On the Construction of Optimal Asymmetric Quantum Codes*** E-mail addresses:
liqiwangg@163.com(L.Wang), zhushixin@hfut.edu.cn(S.Zhu).

Liqi Wang, Shixin Zhu

School of Mathematics, Hefei University of Technology Hefei 230009, Anhui,
P.R.China

Constacyclic codes are important classes of linear codes that have been
applied to the construction of quantum codes. Six new families of asymmetric
quantum codes derived from constacyclic codes are constructed in this paper.
Moreover, the constructed asymmetric quantum codes are optimal and different
from the codes available in the literature.

_Keywords_ : Quantum codes; Asymmetric quantum codes; constacyclic codes

1 Introduction

Quantum error-correcting codes have gained prominence since the initial
discovery of Shor1 and Steane2. In 1998, Calderbank et al.3 presented
systematic methods to construct binary quantum codes, called stabilizer codes
or additive codes, from classical error-correcting codes. Since then the field
has made rapid progress, many good binary quantum codes were constructed by
using classical error-correcting codes, such as BCH codes, Reed-Solomon codes,
Reed-Muller codes, and algebraic geometric codes (see Refs. 4-8). The theory
was later extended to the nonbinary case, since the realization that nonbinary
quantum codes can use fault-tolerant quantum computation (see Refs. 9-13).
Recently, a number of new types of quantum codes, such as convolutional
quantum codes, subsystem quantum codes have been studied and the stabilizer
method has been extended to these variations of quantum codes (see Refs.
14-15).

Asymmetric quantum error-correcting codes(AQECC) are quantum codes defined
over quantum channels where qudit-flip errors and phase-shift errors may have
different probabilities. AQECC was first studied by Steane in [16]. Since
then, the construction of quantum codes have extended to asymmetric quantum
channels. Loffe et al.17 utilize BCH codes to correct qubit-flip errors and
LDPC codes to correct more frequently phase-shift errors. AQECC derived from
LDPC codes and BCH codes were also constructed in [18-21]. Stephens et al.22
consider the investigation of AQECC via code conversion. Wang et al.23
presented the construction of nonadditive AQECC as well as constructions of
asymptotically good AQECC derived from algebraic-geometry codes. Ezerman et
al.24 presented the construction of AQECC under the trace Hermitian inner
product. Ezerman and Ling25 studied two systematic construction of AQECC. Chee
et al.26 constructed pure $q$-ary AQECC and some of these codes attain the
quantum Singleton bound. Recently, Ezerman et al.27 also studied the pure
AQECC and some optimal codes are obtained. A variety of the constructions of
new AQECC were presented in [28-31].

AQECC attain the quantum Singleton bound are called optimal. Until now, just
several families of optimal AQECC have been constructed. Chee et al.26
constructed optimal AQECC with parameters $[[2^{m}+2,2^{m}-4,4/4]]_{2^{m}}$
using generalized RS codes, where $m$ is a positive integer. Guardia 28
constructed optimal AQECC with parameters $[[p-1,p-2d+2,d/(d-1)]]_{p}$, where
$p$ is a prime number. Qian32 constructed optimal AQECC with parameters
$[[q^{2}+1,q^{2}+1-2(k+i+2),(2k+3)/(2i+3)]]_{q^{2}}$, where $0\leq k\leq i\leq
q/2-1$. Recently, Chen et al.33 constructed two families of optimal AQECC
derived from negacyclic codes. In this paper, we constructed six new families
of optimal AQECC derived from constacyclic codes. They are given by

(1) $[[n,n-s-t,(s+1)/(t+1)]]_{q^{2}},$ where $n=(q^{2}-1)/2$ and $1\leq t\leq
s\leq q-1$.

(2) $[[n,n-s-t,(s+1)/(t+1)]]_{q^{2}},$ where $n=\lambda(q-1)$,
$\lambda=(q+1)/r$, $r\neq 2$ is an even divisor of $q+1$ and $1\leq t\leq
s\leq(q-1)/2.$

(3) $[[n,n-s-t,(s+1)/(t+1)]]_{q^{2}},$ where $n=\lambda(q+1)$, $\lambda$ is an
odd divisor of $q-1$ and $1\leq t\leq s\leq(q-1)/2+\lambda.$

(4) $[[n,n-s-t,(s+1)/(t+1)]]_{q^{2}},$ where $n=2\lambda(q+1)$, $\lambda$ is
an odd divisor of $q-1$, $q\ \equiv\ 1\ mod\ 4$, and $1\leq t\leq
s\leq(q-1)/2+2\lambda.$

(5) $[[n,n-2(s+t+1),(2s+2)/(2t+2)]]_{q^{2}}$, where $n=(q^{2}+1)/5$, $q=20m+3$
or $20m+7$ with $m$ a positive integer, and $0\leq t\leq s\leq(q+1)/4.$

(6) $[[n,n-2(s+t+1),(2s+2)/(2t+2)]]_{q^{2}}$, where $n=(q^{2}+1)/5$, $q=20m-3$
or $20m-7$ with $m$ a positive integer, and $0\leq t\leq s\leq(q-1)/4.$

The paper is organized as follows. In Section 2, some definitions and basic
results of constacyclic codes are reviewed. In Section 3, we recall some basic
definitions of asymmetric quantum codes. In Section 4, six classes of optimal
asymmetric quantum codes are constructed. Section 5 concludes the paper.

