then for each fixed $A_{i}\in\mathscr{F}_{i}$, $i=1,3$, $E_{[X_{2},X_{3}\in
A_{3}]}(Y|X_{1}\in A_{1})\in\mathbb{R}$, and

$E_{[X_{2},X_{3}\in A_{3}]}(Y_{n}|X_{1}\in A_{1})\downarrow E_{[X_{2},X_{3}\in
A_{3}]}(Y|X_{1}\in A_{1}).$

Especially,

$E_{[X_{2},X_{3}\in A_{3}]}(Y_{n})\downarrow E_{[X_{2},X_{3}\in A_{3}]}(Y),\ \
and\ \ E_{[X_{2}]}(Y_{n})\downarrow E_{[X_{2}]}(Y).$

######  Corollary 6.8.

Let $(\Omega,\mathscr{F},P)$ be a given probability space,
$(\Omega_{i},\mathscr{F}_{i}),\ i\in\Lambda$ measurable spaces, where
$\Lambda$ is a nonempty index set. Let
$X_{i}(\omega):(\Omega,\mathscr{F})\rightarrow(\Omega_{i},\mathscr{F}_{i}),\
i\in\Lambda$ be random objects. Let $I_{i}\subset\Lambda,\ i=1,2,3$, which are
pairwise disjoint. Let $Y_{n}(\omega_{I_{2}}),n=1,2,3,\cdots$ be random
variables on $(\Omega_{I_{2}},\mathscr{F}_{I_{2}},P[X_{I_{2}}])$ with
$E_{[X_{I_{2}}]}(Y_{n})\in\mathbb{R},\ n=1,2,3,\cdots$.

1\. If $Y_{n}\leq Y_{n+1},\ a.e.\\{P[X_{I_{2}}]\\},\ n=1,2,3,\cdots$,

$\lim\limits_{n\to\infty}Y_{n}=Y,\ a.e.\\{P[X_{I_{2}}]\\},\ \ and\ \
\sup\limits_{n}\\{E_{[X_{I_{2}}]}(Y_{n})\\}\in\mathbb{R},$

then for each fixed $A_{I_{i}}\in\mathscr{F}_{I_{i}}$, $i=1,3$,
$E_{[X_{I_{2}},X_{I_{3}}\in A_{I_{3}}]}(Y|X_{I_{1}}\in
A_{I_{1}})\in\mathbb{R}$, and

$E_{[X_{I_{2}},X_{I_{3}}\in A_{I_{3}}]}(Y_{n}|X_{I_{1}}\in A_{I_{1}})\uparrow
E_{[X_{I_{2}},X_{I_{3}}\in A_{I_{3}}]}(Y|X_{I_{1}}\in A_{I_{1}}).$

Especially,

$E_{[X_{I_{2}},X_{I_{3}}\in A_{I_{3}}]}(Y_{n})\uparrow
E_{[X_{I_{2}},X_{I_{3}}\in A_{I_{3}}]}(Y),\ \ and\ \
E_{[X_{I_{2}}]}(Y_{n})\uparrow E_{[X_{I_{2}}]}(Y).$

2\. If $Y_{n}\geq Y_{n+1},\ a.e.\\{P[X_{I_{2}}]\\},\ n=1,2,3,\cdots$,

$\lim\limits_{n\to\infty}Y_{n}=Y,\ a.e.\\{P[X_{I_{2}}]\\},\ \ and\ \
\inf\limits_{n}\\{E_{[X_{I_{2}}]}(Y_{n})\\}\in\mathbb{R},$

then for each fixed $A_{I_{i}}\in\mathscr{F}_{I_{i}}$, $i=1,3$,
$E_{[X_{I_{2}},X_{I_{3}}\in A_{I_{3}}]}(Y|X_{I_{1}}\in
A_{I_{1}})\in\mathbb{R}$, and

$E_{[X_{I_{2}},X_{I_{3}}\in A_{I_{3}}]}(Y_{n}|X_{I_{1}}\in
A_{I_{1}})\downarrow E_{[X_{I_{2}},X_{I_{3}}\in A_{I_{3}}]}(Y|X_{I_{1}}\in
A_{I_{1}}).$

Especially,

$E_{[X_{I_{2}},X_{I_{3}}\in A_{I_{3}}]}(Y_{n})\downarrow
E_{[X_{I_{2}},X_{I_{3}}\in A_{I_{3}}]}(Y),\ \ and\ \
E_{[X_{I_{2}}]}(Y_{n})\downarrow E_{[X_{I_{2}}]}(Y).$

