\perp _{h}}$ with defining set $T_{1}^{\perp _{h}}=Z_{n} \backslash T^{-q}_{1}=\left\{ 0,1,2,\cdots ,n-1\right\} -\left\{ n-qx_{i} \mid x_{i}\in T_{1}\right\}$.

If $\delta _{1}<\delta _{2}\le min\left\{ n-qx_{i} \mid x_{i}\in T_{1}\right\} =i\cdot q^{(2\,m-3)}-1$, then ${\mathcal {B}}_{2}(n,\delta _{2})$ with defining set $T_{2}=\cup _{t=1}^{\delta _{2}-1} C_{t}$. We can assume $T_{2}=C_{1}\cup C_{2}\cup \cdots C_{\delta _{2}-1}$. If for any $j\in T_{2}$, from Lemma 2.2, we can deduce that $j\not \in T^{-q}_{1}$, that is to say $T_{1}^{-q}\cap T_{2}=\emptyset$, then $j\in Zn\backslash T^{-q}_{1}$, thus one can deduce that $T_{2}\subseteq T^{\perp _{h}}_{1}$. From Lemma 2.1, we can conclude ${\mathcal {B}}_{1}^{\perp _{h}}(n,\delta _{1})\subseteq {\mathcal {B}}_{2}(n,\delta _{2})$ holds.

Theorem 3.2 provides the maximum designed distance of two nested BCH codes satisfying Hermitian dual-containing conditions. However, it is a well-known hard problem to calculate the dimensions of asymmetric quantum codes. Therefore, next, the dimensions of two families of BCH codes with special maximum designed distances can be computed. The following two cases discussed here because the definition set of these BCH codes are influenced by the parity of m. $\square$

Case I. m is even

In this case, let $m\ge 4$ be even, $n=q^{2m}-1$. For $q\le i\le q^{2}-1$, $1\le j_{1}\le i$ and $1\le j_{2}\le q^{2t-2}$, denote the defining set of BCH code is $T(\delta )=\cup _{i=1}^{\delta } C_{i}=T_{[1,\delta ]}$, if

$$\begin{aligned} \mid T(\delta ) \mid =\left\{ \begin{array} {ll} m(\delta -(q^{2t}+\sum _{k=0}^{q^{2}-3}k+2(q^{2}-1))+t &{} \delta =q^{2t+2}+q^{2}\\ m(\delta -((i+1)\cdot q^{2t-2}+\sum _{k=0}^{i-2}k+(i+j_{1}-1)))&{} \delta =(i+1)\cdot q^{2t}+j_{1}\\ m(\delta -(i\cdot q^{2t-2}+\sum _{k=0}^{i-2}k+(i+j_{2}-1))) &{} \delta = i\cdot q^{2t}+j_{2}\cdot q^{2}\\ \end{array} \right. \end{aligned}$$

then there exists a narrow-sense BCH code with parameters $[n,n- \mid T(\delta ) \mid ,\ge \delta ]_{q^{2}}$.

Example 3.1

Take $q=5, m=4$, so $n=390624$. If $T(\delta )=4\cdot (1251-51), 4\cdot (1877-78), 4\cdot (2503-106), 4\cdot (3129-135), 4\cdot (3755-165), 4\cdot (4381-196), 4\cdot (5007-228), 4\cdot (5633-261), 4\cdot (6259-295), 4\cdot (15649-925), 4\cdot (15650-926)+2$, then there exist BCH codes with parameters $[n,385968,d\ge 1252]_{25}, [n,383428,d\ge 1878]_{25}, [n,381036,d\ge 2504]_{25}, [n,378648,d\ge 3130]_{25}, [n,376264,d\ge 3756]_{25}, [n,373884,d\ge 4382]_{25}, [n,371508,d\ge 5008]_{25}, [n,369136,d\ge 5634]_{25}, [n,366768,d\ge 6260]_{25}, [n,331728,d\ge 15650]_{25}, [n,331726,d\ge 15651]_{25}$, respectively.

Summarizing the analysis above, applying Theorem 3.2 and above the corresponding conclusion, then Corollary 3.3 follows.