2 Review of Constacyclic Codes

Let $\mathbb{F}_{q^{2}}$ be the Galois field with $q^{2}$ elements, where $q$
is a power of a prime $p$. A linear $[n,k]$ code $C$ over $\mathbb{F}_{q^{2}}$
is a $k$-dimensional subspace of $\mathbb{F}_{q^{2}}^{n}$. A linear code $C$
of length $n$ over $\mathbb{F}_{q^{2}}$ is called $\eta$-constacyclic if it is
invariant under the $\eta$-constacyclic shift of $\mathbb{F}_{q^{2}}^{n}$:

$(c_{0},c_{1},\ldots,c_{n-1})\rightarrow(\eta c_{n-1},c_{0},\ldots,c_{n-2}),$

where $\eta$ is a nonzero element of $\mathbb{F}_{q^{2}}$. Each codeword
$\textbf{c}=(c_{0},c_{1},\ldots,c_{n-1})$ is customarily identified with its
polynomial representation $c(x)=c_{0}+c_{1}x+\cdots+c_{n-1}x^{n-1}$, and the
code $C$ is in turn identified with the set of all polynomial representations
of its codewords. Then in the ring $\frac{\mathbb{F}_{q^{2}}[x]}{\langle
x^{n}-\eta\rangle},$ $xc(x)$ corresponds to a $\eta$-constacyclic shift of
$c(x)$. It is well known that a linear code $C$ of length $n$ over
$\mathbb{F}_{q^{2}}$ is $\eta$-constacyclic if and only if $C$ is an ideal of
the quotient ring $\frac{\mathbb{F}_{q^{2}}[x]}{\langle x^{n}-\eta\rangle}.$
Moreover, $\frac{\mathbb{F}_{q^{2}}[x]}{\langle x^{n}-\eta\rangle}$ is a
principal ideal ring, whose ideals are generated by monic factors of
$x^{n}-\eta$, i.e., $C=\langle f(x)\rangle$ and $f(x)|(x^{n}-\eta)$.

The Hermitian inner product is defined as

$\langle\textbf{x},\textbf{y}\rangle=x_{0}\bar{y}_{0}+x_{1}\bar{y}_{1}+\cdots+x_{n-1}\bar{y}_{n-1}\in\mathbb{F}_{q^{2}},$

where $\textbf{x}=(x_{0},x_{1},\ldots,x_{n-1})\in\mathbb{F}_{q^{2}}^{n}$,
$\textbf{y}=(y_{0},y_{1},\ldots,y_{n-1})\in\mathbb{F}_{q^{2}}^{n}$, and
$\bar{y}_{i}=y_{i}^{q}$. The vectors x and y are called orthogonal with
respect to the Hermitian inner product if
$\langle\textbf{x},\textbf{y}\rangle=0.$ For a $q^{2}$-ary linear code $C$ of
length $n$, the Hermitian dual code of $C$ is defined as

$C^{\bot_{H}}=\\{\textbf{x}\in\mathbb{F}_{q^{2}}^{n}|\langle\textbf{x},\textbf{y}\rangle=0\
for\ all\ \textbf{y}\in C\\}.$

A linear code $C$ of length $n$ over $\mathbb{F}_{q^{2}}$ is called Hermitian
self-orthogonal if $C\subseteq C^{\bot_{H}}$, and it is called Hermitian self-
dual if $C=C^{\bot_{H}}.$

We assume $\gcd(q,n)=1$. Let $\delta$ be a primitive $rn$th root of unity in
some extension field of $\mathbb{F}_{q^{2}}$ such that $\delta^{n}=\eta$. Let
$\xi=\delta^{r},$ then $\xi$ is a primitive $n$th root of unity. Hence,

$x^{n}-\eta=\Pi_{i=0}^{n-1}(x-\delta\xi^{i})=\Pi_{i=0}^{n-1}(x-\delta^{1+ir}).$

Let $\Omega=\\{1+ir|0\leq i\leq n-1\\}.$ For each $j\in\Omega,$ let $C_{j}$ be
the $q^{2}$-cyclotomic coset modulo $rn$ containing $j$. Let $C$ be an
$\eta$-constacyclic code of length $n$ over $\mathbb{F}_{q^{2}}$ with
generator polynomial $g(x)$. Then the set $Z=\\{j\in\Omega|g(\delta^{j})=0\\}$
is called the defining set of $C$. It is clearly to see the defining set of
$C$ is a union of some $q^{2}$-cyclotomic cosets modulo $rn$ and
$dim(C)=n-|Z|$. It is also easily to see $C^{\bot_{H}}$ has defining set
$Z^{\bot_{H}}=\\{z\in\Omega|-qz\ mod\ rn\not\in Z\\}$(See Ref. 13).

The following results in [34, 35] play an important role in constructing
asymmetric quantum codes.

Theorem 2.1 _(The BCH bound for constacyclic codes) Assume that $\gcd(q,n)=1$.
Let $C=\langle g(x)\rangle$ be an $\eta$-constacyclic code of length $n$ over
$\mathbb{F}_{q^{2}}$ with the roots $\\{\delta^{1+ir}|0\leq i\leq d-2\\}$,
where $\delta$ is a primitive $rn$th root of unity. Then the minimum distance
of $C$ is at least $d$._

Proposition 2.2 _(Singleton bound) Let $C$ be an $[n,k,d]$ linear code over
$\mathbb{F}_{q^{2}}$, then $k\leq n-d+1.$_

Lemma 2.3 _Let $C_{i}$ be an $\eta$-constacyclic code of length $n$ over
$\mathbb{F}_{q^{2}}$ with defining set $Z_{i}$ for $i=1,2$. Then
$C_{1}\subseteq C_{2}$ if and only if $Z_{2}\subseteq Z_{1}.$ _

3 Asymmetric Quantum Codes

In this section, we recall some basic definitions and results of asymmetric
quantum codes. More details we refer to [10, 28-31].

