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# On Euclidean, Hermitian and symplectic quasi-cyclic complementary dual codes

Chaofeng Guan  gcf2020yeah.net Ruihu Li<EMAIL_ADDRESS>Zhi Ma
<EMAIL_ADDRESS>State Key Laboratory of Mathematical Engineering and Advanced
Computing, Zhengzhou, 450001, China.  Henan Key Laboratory of Network
Cryptography Technology, Zhengzhou, 450001, China.  Fundamentals Department,
Air Force Engineering University, Xi’an, 710051, China.

###### Abstract

Linear complementary dual codes (LCD) are codes that intersect trivially with
its dual. LCD codes have recently become a popular topic due to their
applications in data storage, communication systems, and cryptography. In this
paper, we propose a new characterization for LCD codes, which allows us to
judge the complementary duality of linear codes from the codeword level.
Further, we determine the necessary and sufficient conditions for quasi-cyclic
codes to be LCD codes involving Euclidean, Hermitian, and symplectic inner
products. Finally, we give several examples demonstrating that quasi-cyclic
codes can be utilized to construct good Euclidean, Hermitian, and symplectic
LCD codes.

###### keywords:

quasi-cyclic codes, complementary dual codes, necessary and sufficient
conditions, Euclidean, Hermitian, symplectic

##  1 Introduction

Linear complementary dual codes (LCD) intersect their dual codes trivially.
LCD codes have been used extensively in data storage, communication systems,
consumer electronics, and cryptography. In [1], Massey showed that LCD codes
provide an optimal linear coding scheme for a two-user binary adder channel.
In [2], Carlet et al. studied the application of binary LCD codes in
countering side channel attacks and fault injection attacks and proposed
several LCD code constructions. Moreover, it was shown in [3] that LCD codes
satisfy the Gilbert-Varshamov bound. Due to the critical application of LCD
codes, much research has been conducted on LCD codes [4, 5, 6, 7, 8, 9].

Notably, Carlet et al. [10] showed that any code over $\mathbb{F}_{q}$ is
equivalent to some Euclidean LCD code for $q\geq 4$, and any code over
$\mathbb{F}_{q^{2}}$ is equivalent to some Hermitian LCD code for $q\geq 3$.
Therefore, most research on LCD codes is currently focused on small fields.
Bouyuklieva [11], Harada [12], Ishizuka et al. [13, 14], Li et al. [15], and
Liu et al. [16] constructed many good binary, ternary Euclidean LCD codes and
quaternary Hermitian LCD codes. They have established several LCD code tables
with short lengths. In addition, Shi et al. [17] introduced additive
complementary dual codes (ACD) for security applications that still makes
sense. With [18], symplectic inner product and trace Hermitian inner product
are equivalent, so a $\mathbb{F}_{q}^{2n}$-symplectic LCD code is equivalent
to a $\mathbb{F}_{q^{2}}^{n}$-trace Herimtian LCD code. Therefore, the
construction of the symplectic LCD code is also of significant importance. Xu
et al. [19] constructed a class of symplectic LCD MDS codes by employing
Vandermonde matrices. In [20], Huang et al. construct some good low-
dimensional symplectic LCD codes over $\mathbb{F}_{2^{r}}$.

In [21], Yang and Massey provided a necessary and sufficient condition under
which a cyclic code has a Euclidean complementary dual. Esmaeili et al. [22]
studied a sufficient condition for $h$-generated quasi-cyclic codes to be
Euclidean LCD codes and gave a method for constructing quasi-cyclic Euclidean
LCD codes. In [23], Güneri et al. characterized and studied quasi-cyclic codes
with Euclidean and Hermitian complementary duals employing their concatenation
structure and presented a method for constructing quasi-cyclic Euclidean and
Hermitian LCD codes from codes over larger alphabets. Later on, Saleh et al.
[24] studied quasi-twisted codes of specific lengths and proposed some
sufficient conditions for $h$-generator quasi-twisted codes to be Euclidean
LCD. However, many important issues remain regarding developing LCD codes from
quasi-cyclic codes. One of the most critical issues is determining the
sufficient and necessary conditions for quasi-cyclic codes to be LCD codes so
that we can construct quasi-cyclic LCD codes more efficiently.

The main goal of this paper is to investigate quasi-cyclic Euclidean,
Hermitian, and symplectic LCD codes. We ascertain the sufficient and necessary
conditions for quasi-cyclic codes to be Euclidean, Hermitian, and symplectic
LCD codes. More precisely, we answer the following two questions:

1, What polynomials can be applied to construct quasi-cyclic LCD codes?

2, How to use polynomials to construct quasi-cyclic LCD codes?

Firstly, we give a new characterization of LCD codes that allows the treatment
of LCD codes at the codeword level. Then, by decomposing the codeword space of
quasi-cyclic code, we obtain the sufficient and necessary conditions for
quasi-cyclic codes to intersect their dual trivially under Euclidean,
Hermitian, and symplectic inner products. Finally, we present a practical
method for constructing LCD codes using quasi-cyclic codes and give some
examples of good quasi-cyclic Euclidean, Hermitian, and symplectic LCD codes.

The paper is organized as follows. Section 2 gives preliminaries and
background on quasi-cyclic codes, Euclidean, Hermitian, and symplectic LCD
codes. In Section 3 and 4, we redescribe LCD codes in terms of codewords and
identify the sufficient and necessary conditions for the quasi-cyclic codes to
be LCD codes under Euclidean, Hermitian, and symplectic inner products,
respectively. In Section 5, we give several examples to illustrate the method
of constructing quasi-cyclic Euclidean, Hermitian, and symplectic LCD codes.
Finally, we give concluding remarks in Section 6. All calculations in this
paper are done with the algebraic computer system Magma [25].

##  2 Preliminaries

In this section, we introduce some basic concepts of quasi-cyclic codes,
Euclidean, Hermitian, and symplectic LCD codes to facilitate the unfolding of
subsequent sections. For more details, we refer the reader to the standard
handbook [26, 27].