Corollary 3.3

Let $m=2t(t\ge 2)$, $n=q^{2m}-1$ with $q\ge 3$ be a power of a prime. If $\delta _{1}=q^{2t-1}-q$, ${\mathcal {B}}_{1}=[n, n-m\lceil (\delta _{1}-1)(1-q^{-2})\rceil , \ge \delta _{1}]$, then there exist narrow sense BCH codes ${\mathcal {B}}_{2}$ with parameters

• (I) $[n, n-m(\delta _{2}-(q^{2t}+\sum _{k=0}^{q^{2}-3}k+q^{2}-3)), \ge \delta _{2}]$ satisfying ${\mathcal {B}}_{1}^{\perp _{h} }(n,\delta _{1})\subseteq {\mathcal {B}}_{2}(n,\delta _{2})$, where $\delta _{1}< \delta _{2}\le q^{2t+2}-1$.

• (II) $[n, n-m(\delta _{2}-((i+1)\cdot q^{2t-2}+\sum _{k=0}^{i-2}k+(i+j-1))), \ge \delta _{2}]$ satisfying ${\mathcal {B}}_{1}^{\perp _{h} }(n,\delta _{1})\subseteq {\mathcal {B}}_{2}(n,\delta _{2})$, where $q\le i\le q^{2}-1$, $1\le j\le i$ and $\delta _{1}< \delta _{2}\le (i+1)\cdot q^{2t}+j$.

• (III) $[n, n-m(\delta _{2}-(i\cdot q^{2t-2}+\sum _{k=0}^{i-2}k+(i+j-1))), \ge \delta _{2}]$ satisfying ${\mathcal {B}}_{1}^{\perp _{h} }(n,\delta _{1})\subseteq {\mathcal {B}}_{2}(n,\delta _{2})$, where $q\le i\le q^{2}-1$, $1\le j\le q^{2t-2}$ and $\delta _{1}< \delta _{2}\le i\cdot q^{2t}+j\cdot q^{2}$.

Case II. m is odd

In this case, let $m\ge 5$ be odd, $n=q^{2m}-1$. For $\delta _{0}=q^{2t+2}$; $1\le i_{1}\le q^{2}-1$, $1\le j_{1}\le q^{2t-2}-i$ and $\delta _{1}=i_{1}\cdot q^{2t+2}+(i_{1}+j_{1})\cdot q^{2}$; $1\le i_{2}\le q^{2}-2$, $1\le j_{2}\le q^{2}-1$, $1\le s\le q^{2t-2}$ and $\delta _{2}=i_{2}\cdot q^{2t+2}+j_{2}\cdot q^{2t} +s\cdot q^{2}$; $0\le i_{3}\le q^{2}-2$, $1\le j_{3}\le (i_{3}+1)\cdot q^{2}$ and $\delta _{3}=(i_{3}+1)\cdot q^{2t+2}+j_{3}$, denote the defining set of BCH code is $T(\delta )=\cup _{i=1}^{\delta } C_{i}=T_{[1,\delta ]}$, if

$$\begin{aligned} \mid T(\delta ) \mid =\left\{ \begin{array} {ll} m(\delta _{0}-q^{2t})\\ m(\delta _{1}-(i_{1}\cdot q^{2t}+\frac{i_{1}^{2}+i_{1}}{2}\cdot q^{2}+\sum _{k=0}^{i_{1}-1}k\cdot (q^{2}-2)+j_{1}))\\ m(\delta _{2}-(i_{2}\cdot q^{2t}+\frac{i_{2}^{2}+i_{2}}{2}\cdot q^{2}+j_{2}\cdot q^{2t-2}\\ \quad +\sum _{k=0}^{i_{2}-1}k\cdot (q^{2}-2)+i_{2}\cdot (j_{2}-1)+s))\\ m(\delta _{3}-((i_{3}+1)\cdot q^{2t}+\frac{i_{3}^{2}+i_{3}}{2}\cdot q^{2}+(q^{2}-2)\cdot \sum _{k=0}^{i_{3}}k+j_{3}))\\ \end{array} \right. \end{aligned}$$

then there exists a narrow-sense BCH code with parameters $[n,n- \mid T(\delta ) \mid ,\ge \delta ]_{q^{2}}$.