Let $\mathbb{H}$ be the Hilbert space
$\mathbb{H}=\mathbb{C}^{q^{n}}=\mathbb{C}^{q}\otimes\cdots\otimes\mathbb{C}^{q},$
where $\mathbb{C}^{q}$ denotes a $q$-dimensional complex vector space
representing the states of a quantum mechanical system. Let $|x\rangle$ be the
vectors of an orthonormal basis of $\mathbb{C}^{q}$, where the labels $x$ are
elements of $\mathbb{F}_{q}.$ Let $a,b$ be two elements of $\mathbb{F}_{q}.$
The unitary operators $X(a)$ and $Z(b)$ on $\mathbb{C}^{q}$ are defined as
$X(a)|x\rangle=|x+a\rangle$ and $Z(b)|x\rangle=\omega^{tr(bx)}|x\rangle,$
respectively, where $tr$ is the trace map from $\mathbb{F}_{q}$ to the prime
field $\mathbb{F}_{p}$ and $\omega=\textrm{exp}(2\pi i/p)$ is a primitive
$p$th root of unity. Let
$\textbf{a}=(a_{1},a_{2},\ldots,a_{n})\in\mathbb{F}_{q}^{n}$ and
$\textbf{b}=(b_{1},b_{2},\ldots,b_{n})\in\mathbb{F}_{q}^{n}.$ Denote
$X(\textbf{a})=X(a_{1})\otimes X(a_{2})\otimes\cdots\otimes X(a_{n})$ and
$Z(\textbf{b})=Z(b_{1})\otimes Z(b_{2})\otimes\cdots\otimes Z(b_{n})$ by the
tensor products of $n$ error operators. The set
$\varepsilon_{n}=\\{X(\textbf{a})Z(\textbf{b})|\textbf{a},\textbf{b}\in\mathbb{F}_{q}^{n}\\}$
is an error basis on $\mathbb{C}^{q}$ and the set
$G_{n}=\\{\omega^{c}X(\textbf{a})Z(\textbf{b})|\textbf{a},\textbf{b}\in\mathbb{F}_{q}^{n},\
c\in\mathbb{F}_{p}\\}$ is the error group associated with $\varepsilon_{n}.$

For a quantum error $e=\omega^{c}X(\textbf{a})Z(\textbf{b})\in G_{n},$ the
quantum weight $w_{Q}(e),$ the $X$-weight $w_{X}(e)$ and the $Z$-weight
$w_{Z}(e)$ of $e$, are defined respectively as

$w_{Q}(e)=\sharp\\{i:1\leq i\leq n,(a_{i},b_{i})\neq(0,0)\\},$
$w_{X}(e)=\sharp\\{i:1\leq i\leq n,a_{i}\neq 0\\},$ $w_{Z}(e)=\sharp\\{i:1\leq
i\leq n,b_{i}\neq 0\\}.$

Definition 3.1[10,15,16,20] _A $q$-ary asymmetric quantum code $C$, denoted by
$[[n,k,d_{z}/d_{x}]]_{q}$, is a $q^{k}$-dimensional subspace of the Hilbert
space $\mathbb{H}$ and corrects all qudit-flip errors up to
$\lfloor\frac{d_{x}-1}{2}\rfloor$ and all phase-shift errors up to
$\lfloor\frac{d_{z}-1}{2}\rfloor$._

The following well-known CSS construction shown in [18-20] will be utilized in
this paper:

Theorem 3.2 _Let $C_{i}$ be a classical linear code with parameters
$[n,k_{i},d_{i}]_{q^{2}}$, for $i=1,2$. If $C_{1}^{\bot_{H}}\subseteq C_{2}$,
then there exists an asymmetric quantum code with parameters
$[[n,k_{2}+k_{1}-n,d_{z}/d_{x}]]_{q^{2}}$, where $d_{z}=wt(C_{2}\backslash
C_{1}^{\bot_{H}})$ and $d_{x}=wt(C_{1}\backslash C_{2}^{\bot_{H}})$._

For a CSS asymmetric quantum codes $[[n,k,d_{z}/d_{x}]]_{q^{2}}$, the
relations among $n,k,d_{z}$ and $d_{x}$ have the following famous result:

Theorem 3.3 _If a CSS asymmetric quantum code $C$ with parameters
$[[n,k,d_{z}/d_{x}]]_{q^{2}}$ exists, then $C$ satisfies the asymmetric
quantum Singleton bound_

$k\leq n-d_{z}-d_{x}+2.$

_Especially, if $k=n-d_{z}-d_{x}+2,$ then $C$ is called an optimal code._

4 Code Construction

In this section, we construct six classes of asymmetric quantum codes based on
constacyclic codes over $\mathbb{F}_{q^{2}}.$

## 4.1 Construction I

In this section we construct asymmetric quantum codes from constacyclic codes
of length $n=\lambda(q-1)$ with $\lambda$ a divisor of $q+1$ over
$\mathbb{F}_{q^{2}}$, where the classical codes are endowed with the Hermitian
inner product. Let $r=(q+1)/\gcd(v,q-1)$ be even, for some
$v\in\\{1,2,\ldots,q\\}$. Let $\eta=\omega^{v(q-1)}$ and $\lambda=(q+1)/r$.
For each $0\leq j\leq n-1,$ note that the $q^{2}$-cyclotomic coset containing
$1+jr$ modulo $rn$ has only one element $1+jr$. The following Lemma from [13]
plays an important role in the asymmetric quantum codes construction.

Lemma 4.1 [13, Lemma 3.1]_Let $r=(q+1)/\gcd(v,q-1)$ be even, for some
$v\in\\{1,2,\ldots,q\\}$. Let $n=\lambda(q-1)$ with $\lambda=(q+1)/r.$ Suppose
that $C$ is an $\eta$-constacyclic code of length $n$ over
$\mathbb{F}_{q^{2}}$ with defining set $Z=\bigcup_{j=1}^{\delta}C_{1+r(j-1)}.$
Then_

_1) if_ $r=2$ and $1\leq\delta\leq q-1$, then $C^{\bot_{H}}\subseteq C$;

_2) if_ $r\neq 2$ and $1\leq\delta\leq(q-1)/2$, then $C^{\bot_{H}}\subseteq
C$.