###  2.1 Basics of linear codes

Throughout this paper, $p$ is a prime, and $\mathbb{F}_{q}$ is the finite
field of order $q$, where $q=p^{r}$ for some positive integer $r$. A $[\ell
n,k]_{q}$ linear code $\mathscr{C}$ over $\mathbb{F}_{q}$ is a linear subspace
of $\mathbb{F}_{q}^{\ell n}$ with dimension $k$. Let
$\vec{u}=(u_{0},u_{1},\dots,u_{\ell n-1})\in\mathscr{C}$, then Hamming weight
of $\vec{u}$ is $w_{H}(\vec{u})=\\#\\{i\mid u_{i}\neq 0,0\leq i\leq\ell
n-1\\}$. If minimum Hamming distance of $\mathscr{C}$ is
$d_{H}=\min\\{w_{H}(\vec{u})\mid\vec{u}\in\mathscr{C}\setminus\\{\mathbf{0}\\}\\}$,
then $\mathscr{C}$ can be written as $[\ell n,k,d_{H}]_{q}$. If $\ell$ is
even, let $N=\ell n/2$, then symplectic weight of $\vec{u}$ is
$w_{s}(\vec{u})=\\#\\{i\mid(u_{i},u_{N+i})\neq(0,0),0\leq i\leq N-1\\}$.
Analogously, if minimum symplectic weight of $\mathscr{C}$ is
$d_{s}(\mathscr{C})=\min\\{w_{s}(\vec{u})\mid\vec{u}\in\mathscr{C}\setminus\\{\mathbf{0}\\}\\}$,
then we denote $\mathscr{C}$ as $[\ell n,k,d_{s}]_{q}^{s}$.

The Euclidean inner product of $\vec{x}=(x_{0},\ldots,x_{\ell n-1})$,
$\vec{y}=(y_{0},\ldots,y_{\ell n-1})\in\mathbb{F}_{q}^{\ell n}$ is defined as:

$\langle\vec{x},\vec{y}\rangle_{e}=\sum_{i=1}^{\ell n}x_{i}y_{i}.$ (1)

Similarly, the Hermitian inner product of
$\vec{x},\vec{y}\in\mathbb{F}_{q^{2}}^{\ell n}$ is defined as:

$\langle\vec{x},\vec{y}\rangle_{h}=\sum_{i=1}^{\ell n}x_{i}y_{i}^{q}.$ (2)

If $\ell$ is even, then symplectic inner product of
$\vec{x},\vec{y}\in\mathbb{F}_{q}^{\ell n}$ is:

$\langle\vec{x},\vec{y}\rangle_{s}=\sum_{i=0}^{N-1}\left(x_{i}y_{N+i}-x_{N+i}y_{i}\right).$
(3)

The Euclidean, Hermitian and symplectic dual codes of $\mathscr{C}$ can be
separately expressed as
$\mathscr{C}^{\perp_{e}}=\\{\vec{v}\in\mathbb{F}_{q}^{\ell
n}\mid\langle\vec{u},\vec{v}\rangle_{e}=0,\forall\vec{u}\in\mathscr{C}\\}$,
$\mathscr{C}^{\perp_{q}}=\\{\vec{v}\in\mathbb{F}_{q^{2}}^{\ell
n}\mid\langle\vec{u},\vec{v}\rangle_{h}=0,\forall\vec{u}\in\mathscr{C}\\}$ and
$\mathscr{C}^{\perp_{s}}=\\{\vec{v}\in\mathbb{F}_{q}^{\ell
n}\mid\langle\vec{u},\vec{v}\rangle_{s}=0,\forall\vec{u}\in\mathscr{C}\\}.$
$\mathscr{C}$ is an LCD code if and only if
$\mathscr{C}\cap\mathscr{C}^{\perp_{*}}=\\{\mathbf{0}\\}$, where “$\perp_{*}$”
represents one of Euclidean, Hermitian and symplectic dual.

###  2.2 Basics of quasi-cyclic codes

Cyclic codes are a particular type of linear codes that are closed under the
right cyclic shift operator $\tau_{1}$. For
$\vec{x}=(x_{0},x_{1},\ldots,x_{n-1})\in\mathbb{F}_{q}^{n}$, we denote
$\tau_{1}(\vec{x})=\left(x_{n-1},x_{0},\ldots,x_{n-2}\right).$ If
$\mathscr{C}=\tau_{1}(\mathscr{C})$ then $\mathscr{C}$ is called a cyclic
code. Let $\mathbb{R}=\mathbb{F}_{q}[x]/\left\langle x^{n}-1\right\rangle$,
and define a mapping $\varphi_{1}$ as follows,

$\displaystyle\varphi_{1}:\mathbb{F}_{q}^{n}$
$\displaystyle\rightarrow\mathbb{R}$ (4)
$\displaystyle\left(c_{0},c_{1},\ldots,c_{n-1}\right)$ $\displaystyle\mapsto
c_{0}+c_{1}x+\cdots+c_{n-1}x^{n-1}$

Clearly, $\varphi_{1}$ is an isomorphism of $\mathbb{F}_{q}$-modules and a
cyclic code $\mathscr{C}$ of length $n$ is an ideal of the quotient ring
$\mathbb{R}$. Furthermore, a cyclic code $\mathscr{C}$ can be generated by a
monic divisor $g(x)$ of $x^{n}-1$. The polynomial $g(x)$ is called the
generator polynomial of $\mathscr{C}$, and the dimension of $\mathscr{C}$ is
$n-deg(g(x))$. Let $h(x)=x^{n}-1/g(x)$ and
$\tilde{h}(x)=x^{\deg(h(x))}h(x^{-1})$, then Euclidean dual code of
$\mathscr{C}$ is cyclic code with generator polynomial
$g^{\perp_{e}}(x)=\tilde{h}(x)$. Let
$g^{q}(x)=g^{q}_{0}+g^{q}_{1}x+g^{q}_{2}x+\cdots+g^{q}_{n-1}x^{n-1}$. If
$\mathscr{C}$ is a cyclic codes over $\mathbb{F}_{q^{2}}$, then Hermitian dual
code of $\mathscr{C}$ is cyclic code generated by
$g^{\perp_{q}}(x)=\tilde{h}^{q}(x)$.