######  Theorem 6.8.

Let $(\Omega,\mathscr{F},P)$ be a given probability space, and let
$(\Omega_{i},\mathscr{F}_{i}),\ i=1,2,3,4$ be measurable spaces. Let
$X_{i}:(\Omega,\mathscr{F})\rightarrow(\Omega_{i},\mathscr{F}_{i}),\
i=1,2,3,4$ be random objects, and $P[X_{i}]$ the probability on
$(\Omega_{i},\mathscr{F}_{i})$ induced by $X_{i}$, $i=1,2,3,4$. Then
$(\Omega_{i},\mathscr{F}_{i},P[X_{i}]),\ i=1,2,3,4$ are probability spaces.
Let $Y_{n}(\omega_{2}),n=1,2,3,\cdots,$ be random variables on
$(\Omega_{2},\mathscr{F}_{2},P[X_{2}])$ with
$E_{[X_{2}]}(Y_{n})\in\mathbb{R},\ n=1,2,3,\cdots$.

1\. If $Y_{n}\leq Y_{n+1},\ a.e.\\{P[X_{2}]\\},\ n=1,2,\cdots$,

$\lim\limits_{n\to\infty}Y_{n}=Y,\ a.e.\\{P[X_{2}]\\},\ \ and\ \
\sup\limits_{n}\\{E_{[X_{2}]}(Y_{n})\\}\in\mathbb{R},$

then $E_{[X_{2}]}(Y)\in\mathbb{R}$, and for $A_{i}\in\mathscr{F}_{i}$,
$i=3,4$,

$E_{[X_{2},X_{4}\in A_{4}]}(Y_{n}|X_{1},X_{3}\in A_{3})\uparrow
E_{[X_{2},X_{4}\in A_{4}]}(Y|X_{1},X_{3}\in A_{3}),\ a.e.\\{P[X_{1},X_{3}\in
A_{3}]\\}$

provided $P[X_{3}\in A_{3}]>0$ while it is evident provided $P[X_{3}\in
A_{3}]=0$.

2\. If $Y_{n}\geq Y_{n+1},\ a.e.\\{P[X_{2}]\\},\ n=1,2,\cdots$,

$\lim\limits_{n\to\infty}Y_{n}=Y,\ a.e.\\{P[X_{2}]\\},\ \ and\ \
\inf\limits_{n}\\{E_{[X_{2}]}(Y_{n})\\}\in\mathbb{R},$

then $E_{[X_{2}]}(Y)\in\mathbb{R}$, and for $A_{i}\in\mathscr{F}_{i}$,
$i=3,4$,

$E_{[X_{2},X_{4}\in A_{4}]}(Y_{n}|X_{1},X_{3}\in A_{3})\downarrow
E_{[X_{2},X_{4}\in A_{4}]}(Y|X_{1},X_{3}\in A_{3}),\ a.e.\\{P[X_{1},X_{3}\in
A_{3}]\\}$

provided $P[X_{3}\in A_{3}]>0$ while it is evident provided $P[X_{3}\in
A_{3}]=0$.

######  Corollary 6.9.

Let $(\Omega,\mathscr{F},P)$ be a given probability space,
$(\Omega_{i},\mathscr{F}_{i}),\ i\in\Lambda$ measurable spaces, where
$\Lambda$ is a nonempty index set. Let
$X_{i}:(\Omega,\mathscr{F})\rightarrow(\Omega_{i},\mathscr{F}_{i}),\
i\in\Lambda$ be random objects. Let $I_{i}\subset\Lambda,\ i=1,2,3,4$, which
are pairwise disjoint. Let $Y_{n}(\omega_{I_{2}}),n=1,2,3,\cdots$ be random
variables on $(\Omega_{I_{2}},\mathscr{F}_{I_{2}},P[X_{I_{2}}])$ with
$E_{[X_{I_{2}}]}(Y_{n})\in\mathbb{R},\ n=1,2,3,\cdots$.

1\. If $Y_{n}\leq Y_{n+1},\ a.e.\\{P[X_{I_{2}}]\\},\ n=1,2,\cdots$,

$\lim\limits_{n\to\infty}Y_{n}=Y,\ a.e.\\{P[X_{I_{2}}]\\},\ \ and\ \
\sup\limits_{n}\\{E_{[X_{I_{2}}]}(Y_{n})\\}\in\mathbb{R},$

then $E_{[X_{I_{2}}]}(Y)\in\mathbb{R}$, and for
$A_{I_{i}}\in\mathscr{F}_{I_{i}}$, $i=3,4$,

$E_{[X_{I_{2}},X_{I_{4}}\in A_{I_{4}}]}(Y_{n}|X_{I_{1}},X_{I_{3}}\in
A_{I_{3}})\uparrow E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(Y|X_{I_{1}},X_{I_{3}}\in A_{I_{3}}),$

$a.e.\\{P[X_{I_{1}},X_{I_{3}}\in A_{I_{3}}]\\}$ provided $P[X_{I_{3}}\in
A_{I_{3}}]>0$ while it is evident provided $P[X_{I_{3}}\in A_{I_{3}}]=0$.