If $r=2$, then $n=(q^{2}-1)/2$ and $C$ is a negacyclic code over
$\mathbb{F}_{q^{2}}$. Now we give the first construction of this paper:

Theorem 4.2 _Let $q$ be an odd prime power, and $n=(q^{2}-1)/2$. Then there
exist asymmetric quantum codes with parameters
$[[n,n-s-t,(s+1)/(t+1)]]_{q^{2}}$, where $s,t$ are positive integers and
$1\leq t\leq s\leq q-1.$_

Proof. Suppose $C_{2}$ is a negacyclic code over $\mathbb{F}_{q^{2}}$ of
length $n$ with defining set $Z_{2}=\bigcup_{i=1}^{t}C_{2i-1}$, where $1\leq
t\leq q-1$. Then the dimension of $C_{2}$ is $n-t$. Observe that $Z_{2}$
consists of $t$ consecutive odd integers $\\{1,3,\ldots,2t-1\\}$. From the BCH
bound for constacyclic codes, the minimum distance of $C_{2}$ is at least
$t+1$. From Proposition 2.2, we can see that the minimum distance of $C_{2}$
is $t+1$. Hence, $C_{2}$ is a negacyclic code with parameters
$[n,n-t,t+1]_{q^{2}}$.

Now, suppose $C_{1}$ is a negacyclic code over $\mathbb{F}_{q^{2}}$ of length
$n$ with defining set $Z_{1}=\bigcup_{i=1}^{s}C_{2i-1}$, where $1\leq t\leq
s\leq q-1$. By Lemma 4.1, $C_{1}^{\bot_{H}}\subseteq C_{1}$ and the dimension
of $C_{1}$ is $n-s$. Observe that $Z_{2}$ consists of $s$ consecutive odd
integers $\\{1,3,\ldots,2s-1\\}$. From the BCH bound for constacyclic codes,
the minimum distance of $C_{1}$ is at least $s+1$. From Proposition 2.2, we
get the minimum distance of $C_{1}$ is $s+1$. Hence, $C_{1}$ is a negacyclic
code with parameters $[n,n-s,s+1]_{q^{2}}$. It is clearly to see that
$C_{1}^{\bot_{H}}\subseteq C_{2}$. Then from Theorem 3.2, there exist
asymmetric quantum codes with parameters $[[n,n-s-t,(s+1)/(t+1)]]_{q^{2}}.$ ∎

If $r\neq 2$, we can get the following asymmetric quantum codes:

Theorem 4.3 _Let $r\neq 2$ be an even divisor of $q+1$. Let $n=\lambda(q-1)$
with $\lambda=(q+1)/r$. Then there exist asymmetric quantum codes with
parameters $[[n,n-s-t,(s+1)/(t+1)]]_{q^{2}}$, where $s,t$ are positive
integers and $1\leq t\leq s\leq(q-1)/2.$_

Proof. Let $\eta=\omega^{\lambda(q-1)}$, where $\omega$ is a primitive element
of $\mathbb{F}_{q^{2}}.$ Let $C_{2}$ be the $\eta$-constacyclic code over
$\mathbb{F}_{q^{2}}$ of length $n$ with defining set
$Z_{2}=\bigcup_{i=1}^{t}C_{1+r(i-1)}$, where $1\leq t\leq(q-1)/2$. Then the
dimension of $C_{2}$ is $n-t$. Observe that $Z_{2}$ consists of $t$ odd
integers $\\{1,1+r,1+2r,\ldots,1+(t-1)r\\}$. The minimum distance of $C_{2}$
is at least $t+1$ from Theorem 2.1. Furthermore, we can see that the minimum
distance of $C_{2}$ is $t+1$ from Proposition 2.2. Hence, $C_{2}$ is an
$\eta$-constacyclic code with parameters $[n,n-t,t+1]_{q^{2}}$.

Now, suppose $C_{1}$ is an $\eta$-constacyclic code over $\mathbb{F}_{q^{2}}$
of length $n$ with defining set $Z_{1}=\bigcup_{i=1}^{s}C_{1+r(i-1)}$, where
$1\leq t\leq s\leq(q-1)/2$. Similar to the discussion of $C_{2}$, $C_{1}$ is
an $\eta$-constacyclic code with parameters $[n,n-s,s+1]_{q^{2}}$. It is
clearly to see that $C_{1}^{\bot_{H}}\subseteq C_{2}$ by Lemma 2.3 and Lemma
4.1. Then from Theorem 3.2, there exist asymmetric quantum codes with
parameters $[[n,n-s-t,(s+1)/(t+1)]]_{q^{2}}.$ ∎

Example 4.4 Let $q=9$, then $n=(q^{2}-1)/2=40$. Applying Theorem 4.2 produces
asymmetric quantum codes in Table 1.

TABLE 1 Asymmetric quantum codes derived from constacyclic codes of length 40

| $[[40,38,2/2]]_{81}$ |  |  | $[[40,35,4/3]]_{81}$ |  |  | $[[40,31,7/4]]_{81}$ |  |  | $[[40,29,7/6]]_{81}$ |
---|---|---|---|---|---|---|---|---|---|---|---
| $[[40,37,3/2]]_{81}$ |  |  | $[[40,34,5/3]]_{81}$ |  |  | $[[40,30,8/4]]_{81}$ |  |  | $[[40,28,8/6]]_{81}$ |
| $[[40,36,4/2]]_{81}$ |  |  | $[[40,33,6/3]]_{81}$ |  |  | $[[40,29,9/4]]_{81}$ |  |  | $[[40,27,9/6]]_{81}$ |
| $[[40,35,5/2]]_{81}$ |  |  | $[[40,32,7/3]]_{81}$ |  |  | $[[40,32,5/5]]_{81}$ |  |  | $[[40,28,7/7]]_{81}$ |
| $[[40,34,6/2]]_{81}$ |  |  | $[[40,31,8/3]]_{81}$ |  |  | $[[40,31,6/5]]_{81}$ |  |  | $[[40,27,8/7]]_{81}$ |
| $[[40,33,7/2]]_{81}$ |  |  | $[[40,30,9/3]]_{81}$ |  |  | $[[40,30,7/5]]_{81}$ |  |  | $[[40,26,9/7]]_{81}$ |
| $[[40,32,8/2]]_{81}$ |  |  | $[[40,34,4/4]]_{81}$ |  |  | $[[40,29,8/5]]_{81}$ |  |  | $[[40,26,8/8]]_{81}$ |
| $[[40,31,9/2]]_{81}$ |  |  | $[[40,33,5/4]]_{81}$ |  |  | $[[40,28,9/5]]_{81}$ |  |  | $[[40,25,9/8]]_{81}$ |
| $[[40,36,3/3]]_{81}$ |  |  | $[[40,32,6/4]]_{81}$ |  |  | $[[40,30,6/6]]_{81}$ |  |  | $[[40,24,9/9]]_{81}$ |

Example 4.5 Let $q=11$ and $r=4$, then $\lambda=3$ and $n=\lambda(q-1)=30$.
Applying Theorem 4.3 produces asymmetric quantum codes in Table 2.