Let $\vec{x}=\left(x_{0},x_{1},\ldots,x_{\ell
n-1}\right)\in\mathbb{F}_{q}^{\ell n}$, and
$\tau_{2}(\vec{x})=(x_{n-1},x_{0},\ldots,x_{n-2},x_{2n-1},x_{n},\ldots,x_{2n-2},\ldots,x_{n\ell-1},x_{(n-1)\ell},\ldots,x_{n\ell-2}).$
A linear space $\mathscr{C}\subset\mathbb{F}_{q}^{\ell n}$ said to be a quasi-
cyclic code of index $\ell$ if $\mathscr{C}=\tau_{2}(\mathscr{C})$. Define an
$\mathbb{F}_{q}$ -module isomorphism $\varphi_{2}$ from $\mathbb{F}_{q}^{\ell
n}$ to $\mathbb{R}^{\ell}$,

$\displaystyle\varphi_{2}:\mathbb{F}_{q}^{\ell
n}\rightarrow\mathbb{R}^{\ell}=\mathbb{R}\oplus\mathbb{R}\oplus\dots\oplus\mathbb{R}$
(5)
$\displaystyle\left(c_{0},\ldots,c_{n-1},c_{n},\ldots,c_{2n-1},\ldots,c_{\ell(n-1)},\ldots,c_{\ell
n-1}\right)$
$\displaystyle\mapsto\left(c_{0}(x),c_{1}(x),\ldots,c_{\ell-1}(x)\right)$

where $c_{i}(x)=\sum_{t=0}^{n-1}c_{i,t}x^{t},i=0,1,\ldots,\ell-1$.
Algebraically, a quasi-cyclic code $\mathscr{C}$ is an $\mathbb{R}$-submodule
of $\mathbb{R}^{\ell}$. A generator matrix of $h$-generator quasi-cyclic code
with index $\ell$ has the following form:

$M=\left(\begin{array}[]{cccc}A_{1,0}&A_{1,1}&\cdots&A_{1,\ell-1}\\\
A_{2,0}&A_{2,1}&\cdots&A_{2,\ell-1}\\\ \vdots&\vdots&\ddots&\vdots\\\
A_{h,0}&A_{h,1}&\cdots&A_{h,\ell-1}\end{array}\right),$ (6)

where $A_{i,j}$ are circulant matrices defined by some polynomials
$a_{i,j}(x)\in\mathbb{R}$, where $1\leq i\leq h$ and $0\leq j\leq\ell-1$.

##  3 New characterization of complementary dual codes

This section will give a new characterization of LCD codes in terms of
codewords, laying the foundation for further proof. First, we make a
convention for the representation of inner products, where “$l$” denotes one
of Euclidean and Hermitian inner products, and “$*$” denotes one of Euclidean,
Hermitian, and symplectic products.

###### Lemma 1

Let $\mathscr{C}$ be a linear code over $\mathbb{F}_{q}$, then $\mathscr{C}$
is an LCD code under the inner product “$*$” if and only if $\forall
c_{1}\in\mathscr{C}\setminus\\{\mathbf{0}\\}$, $\exists
c_{2}\in\mathscr{C},\langle c_{1},c_{2}\rangle_{*}\neq 0$ holds.

###### Proof 1

It is obvious that $\mathscr{C}\cap\mathscr{C}^{\perp_{*}}=\\{\mathbf{0}\\}$
is equivalent with $\forall c_{1}\in\mathscr{C}\setminus\\{\mathbf{0}\\}$,
$c_{1}\notin\mathscr{C}^{\perp_{*}}$. Moreover,
$c_{1}\notin\mathscr{C}^{\perp_{*}}$ is equivalent with $\exists
c_{2}\in\mathscr{C},\langle c_{1},c_{2}\rangle_{*}\neq 0$, so $\mathscr{C}$ is
LCD equivalent with $\forall c_{1}\in\mathscr{C}\setminus\\{\mathbf{0}\\}$,
$\exists c_{2}\in\mathscr{C},\langle c_{1},c_{2}\rangle_{*}\neq 0$.

For ease of presentation, we give the following definition.

###### Definition 1

Let $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ are linear codes over
$\mathbb{F}_{q}$, if the following conditions hold:

$\left\\{\begin{array}[]{c}\forall
c_{1}\in\mathscr{C}_{1}\setminus\\{\mathbf{0}\\},\exists
c_{2}\in\mathscr{C}_{2},\langle c_{1},c_{2}\rangle_{*}\neq 0,\\\ \forall
c_{2}\in\mathscr{C}_{2}\setminus\\{\mathbf{0}\\},\exists
c_{1}\in\mathscr{C}_{1},\langle c_{1},c_{2}\rangle_{*}\neq
0,\end{array}\right.$ (7)

then we call $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ completely non-orthogonal
to each other under inner product “$*$”.

###### Lemma 2

Let $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ are linear codes over
$\mathbb{F}_{q}$, then $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ completely non-
orthogonal to each other holds if and noly if
$\mathscr{C}_{1}\cap\mathscr{C}_{2}^{\perp_{*}}=\\{\mathbf{0}\\}$ and
$\mathscr{C}_{1}^{\perp_{*}}\cap\mathscr{C}_{2}=\\{\mathbf{0}\\}$.

###### Proof 2

This lemma holds from the definition of dual codes.

###### Lemma 3

Let $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ are two cyclic codes , and
separately generated by $g_{1}(x)$ and $g_{2}(x)$, where $g_{1}(x)\mid
x^{n}-1$ and $g_{2}(x)\mid x^{n}-1$, then $\mathscr{C}_{1}$ and
$\mathscr{C}_{2}$ completely Euclidean non-orthogonal to each other is
equivalent with $g_{1}(x)=\tilde{g}_{1}(x)=g_{2}(x)$, and $\mathscr{C}_{1}$
and $\mathscr{C}_{2}$ completely Hermitian non-orthogonal to each other is
equivalent with $g_{1}(x)=\tilde{g}^{q}_{1}(x)=g_{2}(x)$.