2\. If $Y_{n}\geq Y_{n+1},\ a.e.\\{P[X_{I_{2}}]\\},\ n=1,2,\cdots$,

$\lim\limits_{n\to\infty}Y_{n}=Y,\ a.e.\\{P[X_{I_{2}}]\\},\ \ and\ \
\inf\limits_{n}\\{E_{[X_{I_{2}}]}(Y_{n})\\}\in\mathbb{R},$

then $E_{[X_{I_{2}}]}(Y)\in\mathbb{R}$, and for
$A_{I_{i}}\in\mathscr{F}_{I_{i}}$, $i=3,4$,

$E_{[X_{I_{2}},X_{I_{4}}\in A_{I_{4}}]}(Y_{n}|X_{I_{1}},X_{I_{3}}\in
A_{I_{3}})\downarrow E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(Y|X_{I_{1}},X_{I_{3}}\in A_{I_{3}}),$

$a.e.\\{P[X_{I_{1}},X_{I_{3}}\in A_{I_{3}}]\\}$ provided $P[X_{I_{3}}\in
A_{I_{3}}]>0$ while it is evident provided $P[X_{I_{3}}\in A_{I_{3}}]=0$.

######  comment 6.5.

Theorem 6.7, Theorem 6.8 and their corollarys are called Monotone Convergence
Theorem of $E$-integral.

######  Theorem 6.9.

Let $(\Omega,\mathscr{F},P)$ be a given probability space, and let
$(\Omega_{i},\mathscr{F}_{i}),\ i=1,2,3,4$ be measurable spaces. Let
$X_{i}:(\Omega,\mathscr{F})\rightarrow(\Omega_{i},\mathscr{F}_{i}),\
i=1,2,3,4$ be random objects, and $P[X_{i}]$ the probability on
$(\Omega_{i},\mathscr{F}_{i})$ induced by $X_{i}$, $i=1,2,3,4$. Then
$(\Omega_{i},\mathscr{F}_{i},P[X_{i}]),\ i=1,2,3,4$ are probability spaces.
Let $Y_{n}(\omega_{2}),n=1,2,3,\cdots,$ be random variables on
$(\Omega_{2},\mathscr{F}_{2},P[X_{2}])$ with
$E_{[X_{2}]}(Y_{n})\in\mathbb{R},\ n=1,2,3,\cdots$.

1\. If there exists a random variable $g(\omega_{2})$ on
$(\Omega_{2},\mathscr{F}_{2},P[X_{2}])$ with $E_{[X_{2}]}(g)\in\mathbb{R}$
such that

$Y_{n}\geq g,\ a.e.\\{P[X_{2}]\\},\ n=1,2,3,\cdots,\ \ and\ \
\sup\limits_{n}\\{E_{[X_{2}]}(Y_{n})\\}\in\mathbb{R},$

then for each fixed $A_{i}\in\mathscr{F}_{i},\ i=3,4$, $E_{[X_{2},X_{4}\in
A_{4}]}(\liminf_{n\to\infty}Y_{n}|X_{3}\in A_{3})\in\mathbb{R}$,

$E_{[X_{2},X_{4}\in A_{4}]}(\liminf\limits_{n\to\infty}Y_{n}|X_{3}\in
A_{3})\leq\liminf\limits_{n\to\infty}E_{[X_{2},X_{4}\in A_{4}]}(Y_{n}|X_{3}\in
A_{3}),$

and

$E_{[X_{2},X_{4}\in A_{4}]}(\liminf\limits_{n\to\infty}Y_{n}|X_{1},X_{3}\in
A_{3})\leq\liminf\limits_{n\to\infty}E_{[X_{2},X_{4}\in
A_{4}]}(Y_{n}|X_{1},X_{3}\in A_{3}),$

$a.e.\\{P[X_{1},X_{3}\in A_{3}]\\}$ provided $P[X_{1},X_{3}\in A_{3}]>0$.

2\. If there exists a random variable $g(\omega_{2})$ on
$(\Omega_{2},\mathscr{F}_{2},P[X_{2}])$ with $E_{[X_{2}]}(g)\in\mathbb{R}$
such that

$Y_{n}\leq g,\ a.e.\\{P[X_{2}]\\},\ n=1,2,3,\cdots,\ \ and\ \
\inf\limits_{n}\\{E_{[X_{2}]}(Y_{n})\\}\in\mathbb{R},$

then for each fixed $A_{i}\in\mathscr{F}_{i},\ i=3,4$, $E_{[X_{2},X_{4}\in
A_{4}]}(\limsup_{n\to\infty}Y_{n}|X_{3}\in A_{3})\in\mathbb{R}$,