TABLE 2 Asymmetric quantum codes derived from constacyclic codes of length 30

| $[[30,28,2/2]]_{121}$ |  |  | $[[30,26,3/3]]_{121}$ |  |  | $[[30,23,5/4]]_{121}$ |
---|---|---|---|---|---|---|---|---
| $[[30,27,3/2]]_{121}$ |  |  | $[[30,25,4/3]]_{121}$ |  |  | $[[30,22,6/4]]_{121}$ |
| $[[30,26,4/2]]_{121}$ |  |  | $[[30,24,5/3]]_{121}$ |  |  | $[[30,22,5/5]]_{121}$ |
| $[[30,25,5/2]]_{121}$ |  |  | $[[30,23,6/3]]_{121}$ |  |  | $[[30,21,6/5]]_{121}$ |
| $[[30,24,6/2]]_{121}$ |  |  | $[[30,24,4/4]]_{121}$ |  |  | $[[30,20,6/6]]_{121}$ |

## 4.2 Construction II

In this section we construct asymmetric quantum codes from constacyclic codes
of length $n=\lambda(q+1)$ with $\lambda$ an odd divisor of $q-1$ over
$\mathbb{F}_{q^{2}}$, where the classical codes are endowed with the Hermitian
inner product. Let $v=(q+1)/2$, then $\eta=-1$. It is easy to see that $2n$
divides $q^{2}-1$. Hence, for each odd $i$ in the range $1\leq i\leq 2n$, the
$q^{2}$-cyclotomic coset $C_{i}$ modulo $2n$ is $C_{i}=\\{i\\}$.

Lemma 4.6 [13, Lemma 3.6]_Let $n=\lambda(q+1)$, where $\lambda$ an odd divisor
of $q-1$. If $C$ is a $q^{2}$-ary negacyclic code of length $n$ with defining
set $Z=\bigcup_{j=1}^{\delta}C_{2j-1},$ where
$1\leq\delta\leq(q-1)/2+\lambda$, then $C^{\bot_{H}}\subseteq C$._

Theorem 4.7 _Let $n=\lambda(q+1)$ with $\lambda$ an odd divisor of $q-1$. Then
there exist asymmetric quantum codes with parameters
$[[n,n-s-t,(s+1)/(t+1)]]_{q^{2}}$, where $s,t$ are positive integers and
$1\leq t\leq s\leq(q-1)/2+\lambda.$_

Proof. Suppose $C_{2}$ is a negacyclic code over $\mathbb{F}_{q^{2}}$ of
length $n$ with defining set $Z_{2}=\bigcup_{i=1}^{t}C_{2i-1}$, where $1\leq
t\leq(q-1)/2+\lambda$. Then the dimension of $C_{2}$ is $n-t$. Observe that
$Z_{2}$ consists of $t$ consecutive odd integers $\\{1,3,\ldots,2t-1\\}$. From
Theorem 2.1, the minimum distance of $C_{2}$ is at least $t+1$. From
Proposition 2.2, we can see that the minimum distance of $C_{2}$ is $t+1$.
Hence, $C_{2}$ is a negacyclic code with parameters $[n,n-t,t+1]_{q^{2}}$.

Now, suppose $C_{1}$ is a negacyclic code over $\mathbb{F}_{q^{2}}$ of length
$n$ with defining set $Z_{1}=\bigcup_{i=1}^{s}C_{2i-1}$, where $1\leq t\leq
s\leq(q-1)/2+\lambda$. Similar to the discussion of $C_{2}$, $C_{1}$ is a
negacyclic code with parameters $[n,n-s,s+1]_{q^{2}}$. Then from Lemma 2.3 and
Theorem 3.2, there exist asymmetric quantum codes with parameters
$[[n,n-s-t,(s+1)/(t+1)]]_{q^{2}}.$ ∎

Lemma 4.8 [13, Lemma 3.9]_Assume that $q\equiv 1\ mod\ 4$. Let
$n=2\lambda(q+1)$, where $\lambda$ is an odd divisor of $q-1$. If $C$ is a
$q^{2}$-ary negacyclic code of length $n$ with defining set
$Z=\bigcup_{j=1}^{\delta}C_{2j-1}$, where $1\leq\delta\leq(q-1)/2+2\lambda$,
then $C^{\bot_{H}}\subseteq C$._

Theorem 4.9 _Let $q\equiv 1\ mod\ 4$ and $n=2\lambda(q+1)$ with $\lambda$ an
odd divisor of $q-1$. Then there exist asymmetric quantum codes with
parameters $[[n,n-s-t,(s+1)/(t+1)]]_{q^{2}}$, where $s,t$ are positive
integers and $1\leq t\leq s\leq(q-1)/2+2\lambda.$_

Proof. Suppose $C_{2}$ is a negacyclic code over $\mathbb{F}_{q^{2}}$ of
length $n$ with defining set $Z_{2}=\bigcup_{i=1}^{t}C_{2i-1}$, where $1\leq
t\leq(q-1)/2+2\lambda$. Then the dimension of $C_{2}$ is $n-t$. Observe that
$Z_{2}$ consists of $t$ consecutive odd integers $\\{1,3,\ldots,2t-1\\}$. From
Theorem 2.1, the minimum distance of $C_{2}$ is at least $t+1$. From
Proposition 2.2, we can see that the minimum distance of $C_{2}$ is $t+1$.
Hence, $C_{2}$ is a negacyclic code with parameters $[n,n-t,t+1]_{q^{2}}$.