###### Proof 3

With [26], $\mathscr{C}_{1}\cap\mathscr{C}_{2}^{\perp_{l}}$ and
$\mathscr{C}_{1}^{\perp_{l}}\cap\mathscr{C}_{2}$ are both cyclic codes
generated by $lcm(g_{1}(x),g_{2}^{\perp_{l}}(x))$ and
$lcm(g_{1}^{\perp_{l}}(x),g_{2}(x))$, respectively. Further,
$\mathscr{C}_{1}\cap\mathscr{C}_{2}^{\perp_{l}}=\\{\mathbf{0}\\}$ and
$\mathscr{C}_{1}^{\perp_{l}}\cap\mathscr{C}_{2}=\\{\mathbf{0}\\}$ yield
$lcm(g_{1}(x),g_{2}^{\perp_{l}}(x))\equiv 0\pmod{x^{n}-1}$ and
$lcm(g_{1}^{\perp_{l}}(x),g_{2}(x))\equiv 0\pmod{x^{n}-1}$. Thus, there are
two cases, the first is $\tilde{g}_{2}(x)\mid g_{1}(x)$ and
$\tilde{g}_{1}(x)\mid g_{2}(x)\Longleftrightarrow
g_{1}(x)=\tilde{g}_{1}(x)=g_{2}(x)$. The second is $\tilde{g}^{q}_{2}(x)\mid
g_{1}(x)$ and $\tilde{g}^{q}_{1}(x)\mid g_{2}(x)\Longleftrightarrow
g_{1}(x)=\tilde{g}^{q}_{1}(x)=g_{2}(x)$. Hence, we complete the proof.

##  4 Quasi-cyclic complementary dual codes

In this section, we determine the sufficient and necessary conditions for
quasi-cyclic codes to be LCD codes under Euclidean, Hermitian, and symplectic
inner products, starting from Lemma 1.

Some of the symbols used in this paper are described below for ease of
expression. Let $g(x)=g_{0}+g_{1}x+g_{2}x+\cdots+g_{n-1}x^{n-1}\in\mathbb{R}$,
$[g(x)]$ denote vector defined by coefficients of $g(x)$ in
$\mathbb{F}_{q}^{n}$, i.e.
$[g(x)]=[{{g}_{0}},{{g}_{1}},{{g}_{2}},\cdots,{{g}_{n-1}}]$, and
$\bar{g}(x)=x^{n}g(x^{-1})$.

In order to determine the Euclidean inner product between different
polynomials in coefficient vector form, the following two lemmas are crucial.

###### Lemma 4

([28]) Let $f(x)$, $g(x)$ and $h(x)$ be polynomials in $\mathbb{R}$. Then the
following equation holds for the Euclidean inner product among them:

$\langle[f(x)g(x)],[h(x)]\rangle_{e}=\langle[g(x)],[\bar{f}(x)h(x)]\rangle_{e}.$
(8)

###### Lemma 5

([29]) Let $f(x)$, $g(x)$ and $h(x)$ be monic polynomials in $\mathbb{R}$.
Then the following equality of Hermitian inner product of vectors in
$\mathbb{F}_{q^{2}}^{n}$ holds:

$\langle[f(x)g(x)],[h(x)]\rangle_{h}=\langle[g(x)],[\bar{f}^{q}(x)h(x)]\rangle_{h}.$

###  4.1 One-generator complementary dual quasi-cyclic codes

###### Definition 2

Let $g(x)$ and $f_{j}(x)$ are monic polynomials in $\mathbb{R}$, and
$g(x)\mid(x^{n}-1)$, $0\leq j\leq\ell-1$. If $\mathscr{C}$ is a quasi-cyclic
code generated by $([g(x){{f}_{0}}(x)]$, $[g(x){{f}_{1}}(x)]$, $\cdots$,
$[g(x){{f}_{\ell-1}}(x)])$, then $\mathscr{C}$ is called $1$-generator quasi-
cyclic code with index $\ell$. A genrartor matrix $G$ of $\mathscr{C}$ have
the following form:

$G=\left(G_{0},G_{1},\cdots,G_{\ell-1}\right),$ (9)

where $G_{j}$ are $n\times n$ circulant matrices generated by
$[g(x)f_{j}(x)]$, respectively.

###### Theorem 6

If $\mathscr{C}$ is a $1$-generator quasi-cyclic code in Definition 2, then
the sufficient and necessary conditions for $\mathscr{C}$ to be Euclidean LCD
code are

$\left\\{\begin{array}[]{c}g(x)=\tilde{g}(x),\\\
gcd(\sum\limits_{i=0}^{\ell-1}f_{i}(x)\bar{f}_{i}(x),\frac{x^{n}-1}{g(x)})=1.\end{array}\right.$
(10)

###### Proof 4

Suppose $a(x)$,$b(x)$ are any polynomials in $\mathbb{R}$, then any two
codewords in $\mathscr{C}$ can be represented as
$c_{1}=([a(x)g(x){{f}_{0}}(x)],[a(x)g(x){{f}_{1}}(x)],\cdots,[a(x)g(x){{f}_{\ell-1}}(x)])$
and
$c_{2}=([b(x)g(x){{f}_{0}}(x)],[b(x)g(x){{f}_{1}}(x)],\cdots,[b(x)g(x){{f}_{\ell-1}}(x)])$,
respectively. The Euclidean inner product of $c_{1}$ and $c_{2}$ can be
expressed as:

$\begin{array}[]{rl}\langle
c_{1},c_{2}\rangle_{e}=&\sum\limits_{i=0}^{\ell-1}\langle[a(x)g(x){{f}_{i}}(x)],[b(x)g(x){{f}_{i}}(x)]\rangle_{e}\\\
=&\sum\limits_{i=0}^{\ell-1}\langle[a(x)g(x){{f}_{i}}(x)\bar{f}_{i}(x)],[b(x)g(x)]\rangle_{e}\\\
=&\langle[a(x)g(x)\sum\limits_{i=0}^{\ell-1}{{f}_{i}}(x)\bar{f}_{i}(x)],[b(x)g(x)]\rangle_{e}.\\\
\end{array}$

From Lemma 1 and 3, it is clear that the sufficient and necessary conditions
for $\mathscr{C}$ to be Euclidean LCD code is that $g(x)=\tilde{g}(x)$ and
$gcd(\sum\limits_{i=0}^{\ell-1}f_{i}(x)\bar{f}_{i}(x),\frac{x^{n}-1}{g(x)})=1$,
so this theorem is proved.

Since the Hermitian inner product and the Euclidean inner product have a
similar form, an analogous approach yields the sufficient and necessary
conditions for 1-generator quasi-cyclic code to be a Hermitian LCD code.
Therefore, we give the following theorem without proof.