Example 3.2

Take $q=4, m=5$, so $n=1048575$. If $T(\delta )=5\cdot (4112-272), 5\cdot (4353-288), 5\cdot (4609-305), 5\cdot (4865-322), 5\cdot (5121-339), 5\cdot (5377-356), 5\cdot (5633-373), 5\cdot (5889-390), 5\cdot (6145-407), 5\cdot (9986-698), 5\cdot (10498-734), 5\cdot (11010-770), 5\cdot (11266-788), 5\cdot (11522-806, 5\cdot (11778-824), 5\cdot (12336-906), 5\cdot (61680-7230)$, then there exist BCH codes with parameters $[n,1029375,d\ge 4113]_{16}, [n,1028250,d\ge 4354]_{16}, [n,1027055,d\ge 4610]_{16}, [n,1025860,d\ge 4866]_{16}, [n,1024665,d\ge 5122]_{16}, [n,1023470,d\ge 5378]_{16}, [n,1022275,d\ge 5634]_{16}, [n,1021080,d\ge 5890]_{16}, [n,1019885,d\ge 6146]_{16}, [n,1002135,d\ge 9987]_{16}, [n,999755,d\ge 10499]_{16}, [n,997375,d\ge 11011]_{16}, [n,996185,d\ge 11267]_{16}, [n,994995,d\ge 11523]_{16}, [n,993805,d\ge 11779]_{16}, [n,991425,d\ge 12337]_{16}, [n,776325,d\ge 61681]_{16}$, respectively.

Summarizing the analysis above, applying Theorem 3.2 and above the corresponding conclusion, then Corollary 3.4 follows.

Corollary 3.4

Let $m=2t+1(t\ge 2)$, $n=q^{2m}-1$ with $q\ge 3$ be a power of a prime. If $\delta _{1}=q^{2t-1}-q$, ${\mathcal {B}}_{1}=[n, n-m\lceil (\delta _{1}-1)(1-q^{-2})\rceil , \ge \delta _{1}]$, then there exist narrow sense BCH codes ${\mathcal {B}}_{2}$ with parameters

• (I) $[n, n-m(\delta _{2}-q^{2t}), \ge \delta _{2}]$ satisfying ${\mathcal {B}}_{1}^{\perp _{h} }(n,\delta _{1})\subseteq {\mathcal {B}}_{2}(n,\delta _{2})$, where $\delta _{1}< \delta _{2}\le q^{2t+2}$.

• (II) $[n, n-m(\delta _{2}-(i\cdot q^{2t}+\frac{i^{2}+i}{2}\cdot q^{2}+\sum _{k=0}^{i-1}k\cdot (q^{2}-2)+j)), \ge \delta _{2}]$ satisfying ${\mathcal {B}}_{1}^{\perp _{h} }(n,\delta _{1})\subseteq {\mathcal {B}}_{2}(n,\delta _{2})$, where $1\le i\le q^{2}-1$, $1\le j\le q^{2t-2}-i$ and $\delta _{1}< \delta _{2}\le i\cdot q^{2t+2}+(i+j)\cdot q^{2}$.

• (III) $[n, n-m(\delta _{2}-(i\cdot q^{2t}+\frac{i^{2}+i}{2}\cdot q^{2}+j\cdot q^{2t-2}+\sum _{k=0}^{i-1}k\cdot (q^{2}-2)+i\cdot (j-1)+s)), \ge \delta _{2}]$ satisfying ${\mathcal {B}}_{1}^{\perp _{h} }(n,\delta _{1})\subseteq {\mathcal {B}}_{2}(n,\delta _{2})$, where $1\le i\le q^{2}-2$, $1\le j\le q^{2}-1$, $1\le s\le q^{2t-2}$ and $\delta _{1}< \delta _{2}\le i\cdot q^{2t+2}+j\cdot q^{2t} +s\cdot q^{2}$.

• (IV) $[n, n-m(\delta _{2}-((i+1)\cdot q^{2t}+\frac{i^{2}+i}{2}\cdot q^{2}+(q^{2}-2)\cdot \sum _{k=0}^{i}k+j)), \ge \delta _{2}]$ satisfying ${\mathcal {B}}_{1}^{\perp _{h} }(n,\delta _{1})\subseteq {\mathcal {B}}_{2}(n,\delta _{2})$, where $0\le i\le q^{2}-2$, $1\le j\le (i+1)\cdot q^{2}$, and $\delta _{1}< \delta _{2}\le (i+1)\cdot q^{2t+2}+j$.

4 Construction of asymmetric quantum codes

In this section, applying CSS-like construction and the main results above section, some new asymmetric quantum codes can be constructed, and their dimension can be exactly calculated. In the following Theorem 4.1, our main construction results can be provided.

Theorem 4.1

Let $n=q^{2m}-1$, where $q\ge 3$ is a prime power and $m\ge 4$.