Now, suppose $C_{1}$ is a negacyclic code over $\mathbb{F}_{q^{2}}$ of length
$n$ with defining set $Z_{1}=\bigcup_{i=1}^{s}C_{2i-1}$, where $1\leq t\leq
s\leq(q-1)/2+\lambda$. Similar to the discussion of $C_{2}$, $C_{1}$ is a
negacyclic code with parameters $[n,n-s,s+1]_{q^{2}}$. Then from Lemma 2.3 and
Theorem 3.2, there exist asymmetric quantum codes with parameters
$[[n,n-s-t,(s+1)/(t+1)]]_{q^{2}}.$ ∎

Remark From Theorem 4.2, Theorem 4.3, Theorem 4.7 and Theorem 4.9,
$d_{z}+d_{x}=s+t+2$. Then from Theorem 3.3, the constructed asymmetric quantum
codes with parameters $[[n,n-s-t,(s+1)/(t+1)]]_{q^{2}}$ attain asymmetric
quantum Singleton bound. Hence, these asymmetric quantum codes are optimal.

Example 4.10 Let $q=7$ and $\lambda=3$, then $n=\lambda(q+1)=24$. Applying
Theorem 4.7 produces asymmetric quantum codes in Table 3.

TABLE 3 Asymmetric quantum codes derived from constacyclic codes of length 24

| $[[24,22,2/2]]_{49}$ |  |  | $[[24,19,4/3]]_{49}$ |  |  | $[[24,15,7/4]]_{49}$ |
---|---|---|---|---|---|---|---|---
| $[[24,21,3/2]]_{49}$ |  |  | $[[24,18,5/3]]_{49}$ |  |  | $[[24,16,5/5]]_{49}$ |
| $[[24,20,4/2]]_{49}$ |  |  | $[[24,17,6/3]]_{49}$ |  |  | $[[24,15,6/5]]_{49}$ |
| $[[24,19,5/2]]_{49}$ |  |  | $[[24,16,7/3]]_{49}$ |  |  | $[[24,14,7/5]]_{49}$ |
| $[[24,18,6/2]]_{49}$ |  |  | $[[24,18,4/4]]_{49}$ |  |  | $[[24,14,6/6]]_{49}$ |
| $[[24,17,7/2]]_{49}$ |  |  | $[[24,17,5/4]]_{49}$ |  |  | $[[24,13,7/6]]_{49}$ |
| $[[24,20,3/3]]_{49}$ |  |  | $[[24,16,6/4]]_{49}$ |  |  | $[[24,12,7/7]]_{49}$ |

Example 4.11 Let $q=9$ and $\lambda=1$, then $n=2\lambda(q+1)=20$. Applying
Theorem 4.9 produces asymmetric quantum codes in Table 4.

TABLE 4 Asymmetric quantum codes derived from constacyclic codes of length 20

| $[[20,18,2/2]]_{81}$ |  |  | $[[20,15,4/3]]_{81}$ |  |  | $[[20,11,7/4]]_{81}$ |
---|---|---|---|---|---|---|---|---
| $[[20,17,3/2]]_{81}$ |  |  | $[[20,14,5/3]]_{81}$ |  |  | $[[20,12,5/5]]_{81}$ |
| $[[20,16,4/2]]_{81}$ |  |  | $[[20,13,6/3]]_{81}$ |  |  | $[[20,11,6/5]]_{81}$ |
| $[[20,15,5/2]]_{81}$ |  |  | $[[20,12,7/3]]_{81}$ |  |  | $[[20,10,7/5]]_{81}$ |
| $[[20,14,6/2]]_{81}$ |  |  | $[[20,14,4/4]]_{81}$ |  |  | $[[20,10,6/6]]_{81}$ |
| $[[20,13,7/2]]_{81}$ |  |  | $[[20,13,5/4]]_{81}$ |  |  | $[[20,9,7/6]]_{81}$ |
| $[[20,16,3/3]]_{81}$ |  |  | $[[20,12,6/4]]_{81}$ |  |  | $[[20,8,7/7]]_{81}$ |

## 4.3 Construction III

In this section, we construct asymmetric quantum codes with a special length
$n=(q^{2}+1)/5$ over $\mathbb{F}_{q^{2}}$.

Lemma 4.12 [13, Lemma 3.12]_Let $n=(q^{2}+1)/5$ and $k=(q^{2}+1)/2$. Then, for
any integer $i\in\Omega=\\{1+(q+1)j|0\leq j\leq n-1\\},$ the
$q^{2}$-cyclotomic coset $C_{i}$ modulo $(q+1)n$ is given by_

_1)_ $C_{k}=\\{k\\}$ and $C_{k+n(q+1)/2}=\\{k+n(q+1)/2\\}.$

_2)_ $C_{k-(q+1)j}=\\{k-(q+1)j,k+(q+1)j\\}$ for $1\leq j\leq n/2-1.$

Lemma 4.13 [13, Lemma 3.13]_Let $q$ be an odd prime power with the form
$20m+3$ or $20m+7$, where $m$ is a positive integer. Let $n=(q^{2}+1)/5$ and
$k=(q^{2}+1)/2$. If $C$ is an $\omega^{q-1}$-constacyclic code over
$\mathbb{F}_{q^{2}}$ of length $n$ with defining set
$Z=\bigcup_{j=0}^{\delta}C_{k-(q+1)j}$, where $0\leq\delta\leq(q+1)/4$, then
$C^{\bot_{H}}\subseteq C.$ _

Theorem 4.14 _Let $q$ be an odd prime power with the form $20m+3$ or $20m+7$,
where $m$ is a positive integer. Let $n=(q^{2}+1)/5$, then there exist
asymmetric quantum codes with parameters
$[[n,n-2(s+t+1),(2s+2)/(2t+2)]]_{q^{2}}$, where $s,t$ are positive integers
and $0\leq t\leq s\leq(q+1)/4.$_