###### Theorem 7

If $\mathscr{C}$ is a $1$-generator quasi-cyclic code in Definition 2, then
the sufficient and necessary conditions for $\mathscr{C}$ to be Hermitian LCD
code are

$\left\\{\begin{array}[]{c}g(x)=\tilde{g}^{q}(x),\\\
gcd(\sum\limits_{i=0}^{\ell-1}f_{i}(x)\bar{f}^{q}_{i}(x),\frac{x^{n}-1}{g(x)})=1.\end{array}\right.$
(11)

###### Theorem 8

If $\mathscr{C}$ is a $1$-generator quasi-cyclic code in Definition 2, and
$\ell$ is even. Let $m=\ell/2$, then $\mathscr{C}$ is a symplectic LCD code if
and only if the following equations hold.

$\left\\{\begin{array}[]{c}g(x)=\tilde{g}(x),\\\
gcd(\sum\limits_{j=0}^{m-1}({{f}_{j}}(x){{\bar{f}}_{m+j}}(x)-{{f}_{m+j}}(x){{\bar{f}}_{j}}(x)),\frac{x^{n}-1}{g(x)})=1.\end{array}\right.$
(12)

###### Proof 5

Suppose $a(x)$,$b(x)$ are any polynomials in $\mathbb{R}$, then any two
codewords in $\mathscr{C}$ can be represented as
$c_{1}=([a(x)g(x){{f}_{0}}(x)],[a(x)g(x){{f}_{1}}(x)],\cdots,[a(x)g(x){{f}_{\ell-1}}(x)])$
and
$c_{2}=([b(x)g(x){{f}_{0}}(x)],[b(x)g(x){{f}_{1}}(x)],\cdots,[b(x)g(x){{f}_{\ell-1}}(x)])$,
respectively. The symplectic inner product of $c_{1}$ and $c_{2}$ can be
expressed as:

$\begin{array}[]{rl}\langle
c_{1},c_{2}\rangle_{s}=&c_{1}\cdot\left(\begin{array}[]{cc}0&I_{mn}\\\
-I_{mn}&0\end{array}\right)\cdot c_{2}^{T}\\\
=&\sum\limits_{j=0}^{m-1}{{{\langle{[{a(x)g(x){f_{j}}(x)}],[{b(x)g(x){f_{m+j}}(x)}]}\rangle}_{e}}}\\\
&-\sum\limits_{j=0}^{m-1}{{{\langle{[{a(x)g(x){f_{m+j}}(x)}],[{b(x)g(x){f_{j}}(x)}]}\rangle}_{e}}}\\\
=&\sum\limits_{j=0}^{m-1}{{{\langle{[{a(x)g(x){f_{j}}(x){{\bar{f}}_{m+j}}(x)}],[{b(x)g(x)}]}\rangle}_{e}}}\\\
&-\sum\limits_{j=0}^{m-1}{{{\langle{[{a(x)g(x){f_{m+j}}(x){{\bar{f}}_{j}}(x)}],[{b(x)g(x)}]}\rangle}_{e}}}\\\
=&\langle[a(x)g(x)\sum\limits_{j=0}^{m-1}({f_{j}}(x){{\bar{f}}_{m+j}}(x)-{f_{m+j}}(x){{\bar{f}}_{j}}(x))],[b(x)g(x)]\rangle_{e}.\end{array}$

From Lemma 1 and 3, it is clear that the sufficient and necessary conditions
for $\mathscr{C}$ to be symplectic LCD code is that $g(x)=\tilde{g}(x)$ and
$gcd(\sum\limits_{j=0}^{m-1}({{f}_{j}}(x){{\bar{f}}_{m+j}}(x)-{{f}_{m+j}}(x){{\bar{f}}_{j}}(x)),\frac{x^{n}-1}{g(x)})=1$,
so this theorem is proved.

###  4.2 Multi-generator complementary dual quasi-cyclic codes

A multi-generator quasi-cyclic code is a quasi-cyclic code with more than two
generators which has a richer algebraic structure than a 1-generator quasi-
cyclic code. The specific definition of multi-generator quasi-cyclic code is
given below.

###### Definition 3

Let $g_{i}(x)$, $f_{i,j}(x)$ are monic polynomials in $\mathbb{R}$, where
$g_{i}(x)\mid(x^{n}-1)$, and $1\leq i\leq h$, $0\leq j\leq\ell-1$. If
$\mathscr{C}$ is a quasi-cyclic code with $h$-generators:
$([g_{1}(x)f_{1,0}(x)],[g_{1}(x)f_{1,1}(x)],\cdots,[g_{1}(x)f_{1,\ell-1}(x)])$,
$([g_{2}(x)f_{2,0}(x)],[g_{2}(x)f_{2,1}(x)]$, $\cdots$,
$[g_{2}(x)f_{2,\ell-1}(x)]),\ldots,([g_{h}(x)f_{h,0}(x)],[g_{h}(x)f_{h,1}(x)]$,
$\cdots$,$[g_{h}(x)f_{h,\ell-1}(x)])$, then $\mathscr{C}$ is called
$h$-generator quasi-cyclic code with index $\ell$, whose generator matrix $G$
has the following form:

$G=\left(\begin{array}[]{cccc}G_{1,0}&G_{1,1}&\cdots&G_{1,\ell-1}\\\
G_{2,0}&G_{2,1}&\cdots&G_{2,\ell-1}\\\ \vdots&\vdots&\ddots&\vdots\\\
G_{h,0}&G_{h,1}&\cdots&G_{h,\ell-1}\end{array}\right),$ (13)

where $G_{i,j}$ are $n\times n$ circulant matrices generated by
$[g_{i}(x)f_{i,j}(x)]$, respectively.

It follows from Definition 3 that a multi-generator quasi-cyclic code can be
considered a code generated by juxtaposing multiple 1-generator quasi-cyclic
codes up and down. Therefore, before determining the overall complete non-
orthogonality of multi-generator quasi-cyclic code, the relationship between
the different 1-generator quasi-cyclic codes as constituents needs to be
determined.

###### Theorem 9

Let $g_{i}(x)$, $f_{i,j}(x)$ are monic polynomials in $\mathbb{R}$, and
$g_{i}(x)\mid(x^{n}-1)$, $1\leq i\leq 2$, $0\leq j\leq\ell-1$.
$\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ are 1-generator quasi-cyclic codes
with generators
$([g_{1}(x)f_{1,0}(x)],[g_{1}(x)f_{1,1}(x)],\cdots,[g_{1}(x)f_{1,\ell-1}(x)])$
and
$([g_{2}(x)f_{2,0}(x)],[g_{2}(x)f_{2,1}(x)],\cdots,[g_{2}(x)f_{2,\ell-1}(x)])$,
respectively. Then, $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ complete Euclidean
non-othogaonal with each other is equivalent with

$\left\\{\begin{array}[]{c}g_{1}(x)=g_{2}(x)=\tilde{g}_{1}(x),\\\
gcd(\sum\limits_{j=0}^{\ell-1}{{f}_{1,j}}(x)\bar{f}_{2,j}(x),\frac{x^{n}-1}{g_{1}(x)})=1,\end{array}\right.$
(14)

where $G_{i,j}$ are $n\times n$ circulant matrices generated by
$[g_{i}(x)f_{i,j}(x)]$, respectively.