• (I) For $\delta _{1}=(i\cdot q+j)\cdot q-1$, $\delta _{1}< \delta _{2}\le q^{(2\,m-1)}-(i\cdot q+j)$, then there exist asymmetric quantum codes $[[n,n- \mid T(\delta _{1}) \mid - \mid T({\delta _{2}}) \mid ,d_{z}\ge \delta _{2}/d_{x}\ge \delta _{1}]]_{q^{2}}$, where $1\le i\le q-2$, $2\le j\le q$.

• (II) For $\delta _{1}=q^{3}-i$, $\delta _{1}< \delta _{2}\le i\cdot q^{(2\,m-3)}-1$, then there exist asymmetric quantum codes $[[n,n- \mid T(\delta _{1}) \mid - \mid T(\delta _{2}) \mid ,d_{z}\ge \delta _{2}/d_{x}\ge \delta _{1}]]_{q^{2}}$, where $2\le i\le q^{2}-q$.

• (III) If $1\le j\le q^{2}-1$.

• (1) For $\delta _{1}=j\cdot q^{2\,s+1}-1$, $\delta _{1}<\delta _{2}\le q^{2\,m-2\,s-1}-j$, then there exist asymmetric quantum codes $[[n,n- \mid T(\delta _{1}) \mid - \mid T(\delta _{2}) \mid ,d_{z}\ge \delta _{2}/d_{x}\ge \delta _{1}]]_{q^{2}}$, where $1\le s\le [\frac{m}{2}]-1$;

• (2) For $\delta _{1}= q^{2\,s+1}-j$, $\delta _{1}<\delta _{2}\le j\cdot q^{2\,m-2\,s-1}-1$, then there exist asymmetric quantum codes $[[n,n- \mid T(\delta _{1}) \mid - \mid T(\delta _{2}) \mid ,d_{z}\ge \delta _{2}/d_{x}\ge \delta _{1}]]_{q^{2}}$, where $2\le s\le [\frac{m+1}{2}]-1$.

Proof

We only prove item (II) since the other constructions are similar.

Let ${\mathcal {B}}_{1}$ be the narrow-sense BCH code over $\textbf{F}_{q^{2}}$ of length $n=q^{2m}-1$. Using $T(\delta _{1})=\bigcup _{i=1}^{\delta _{1}-1}C_{i}$ to denote the defining set of BCH code, and the cardinality of $T(\delta _{1})$ as $\mid T(\delta _{1}) \mid$. For $\delta _{1}=q^{3}-i$ where $2\le i\le q^{2}-q$, then there exists a narrow sense BCH code with parameters $[n,n- \mid T(\delta _{1}) \mid ,q^{3}-i]$. Next, consider another BCH code ${\mathcal {B}}_{2}$ with parameters $[n,n- \mid T(\delta _{2}) \mid ,\delta _{2}]$.

According to Theorem 3.2, we know that if $\delta _{1}< \delta _{2}\le i\cdot q^{(2\,m-3)}-1$, then there exist narrow-sense BCH codes satisfying ${\mathcal {B}}_{1}^{\perp _{h} }(n,\delta _{1})\subseteq {\mathcal {B}}_{2}(n,\delta _{2})$. Hence, applying the CSS-Like construction in Theorem 2.3, and use the parameters of $B_{1}(n,\delta _{1})$ and $B_{2}(n,\delta _{2})$, $q^{2}$-ary asymmetric quantum codes $[[n,n- \mid T(\delta _{1}) \mid - \mid T(\delta _{2}) \mid ,d_{z}\ge \delta _{2}/d_{x}\ge \delta _{1}]]_{q^{2}}$ can be constructed. $\square$

Summarizing the above discussion, we can conclude (II) holds.

Remark 1

We presented the lower bound $\delta$ on Z-distance and X-distance of asymmetric quantum codes of length $n=q^{2m}-1$ where $q\ge 3$. Obviously, our Z-distance are much larger than X-distance as well as are much larger than $\delta _{max}+1$ in [22]. However, Theorem 4.1 does not give the exact dimensions of these code, owning to complex to calculate the exact dimensions for all $\delta$. So we denote the cardinality of $T(\delta )$ as $\mid T(\delta ) \mid$, and the dimension as $n-\mid T(\delta ) \mid$. But, for fixed the special values of $d_{z}$ and $d_{x}$, we will calculate the exact parameters of two families of asymmetric quantum codes in the following Corollary.