Proof. Let $k=(q^{2}+1)/2$. Suppose $C_{2}$ is a $q^{2}$-ary
$\omega^{q-1}$-constacyclic code of length $n=(q^{2}+1)/5$ with defining set
$Z_{2}=\bigcup_{i=0}^{t}C_{k-(q+1)i}$, where $0\leq t\leq(q+1)/4$. Then the
dimension of $C_{2}$ is $n-(2t+1)$. Observe that $Z_{2}$ consists of $2t+1$
consecutive integers
$\\{k-(q+1)t,\ldots,k-(q+1),k,k+(q+1),\ldots,k+(q+1)t\\}$. From Theorem 2.1,
the minimum distance of $C_{2}$ is at least $2t+2$. From Proposition 2.2, we
can see that the minimum distance of $C_{2}$ is $2t+2$. Hence, $C_{2}$ is a
$q^{2}$-ary $\omega^{q-1}$-constacyclic code with parameters
$[n,n-(2t+1),2t+2]_{q^{2}}$.

Now, suppose $C_{1}$ is a $q^{2}$-ary $\omega^{q-1}$-constacyclic code of
length $n=(q^{2}+1)/5$ with defining set
$Z_{1}=\bigcup_{i=0}^{s}C_{k-(q+1)i}$, where $0\leq t\leq s\leq(q+1)/4$.
Similar to the discussion of $C_{2}$, $C_{1}$ has parameters
$[n,n-(2s+1),2s+2]_{q^{2}}$. Then from Lemma 2.3 and Theorem 3.2, there exist
asymmetric quantum codes with parameters
$[[n,n-2(s+t+1),(2s+2)/(2t+2)]]_{q^{2}}.$ ∎

Similar to Theorem 4.14, we have the following result:

Theorem 4.15 _Let $q$ be an odd prime power with the form $20m-3$ or $20m-7$,
where $m$ is a positive integer. Let $n=(q^{2}+1)/5$, then there exist
asymmetric quantum codes with parameters
$[[n,n-2(s+t+1),(2s+2)/(2t+2)]]_{q^{2}}$, where $s,t$ are positive integers
and $0\leq t\leq s\leq(q-1)/4.$_

Remark From Theorem 4.14, and Theorem 4.15, $d_{z}+d_{x}=2s+2t+4$. Then from
Theorem 3.3, the constructed asymmetric quantum codes with parameters
$[[n,n-2(s+t+1),(2s+2)/(2t+2)]]_{q^{2}}$ attain asymmetric quantum Singleton
bound. Hence, these asymmetric quantum codes are optimal.

Example 4.16 Let $q=23$, then $n=(q^{2}+1)/5=106$. Suppose the defining set of
$\omega^{22}$-constacyclic code $C_{1}$ is given by $Z_{1}=C_{265}=\\{265\\}$.
Then $C_{1}$ is a MDS code with parameters $[106,105,2]_{529}$. We also
suppose the defining set of $\omega^{22}$-constacyclic code $C_{2}$ is given
by $Z_{2}=C_{265}\cup C_{241}\cup C_{217}\cup C_{193}\cup C_{169}\cup
C_{145}\cup
C_{121}=\\{121,145,169,193,217,241,265,289,313,337,361,385,409\\}$. Then,
$C_{2}$ is a MDS code with parameters $[106,93,14]_{529}$. From Theorem 4.14,
there exists an optimal asymmetric quantum code with parameters
$[[106,92,14/2]]_{529}$. By taking the different defining sets of $C_{1}$ and
$C_{2}$, we can get the optimal asymmetric quantum codes in Table 5.

TABLE 5 Asymmetric quantum codes derived from constacyclic codes of length 106

| $[[106,104,2/2]]_{529}$ |  |  | $[[106,100,4/4]]_{529}$ |  |  | $[[106,94,8/6]]_{529}$ |  |  | $[[106,86,14/8]]_{529}$ |
---|---|---|---|---|---|---|---|---|---|---|---
| $[[106,102,4/2]]_{529}$ |  |  | $[[106,98,6/4]]_{529}$ |  |  | $[[106,92,10/6]]_{529}$ |  |  | $[[106,88,10/10]]_{529}$ |
| $[[106,100,6/2]]_{529}$ |  |  | $[[106,96,8/4]]_{529}$ |  |  | $[[106,90,12/6]]_{529}$ |  |  | $[[106,86,12/10]]_{529}$ |
| $[[106,98,8/2]]_{529}$ |  |  | $[[106,94,10/4]]_{529}$ |  |  | $[[106,88,14/6]]_{529}$ |  |  | $[[106,84,14/10]]_{529}$ |
| $[[106,96,10/2]]_{529}$ |  |  | $[[106,92,12/4]]_{529}$ |  |  | $[[106,92,8/8]]_{529}$ |  |  | $[[106,84,12/12]]_{529}$ |
| $[[106,94,12/2]]_{529}$ |  |  | $[[106,90,14/4]]_{529}$ |  |  | $[[106,90,10/8]]_{529}$ |  |  | $[[106,82,14/12]]_{529}$ |
| $[[106,92,14/2]]_{529}$ |  |  | $[[106,96,6/6]]_{529}$ |  |  | $[[106,88,12/8]]_{529}$ |  |  | $[[106,80,14/14]]_{529}$ |

Example 4.17 Let $q=17$, then $n=(q^{2}+1)/5=58$. Suppose the defining set of
$\omega^{16}$-constacyclic code $C_{1}$ is given by $Z_{1}=C_{145}=\\{145\\}$.
Then $C_{1}$ is a MDS code with parameters $[58,57,2]_{289}$. We also suppose
the defining set of $\omega^{16}$-constacyclic code $C_{2}$ is given by
$Z_{2}=C_{145}\cup C_{127}\cup C_{109}\cup C_{91}\cup
C_{73}=\\{73,91,109,127,145,163,181,199,217\\}$. Then, $C_{2}$ is a MDS code
with parameters $[58,49,10]_{289}$. From Theorem 4.15, there exists an optimal
asymmetric quantum code with parameters $[[58,48,10/2]]_{289}$. By taking the
different defining sets of $C_{1}$ and $C_{2}$, we can get the optimal
asymmetric quantum codes in Table 6.