###### Proof 6

Assume that $a(x)$ and $b(x)$ are arbitrary polynomials in $\mathbb{R}$, then
any codewords in $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ can be represented as
$c_{1}=([a(x)g_{1}(x)f_{1,0}(x)],[a(x)g_{1}(x)f_{1,1}(x)],\cdots,[a(x)g_{1}(x)f_{1,\ell-1}(x)])$
and $c_{2}=$ $([b(x)g_{2}(x)f_{2,0}(x)],$ $[b(x)g_{2}(x)f_{2,1}(x)],$
$\cdots,[b(x)g_{2}(x)f_{2,\ell-1}(x)])$, respectively. The Euclidean inner
product of $c_{1}$ and $c_{2}$ can be expressed as:

$\begin{array}[]{rl}\langle
c_{1},c_{2}\rangle_{e}=&\sum\limits_{j=0}^{\ell-1}\langle[a(x)g_{1}(x){{f}_{1,j}}(x)],[b(x)g_{2}(x){{f}_{2,j}}(x)]\rangle_{e}\\\
=&\sum\limits_{j=0}^{\ell-1}\langle[a(x)g_{1}(x){{f}_{1,j}}(x)\bar{f}_{2,j}(x)],[b(x)g_{2}(x)]\rangle_{e}\\\
=&\langle[a(x)g_{1}(x)\sum\limits_{j=0}^{\ell-1}{{f}_{1,j}}(x)\bar{f}_{2,j}(x)],[b(x)g_{2}(x)]\rangle_{e}.\end{array}$

From Lemma 3, it is clear that $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$
completely Euclidean non-othogaonal with each other is equivalent with
$g_{1}(x)=g_{2}(x)=\tilde{g}_{1}(x)$ and
$gcd(\sum\limits_{j=0}^{\ell-1}{{f}_{1,j}}(x)\bar{f}_{2,j}(x),\frac{x^{n}-1}{g_{1}(x)})=1$,
so this theorem is proved.

There is a similar result under Hermitian inner product, so we give the
following theorem without proof.

###### Theorem 10

Let $g_{i}(x)$, $f_{i,j}(x)$ are monic polynomials in $\mathbb{R}$, and
$g_{i}(x)\mid\left(x^{n}-1\right)$, $1\leq i\leq 2$, $0\leq j\leq\ell-1$.
$\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ are 1-generator quasi-cyclic codes
with generators
$\left([g_{1}(x)f_{1,0}(x)],[g_{1}(x)f_{1,1}(x)],\cdots,[g_{1}(x)f_{1,\ell-1}(x)]\right)$
and
$\left([g_{2}(x)f_{2,0}(x)],[g_{2}(x)f_{2,1}(x)],\cdots,[g_{2}(x)f_{2,\ell-1}(x)]\right)$,
respectively. Then, $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ complete Hermitian
non-othogaonal with each other is equivalent with

$\left\\{\begin{array}[]{c}g_{1}(x)=g_{2}(x)=\tilde{g}_{1}^{q}(x),\\\
gcd(\sum\limits_{j=0}^{\ell-1}f_{1,j}(x)\bar{f}^{q}_{2,j}(x),\frac{x^{n}-1}{g_{1}(x)})=1,\end{array}\right.$
(15)

where $G_{i,j}$ are $n\times n$ circulant matrices generated by
$[g_{i}(x)f_{i,j}(x)]$, respectively.

###### Theorem 11

Let $\ell$ be even, $m=\ell/2$, $g_{i}(x)$, $f_{i,j}(x)$ are monic polynomials
in $\mathbb{R}$, and $g_{i}(x)\mid(x^{n}-1)$, $1\leq i\leq 2$, $0\leq
j\leq\ell-1$. $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ are 1-generator quasi-
cyclic codes with generators
$([g_{1}(x)f_{1,0}(x)],[g_{1}(x)f_{1,1}(x)],\cdots,[g_{1}(x)f_{1,\ell-1}(x)])$
and
$([g_{2}(x)f_{2,0}(x)],[g_{2}(x)f_{2,1}(x)],\cdots,[g_{2}(x)f_{2,\ell-1}(x)])$,
respectively. Then, $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ complete
symplectic non-orthogonal with each other is equivalent with

$\left\\{\begin{array}[]{c}g_{1}(x)=g_{2}(x)=\tilde{g}_{1}(x),\\\
gcd(\sum\limits_{j=0}^{m-1}({f_{1,j}}(x){{\bar{f}}_{2,m+j}}(x)-{f_{1,m+j}}(x){{\bar{f}}_{2,j}}(x)),\frac{x^{n}-1}{g_{1}(x)})=1,\end{array}\right.$
(16)

where $G_{i,j}$ are $n\times n$ circulant matrices generated by
$[g_{i}(x)f_{i,j}(x)]$, respectively.