Corollary 4.2

Let $m=2t(t\ge 2)$, $n=q^{2m}-1$ with $q\ge 3$ be a power of a prime. Then there exist asymmetric quantum codes with parameters

• (I) $[[n,n-m(\lceil (q^{2t-1}-q)(1-q^{-2})\rceil +(q^{2t-1}-1)(q^{2}-1)-\sum _{k=0}^{q^{2}-3}k+1), d_{z}\ge q^{2t+2}-1/d_{x}\ge q^{2t-1}-q]]_{q^{2}}$.

• (II) $[[n,n-m(\lceil (q^{2t-1}-q)(1-q^{-2})\rceil +(i+1)\cdot (q^{2t}-q^{2t-2})-\sum _{k=0}^{i-2}k-i+1), d_{z}\ge (i+1)\cdot q^{2t}+j/d_{x}\ge q^{2t-1}-q]]_{q^{2}}$, where $q\le i\le q^{2}-1$, $1\le j\le i$.

• (III) $[[n,n-m(\lceil (q^{2t-1}-q)(1-q^{-2})\rceil +i\cdot (q^{2t}-q^{2t-2}-1)+j\cdot (q^{2}-1)-\sum _{k=0}^{i-2}k+1), d_{z}\ge i\cdot q^{2t}+j\cdot q^{2}/d_{x}\ge q^{2t-1}-q]]_{q^{2}}$, where $q\le i\le q^{2}-1$, $1\le j\le q^{2(t-1)}$.

Corollary 4.3

Let $m=2t+1(t\ge 2)$, $n=q^{2m}-1$ with $q\ge 3$ be a power of a prime. Then there exist asymmetric quantum codes with parameters

• (I) $[[n,n-m(\lceil (q^{2t-1}-q)(1-q^{-2})\rceil + (q^{2t+2}-q^{2t})), d_{z}\ge q^{2t+2}/d_{x}\ge q^{2t-1}-q]]_{q^{2}}$.

• (II) $[[n,n-m(\lceil (q^{2t-1}-q)(1-q^{-2})\rceil +i\cdot q^{2t}(q^{2}-1)+(i+j-\frac{i^{2}+i}{2})\cdot q^{2}-(q^{2}-2)\sum _{k=0}^{i-1}k-j), d_{z}\ge i\cdot q^{2t+2}+(i+j)\cdot q^{2}/d_{x}\ge q^{2t-1}-q]]_{q^{2}}$, where $1\le i\le q^{2}-1$, $1\le j\le q^{2t-2}-i$.

• (III) $[[n,n-m(\lceil (q^{2t-1}-q)(1-q^{-2})\rceil +(i\cdot q^{2t}+j\cdot q^{2t-2})(q^{2}-1)+(s-\frac{i^{2}+i}{2})\cdot q^{2}-(q^{2}-2)\sum _{k=0}^{i-1}k+i\cdot (j-1)-s, d_{z}\ge i\cdot q^{2t+2}+j\cdot q^{2t}+s\cdot q^{2}/d_{x}\ge q^{2t-1}-q]]_{q^{2}}$, where $1\le i\le q^{2}-2$, $1\le j\le q^{2}-1$, $1\le s\le q^{2t-2}$.

• (IV) $[[n,n-m(\lceil (q^{2t-1}-q)(1-q^{-2})\rceil +(i+1)\cdot q^{2t}(q^{2}-1)-\frac{i^{2}+i}{2}\cdot q^{2}-(q^{2}-2)\sum _{k=0}^{i}k), d_{z}\ge (i+1)\cdot q^{2t+2}+j/d_{x}\ge q^{2t-1}-q]]_{q^{2}}$, where $1\le i\le q^{2}-2$, $1\le j\le (i+1)q^{2}$.

Example 4.1

Take $q=3, m=4$, so $n=6560$. The following Table 1 lists some asymmetric quantum BCH codes derived from Corollary 4.2.

Table 1 Sample parameters of asymmetric quantum codes for $m=4$

Example 4.2

Take $q=3, m=5$, so $n=59048$. Table 2 lists some asymmetric quantum BCH codes derived from Corollary 4.3.