TABLE 6 Asymmetric quantum codes derived from constacyclic codes of length 58

| $[[58,56,2/2]]_{289}$ |  |  | $[[58,52,4/4]]_{289}$ |  |  | $[[58,46,8/6]]_{289}$ |
---|---|---|---|---|---|---|---|---
| $[[58,54,4/2]]_{289}$ |  |  | $[[58,50,6/4]]_{289}$ |  |  | $[[58,44,10/6]]_{289}$ |
| $[[58,52,6/2]]_{289}$ |  |  | $[[58,48,8/4]]_{289}$ |  |  | $[[58,44,8/8]]_{289}$ |
| $[[58,50,8/2]]_{289}$ |  |  | $[[58,46,10/4]]_{289}$ |  |  | $[[58,42,10/8]]_{289}$ |
| $[[58,48,10/2]]_{289}$ |  |  | $[[58,48,6/6]]_{289}$ |  |  | $[[58,40,10/10]]_{289}$ |

5 Conclusion

In this paper we have constructed six new families of asymmetric quantum codes
based on constacyclic codes by applying the CSS construction. The new codes
achieve the asymmetric quantum Singleton bound and different from the codes
available in the literature. Additionally, the quantum codes constructed in
this paper can be utilized in quantum channels with great asymmetry.

## References

  * [1] P. W. Shor, Phys. Rev. A 52 (1995) 2493.
  * [2] A. M. Steane, Phys. Rev. Lett. 77 (1996) 793.
  * [3] A. R. Calderbank and P. W. Shor, Phys. Rev. A 54 (1996) 1098.
  * [4] G. Cohen, S. Encheva and S. Litsyn, IEEE Trans. Inf. Theory 45 (1999) 2495.
  * [5] Z. Li, L. Xing and X. Wang, Phys. Rev. A 77 (2008) 012308.
  * [6] A. M. Steane, IEEE Trans. Inf. Theory 45 (1999) 1701.
  * [7] A. M. Steane, IEEE Trans. Inf. Theory 45 (1999) 2492.
  * [8] H. Chen, S. Ling and C. Xing, IEEE Trans. Inf. Theory 51 (2005) 2915.
  * [9] A. Ashikhmin and E. Knill, IEEE Trans. Inf. Theory 47 (2001) 3065.
  * [10] A. Ketkar, A. Klappenecker, S. Kumar and P. K. Sarvepalli, IEEE Trans. Inf. Theory 52 (2006) 4892.
  * [11] S. A. Aly, A. Klappenecker and P. K. Sarvepalli, IEEE Trans. Inf. Theory 53 (2007) 1183.
  * [12] X. S. Kai and S. X. Zhu, IEEE Trans. Inf. Theory 59 (2013) 1193.
  * [13] X. S. Kai and S. X. Zhu, IEEE Trans. Inf. Theory 60 (2014) 2080.
  * [14] G. G. La Guardia, IEEE Trans. Inf. Theory 60 (2014) 304.
  * [15] S. A. Aly and A. Ashikhmin, In: IEEE Information Theory Workshop, pp. 1-5 (2010).
  * [16] A. M. Steane, Phys. Rev. A 54 (1996) 4741.
  * [17] L. Ioffe and M. Mezard, Phys. Rev. A 75 (2007) 032345.
  * [18] P. K. Sarvepalli, M. Rotteler and A. Klappenecker, In: Proceedings International Symposium Information Theory, pp. 305-309 (2008).
  * [19] S. A. Aly, In: Proceedings IEEE International Conference on Computer Engineering and Systems, pp. 157-162 (2008)
  * [20] P. K. Sarvepalli, M. Rotteler and A. Klappenecker,In: Proceedings of the Royal Society A, pp. 1645-1672 (2009).
  * [21] G. G. La Guardia, Quantum Inf. Comput. 11 (2011) 0239.
  * [22] A. M. Stephens, Z. W. E. Evans, S. J. Devitt and L. C. L. Hollenberg, Phys. Rev. A 77 (2008) 062335.
  * [23] L. Wang, L. Q. Feng, S. Ling and C. P. Xing, IEEE Trans. Inf. Theory 56 (2010) 2938.
  * [24] M. F. Ezerman, S. Ling and P. Solé, IEEE Trans. Inf. Theory 57 (2011) 5536.
  * [25] M. F. Ezerman and S. Ling, Adv. Math. Commun. 5 (2011) 41.
  * [26] Y. Chee, S. Jitman and M. F. Ezerman, In: 3rd Int. Castle Meeting on Coding Theroy and Applications, pp. 97-102 (2011).
  * [27] M. F. Ezerman, S. Jitman, H. M. Kiah and S. Ling, Int. J. Quantum Inf. 11 (2013) 1350027.
  * [28] G. G. La Guardia, Quantum Inf. Process 11 (2012) 591.
  * [29] G. G. La Guardia, Int. J. Quantum Inf. 10 (2012) 1250005.
  * [30] G. G. La Guardia, Quantum Inf. Process 12 (2013) 2771.
  * [31] G. G. La Guardia, Int. J. Theory Phys. (2014) doi: 10.1007/s10773-014-2031-y.
  * [32] J. F. Qian, L. N. Zhang, Mod. Phys. Lett. B 27 (2013) 1350010.
  * [33] J. Z. Chen, J. P. Li and J. Lin, Int. J. Theory Phys. 53 (2014) 72.
  * [34] N. Aydin, I. Siap and D. K. Ray-Chaudhuri, Des. Codes Crypt. 24 (2001) 313.
  * [35] F. J. MacWilliams and N. J. A. Sloane, Amsterdam, The Netherlands: North-Holland, 1997.