###### Proof 7

Suppose $a(x)$ and $b(x)$ are any polynomials in $\mathbb{R}$, then any two
codewords in $\mathscr{C}$ can be represented as
$c_{1}=([a(x)g_{1}(x)f_{1,0}(x)],[a(x)g_{1}(x)f_{1,1}(x)],$
$\cdots,[a(x)g_{1}(x)f_{1,\ell-1}(x)])$ and
$c_{2}=([b(x)g_{2}(x)f_{2,0}(x)],[b(x)g_{2}(x)f_{2,1}(x)],$
$\cdots,[b(x)g_{2}(x)f_{2,\ell-1}(x)])$, respectively. The symplectic inner
product of $c_{1}$ and $c_{2}$ can be expressed as:

$\begin{array}[]{rl}\langle
c_{1},c_{2}\rangle_{s}=&c_{1}\cdot\left(\begin{array}[]{cc}0&I_{mn}\\\
-I_{mn}&0\end{array}\right)\cdot c_{2}^{T}\\\
=&\sum\limits_{j=0}^{m-1}{{{\langle{[{a(x)g_{1}(x)f_{1,j}(x)}],[{b(x)g_{2}(x){f_{2,m+j}}(x)}]}\rangle}_{e}}}\\\
&-\sum\limits_{j=0}^{m-1}{{{\langle{[{a(x)g_{1}(x){f_{1,m+j}}(x)}],[{b(x)g_{2}(x){f_{2,j}}(x)}]}\rangle}_{e}}}\\\
=&\sum\limits_{j=0}^{m-1}{{{\langle{[{a(x)g_{1}(x){f_{1,j}}(x){{\bar{f}}_{2,m+j}}(x)}],[{b(x)g_{2}(x)}]}\rangle}_{e}}}\\\
&-\sum\limits_{j=0}^{m-1}{{{\langle{[{a(x)g_{1}(x){f_{1,m+j}}(x){{\bar{f}}_{2,j}}(x)}],[{b(x)g_{2}(x)}]}\rangle}_{e}}}\\\
=&\langle[a(x)g_{1}(x)\sum\limits_{j=0}^{m-1}({f_{1,j}}(x){{\bar{f}}_{2,m+j}}(x)-{f_{1,m+j}}(x){{\bar{f}}_{2,j}}(x))],[b(x)g_{2}(x)]\rangle_{e}.\end{array}$

From Lemma 1 and 3, it is clear that the sufficient and necessary conditions
for $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ complete symplectic non-othogaonal
with each other is is that $g_{1}(x)=g_{2}(x)=\tilde{g}_{1}(x)$ and
$gcd(\sum\limits_{j=0}^{m-1}({f_{1,j}}(x){{\bar{f}}_{2,m+j}}(x)-{f_{1,m+j}}(x){{\bar{f}}_{2,j}}(x)),\frac{x^{n}-1}{g_{1}(x)})=1$,
so this theorem is proved.

###### Theorem 12

Let $\mathscr{C}$ be an $h$-generator quasi-cyclic code in Definition 3 and
$\mathscr{C}_{i}$ are 1-generator quasi-cyclic codes generated by
$([g_{i}(x)f_{i,0}(x)],[g_{i}(x)f_{i,1}(x)],\cdots,[g_{i}(x)f_{i,\ell-1}(x)])$,
$i\in\\{1,2,\cdots,h\\}$. Then $\mathscr{C}$ is LCD code if and only if
$\forall r,s\in\\{1,2,\cdots,h\\}$, there exist
$\mathscr{C}_{r}\cap\mathscr{C}_{s}^{\perp_{*}}=\\{\mathbf{0}\\}$.

###### Proof 8

With Definition 3,
$\mathscr{C}=\mathscr{C}_{1}+\mathscr{C}_{2}+\cdots+\mathscr{C}_{h}$.
$\mathscr{C}$ is LCD if and only if
$\mathscr{C}\cap\mathscr{C}^{\perp_{*}}=\\{\mathbf{0}\\}$, i.e., $\forall
c_{1}\in\mathscr{C}$, there exist $c_{2}\in\mathscr{C}$, $\langle
c_{1},c_{2}\rangle_{*}\neq 0$. Therefore, $\forall r,s\in\\{1,2,\cdots,h\\}$,
there is $\mathscr{C}_{r}\cap\mathscr{C}_{s}^{\perp_{*}}=\\{\mathbf{0}\\}$.

In addition,
$\mathscr{C}^{\perp_{*}}=\mathscr{C}_{1}^{\perp_{*}}\cap\mathscr{C}_{2}^{\perp_{*}}\cap\cdots\cap\mathscr{C}_{h}^{\perp_{*}}$.
For the reason that $\forall r,s\in\\{1,2,\cdots,h\\}$, there is
$\mathscr{C}_{r}\cap\mathscr{C}_{s}^{\perp_{*}}=\\{\mathbf{0}\\}$, so
$\mathscr{C}\cap\mathscr{C}^{\perp_{*}}=\\{\mathbf{0}\\}$.

It is evident that by combining Theorems 6-12, we have the following theorems.

###### Theorem 13

Let $\mathscr{C}$ be $h$-generator quasi-cyclic code with index $\ell$ over
$\mathbb{F}_{q}$ in Definition 3, then the sufficient and necessary conditions
for $\mathscr{C}$ to be Euclidean LCD code are

$\left\\{\begin{array}[]{c}g_{1}(x)=g_{2}(x)=\dots=g_{h}(x)=\tilde{g}_{1}(x),\\\
\gcd(\sum\limits_{j=0}^{\ell-1}f_{r,j}(x)\bar{f}_{s,j}(x),\frac{x^{n}-1}{g_{1}(x)})=1,\forall
r,s\in\\{1,2,\cdots,h\\}.\end{array}\right.$ (17)

###### Theorem 14

Let $\mathscr{C}$ be $h$-generator quasi-cyclic code with index $\ell$ over
$\mathbb{F}_{q^{2}}$ in Definition 3, then the sufficient and necessary
conditions for $\mathscr{C}$ to be Hermitian LCD code are

$\left\\{\begin{array}[]{c}g_{1}(x)=g_{2}(x)=\dots=g_{h}(x)=\tilde{g}_{1}^{q}(x),\\\
gcd(\sum\limits_{j=0}^{\ell-1}f_{r,j}(x)\bar{f}^{q}_{s,j}(x),\frac{x^{n}-1}{g_{1}(x)})=1,\forall
r,s\in\\{1,2,\cdots,h\\}.\end{array}\right.$ (18)

###### Theorem 15

Let $\ell$ be even, $m=\ell/2$, $\mathscr{C}$ be $h$-generator quasi-cyclic
code with index $\ell$ over $\mathbb{F}_{q}$ in Definition 3, then the
sufficient and necessary conditions for $\mathscr{C}$ to be symplectic LCD
code are

$\left\\{\begin{array}[]{c}g_{1}(x)=g_{2}(x)=\dots=g_{h}(x)=\tilde{g}_{1}(x),\\\
gcd(\sum\limits^{m-1}_{i=0}(f_{s,i}(x)\bar{f}_{r,m+i}(x)-f_{s,m+i}(x)\bar{f}_{r,i}(x)),\frac{x^{n}-1}{g_{1}(x)})=1,\forall
r,s\in\\{1,2,\cdots,h\\}.\end{array}\right.$ (19)

##  5 Some examples of good complementary dual quasi-cyclic codes

In this section, we provide some examples from our construction of good LCD
codes to help the reader better understand the approach of this paper.