Table 2 Sample parameters of asymmetric quantum codes for $m=5$

Table 3 Compare the real Z-distances $d_{z}$ and $\delta _{max}$

Table 4 Sample parameters of asymmetric quantum codes $[[n,k,d_{z}/d_{x}]]_{q^{2}}$

Remark 2

Tables 1 and 2 listed some new asymmetric quantum codes which given in Corollary 4.2 and Corollary 4.4. For $q=3$ and $m=4,5$, some of the Z-distances of our asymmetric quantum codes are much larger than X-distances. In general, researchers use the code rate to characterize the performance of a code. The so-called code rate is the ratio $\frac{k}{n}$, where n is code length, k is the dimension. However, in order to show the error-correcting ability to the phase-flip errors and qubit-flip errors, here we use the factor $\rho =\frac{d_{z}}{d_{x}}$ given in [5] to compare $d_{z}$ and $d_{x}$. Therefore, if $d_{z}>d_{x}$, then the asymmetric quantum codes has a factor great than one. Hence, the phase-flip errors affect the quantum system more than qubit-flip errors do. In this paper, we would like to increase the factor $\rho$ and dimension k of the codes. For example, for $n=6560$, if $d_{x}\ge 6,10,15,19,22,23,24,25,26$, our Z-distance can reach $d_{z}\ge 2185/d_{x}\ge 6, d_{z}\ge 2184/d_{x}\ge 10, d_{z}\ge 2182/d_{x}\ge 15, d_{z}\ge 2181/d_{x}\ge 19, d_{z}\ge 1457/d_{x}\ge 22, d_{z}\ge 1214/d_{x}\ge 23, d_{z}\ge 971/d_{x}\ge 24, d_{z}\ge 728/d_{x}\ge 25, d_{z}\ge 485/d_{x}\ge 26$. It is obvious that the factor $\rho$ can reach 364.16, 218.40, 145.46, 114.78, 66.22, 52.78, 40.45, 29.12, 18.65, respectively. That is to say, the error-correcting ability to the phase-flip errors of our asymmetric quantum BCH codes can be much better then qubit-flip errors.

5 Parameters analysis

In this section, we compare the parameters of the new asymmetric quantum codes and the ones available in the literature. In the following tables, the parameters of the asymmetric quantum codes shown in [14](Theorem 4.7) are denoted by $[[n,k',d_{z'}/d_{x'}]]_{q^{2}}$, and the new code parameters are denoted by $[[n,k,d_{z}/d_{x}]]_{q^{2}}$. Furthermore, our Z-distances are denoted by $d_{z}$, $\delta _{max}$ are the maximal designed distance of dual containing narrow-sense BCH code in [24].

Remark 3

Table 3 has showed some Z-distances which given in Theorem 4.1. For example, for $q=3,m=4,5,6$, if $\delta _{max}=235$, 242, 2179, our Z-distance can reach 2185, 19681 and 177145, respectively, it is obviously larger than $\delta _{max}$. In a word, we use Table 4 to present evidences of the real Z-distance of our asymmetric quantum codes, which are much larger than $\delta _{max}+1$.

Remark 4

Table 4 listed some new asymmetric quantum codes which given in Corollary 4.2 and Corollary 4.3. For $m=3,4,5$ and $q=4,5$, some of the parameters of our asymmetric quantum codes are better than those available in [14]. For example, let $q=5,m=4,n=390624$, for $d_{x}\ge 26$ and $d_{z}\ge 27$, 28, 29, 30, 31, 46, 47, 48, the dimensions of our asymmetric quantum codes are larger than those available in [14]. What is more, some of the asymmetric quantum codes are new ones and are not included in the literature. For example, if $d_{x}\ge 26$, our Z-distance can reach 77353, which are new and are much greater than the results in the literature. Hence, the error-correcting ability to the phase-flip errors can be further improved. Furthermore, in order to calculate the dimensions, we restrict $d_{z}\ge 77353/d_{x}\ge 35$ and $d_{z}\ge 77353/d_{x}\ge 45$, then one can easily construct two asymmetric quantum codes $[[n,161720,d_{z}\ge 77353/d_{x}\ge 35]]_{25}$ and $[[n,161680,d_{z}\ge 77353/d_{x}\ge 45]]_{25}$. However, if the value of $d_{x}\ge 35,45$, then our Z-distance can reach 78118, 78116, respectively.

6 Conclusion

Using the CSS-like construction, we have constructed several families of $q^{2}$-ary asymmetric quantum codes of length $n=q^{2m}-1$ derived from Hermitian dual-containing primitive narrow-sense BCH codes. Some of these codes have parameters better than the ones available in the literature. Furthermore, the real Z-distance are much larger than X-distance and $\delta _{max}+1$. And others are not included in the literature, which are new ones. Unfortunately, for fixed values of the length n, we only give partial results, the discussions of asymmetric quantum codes constructed from pairs of nested BCH codes for all $\delta$ may be a little complex. It would be interesting to construct good asymmetric quantum codes from cyclic codes of other lengths.