Specifically, Examples 1, 2, and 3 explain the construction methods of the
Euclidean, Hermitian, and symplectic LCD codes in Section 4, respectively.

###### Example 1

Set $q=2$, $n=13$. Let $g(x)=1$, $f_{0}(x)=1$,
$f_{1}(x)=x^{12}+x^{7}+x^{3}+x+1$, and
$f_{2}(x)=x^{12}+x^{11}+x^{9}+x^{8}+x^{5}+x^{3}+x^{2}$. One can easy to check
that $g(x)=\tilde{g}(x)$,
$gcd(\sum_{i=0}^{2}f_{i}(x)\bar{f}_{i}(x),\frac{x^{n}-1}{g(x)})=1$, so
$([g(x)f_{0}(x)],[g(x)f_{1}(x)],[g(x)f_{2}(x)])$ can generate a 1-generator
quasi-cyclic Euclidean LCD code. Then, using Magma [25] we can compute this
code have parameters $[39,13,12]_{2}$, whose weight distribution is
$w(z)=1+39z^{12}+208z^{13}+286z^{14}+325z^{15}+546z^{16}+\dots+z^{39}$.
Observe that a code with parameters $[39,13,11]_{2}$ is the best-known binary
Euclidean LCD with length $39$ and dimension $13$ in [15]. Therefore, the
corresponding minimum distance record can be improved to $12$.

###### Example 2

Set $q^{2}=4$, $n=19$. Let $w$ be a primitive element of $F_{4}$. Let
$g(x)=x+1$, and
$f_{0}(x)=w^{2}x^{18}+x^{17}+w^{2}x^{16}+x^{15}+x^{14}+w^{2}x^{13}+wx^{12}+wx^{11}+w^{2}x^{10}+x^{9}+x^{8}+w^{2}x^{7}+wx^{6}+x^{5}+wx^{4}+x^{3}+x^{2}+wx+1$,
$f_{1}(x)=x^{15}+w^{2}x^{14}+wx^{12}+w^{2}x^{10}+wx^{7}+x^{6}+wx^{5}+wx^{4}+wx^{3}+wx^{2}+w^{2}x+1$.
One can easy to check that $g(x)=\tilde{g}^{q}(x)$,
$gcd(f_{0}(x)\bar{f}^{q}_{0}(x)+f_{1}(x)\bar{f}^{q}_{1}(x),\frac{x^{n}-1}{g(x)})=1$,
so $([g(x)f_{0}(x)],[g(x)f_{1}(x)])$ can generate a 1-generator quasi-cyclic
Hermitian LCD code. Then, using Magma [25] we can compute this code have
parameters $[38,18,12]_{4}$, and it’s dual is $[38,20,11]_{4}$, their weight
distributions are
$w(z)=1+912z^{12}+7296z^{13}+44859z^{14}+199842z^{15}+886977z^{16}+\dots+1229205z^{38}$
and
$w(z^{\perp})=1+2736z^{11}+21888z^{12}+127224z^{13}+684171z^{14}+3211722z^{15}+\dots+19686954z^{38}$,
respectively. It is worth stating that both $[38,18,12]_{4}$ and
$[38,20,11]_{4}$ reach the minimum distance lower bounds in code table [30],
so they are both best codes.

###### Example 3

Set $q=2$, $n=21$. Let $g(x)=x^{3}+1$,
$f_{0}(x)=x^{18}+x^{16}+x^{15}+x^{14}+x^{13}+x^{12}+x^{8}+x^{7}+x^{3}+1$,
$f_{1}(x)=x^{20}+x^{19}+x^{18}+x^{15}+x^{14}+x^{9}+x^{7}+x^{5}+x^{3}+x^{2}+1$.
One can easy to check that $g(x)=\tilde{g}(x)$,
$gcd(f_{0}(x)\bar{f}_{1}(x)-f_{1}(x)\bar{f}_{0}(x),\frac{x^{n}-1}{g(x)})=1$,
so $([g(x)f_{0}(x)],[g(x)f_{1}(x)])$ can generate a 1-generator quasi-cyclic
symplectic LCD code. Then, using Magma [25] we can calculate this code have
parameters $[42,18,9]_{2}^{s}$, whose symplectic weight distribution is
$w(z)=1+448z^{9}+1344z^{10}+3906z^{11}+9051z^{12}+18753z^{13}+\dots+609z^{21}$.
Therefore, there exists a trace Hermitian ACD code with parameters
$(21,9,9)_{4}$. It should be noted that the best known Hermitian LCD code in
[14] with length 21 and dimension 9 have parameters $[21,9,8]_{4}$, so our
symplectic construction has better performance.

##  6 Conclusion

In this work, we propose a new characterization for LCD codes, which allows us
to determine the complementary duality of linear codes from the codeword
level. Furthermore, depending on this result, we determine the necessary and
sufficient conditions for quasi-cyclic codes to be LCD codes concerning
Euclidean, Hermitian, and symplectic inner products. Finally, we give some
specific examples of the construction of Euclidean, Hermitian, and symplectic
quasi-cyclic LCD codes to show that quasi-cyclic codes can be utilized to
construct good LCD codes.

However, by Theorems 13, 14 and 15, the reader can find that when the quasi-
cyclic codes are multi-generated, different generators can only pick the same
self-reciprocal polynomial, which significantly limits the performance of the
quasi-cyclic LCD codes. Therefore, research on quasi-cyclic LCD codes should
mainly focus on 1-generator quasi-cyclic codes, which will help to construct
quasi-cyclic LCD codes of low dimensions. In the case of LCD codes with
dimensions close to $\ell n/2$, it is required to use 1-generator quasi-cyclic
codes with a small index or other mathematical tools. We hope this will
attract scholars’ interest in research related to quasi-cyclic codes and
advance this area together.

##  7 Acknowledgments

This work is supported by the National Natural Science Foundation of China
under Grant No.U21A20428, 61972413, 61901525, 62002385, Natural Science
Foundation of Shaanxi under Grant No.2021JM-216, 2021JQ-335.

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