$E_{[X_{2},X_{4}\in A_{4}]}(\limsup\limits_{n\to\infty}Y_{n}|X_{3}\in
A_{3})\geq\limsup\limits_{n\to\infty}E_{[X_{2},X_{4}\in A_{4}]}(Y_{n}|X_{3}\in
A_{3}),$

and

$E_{[X_{2},X_{4}\in A_{4}]}(\limsup\limits_{n\to\infty}Y_{n}|X_{1},X_{3}\in
A_{3})\geq\limsup\limits_{n\to\infty}E_{[X_{2},X_{4}\in
A_{4}]}(Y_{n}|X_{1},X_{3}\in A_{3}),$

$a.e.\\{P[X_{1},X_{3}\in A_{3}]\\}$ provided $P[X_{1},X_{3}\in A_{3}]>0$.

######  Corollary 6.10.

Let $(\Omega,\mathscr{F},P)$ be a given probability space,
$(\Omega_{i},\mathscr{F}_{i}),\ i\in\Lambda$ measurable spaces, where
$\Lambda$ is a nonempty index set. Let
$X_{i}:(\Omega,\mathscr{F})\rightarrow(\Omega_{i},\mathscr{F}_{i}),\
i\in\Lambda$ be random objects. Let $I_{i}\subset\Lambda,\ i=1,2,3,4$, which
are pairwise disjoint. Let $Y_{n}(\omega_{I_{2}}),n=1,2,3,\cdots$ be random
variables on $(\Omega_{I_{2}},\mathscr{F}_{I_{2}},P[X_{I_{2}}])$ with
$E_{[X_{I_{2}}]}(Y_{n})\in\mathbb{R},\ n=1,2,3,\cdots$.

1\. If there exists a random variable $g(\omega_{I_{2}})$ on
$(\Omega_{I_{2}},\mathscr{F}_{I_{2}},P[X_{I_{2}}])$ with
$E_{[X_{I_{2}}]}(g)\in\mathbb{R}$ such that

$Y_{n}\geq g,\ a.e.\\{P[X_{I_{2}}]\\},\ n=1,2,3,\cdots,\ \ and\ \
\sup\limits_{n}\\{E_{[X_{I_{2}}]}(Y_{n})\\}\in\mathbb{R},$

then for each fixed $A_{I_{i}}\in\mathscr{F}_{I_{i}},\ i=3,4$,
$E_{[X_{I_{2}},X_{I_{4}}\in A_{I_{4}}]}(\liminf_{n\to\infty}Y_{n}|X_{I_{3}}\in
A_{I_{3}})\in\mathbb{R}$,

$E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(\liminf\limits_{n\to\infty}Y_{n}|X_{I_{3}}\in
A_{I_{3}})\leq\liminf\limits_{n\to\infty}E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(Y_{n}|X_{I_{3}}\in A_{I_{3}}),$

and

$E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(\liminf\limits_{n\to\infty}Y_{n}|X_{I_{1}},X_{I_{3}}\in
A_{I_{3}})\leq\liminf\limits_{n\to\infty}E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(Y_{n}|X_{I_{1}},X_{I_{3}}\in A_{I_{3}}),$

$a.e.\\{P[X_{I_{1}},X_{I_{3}}\in A_{I_{3}}]\\}$ provided
$P[X_{I_{1}},X_{I_{3}}\in A_{I_{3}}]>0$.

2\. If there exists a random variable $g(\omega_{I_{2}})$ on
$(\Omega_{I_{2}},\mathscr{F}_{I_{2}},P[X_{I_{2}}])$ with
$E_{[X_{I_{2}}]}(g)\in\mathbb{R}$ such that

$Y_{n}\leq g,\ a.e.\\{P[X_{I_{2}}]\\},\ n=1,2,3,\cdots,\ \ and\ \
\inf\limits_{n}\\{E_{[X_{I_{2}}]}(Y_{n})\\}\in\mathbb{R},$

then for each fixed $A_{I_{i}}\in\mathscr{F}_{I_{i}},\ i=3,4$,
$E_{[X_{I_{2}},X_{I_{4}}\in A_{I_{4}}]}(\limsup_{n\to\infty}Y_{n}|X_{I_{3}}\in
A_{I_{3}})\in\mathbb{R}$,

$E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(\limsup\limits_{n\to\infty}Y_{n}|X_{I_{3}}\in
A_{I_{3}})\geq\limsup\limits_{n\to\infty}E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(Y_{n}|X_{I_{3}}\in A_{I_{3}}),$

and

$E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(\limsup\limits_{n\to\infty}Y_{n}|X_{I_{1}},X_{I_{3}}\in
A_{I_{3}})\geq\limsup\limits_{n\to\infty}E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(Y_{n}|X_{I_{1}},X_{I_{3}}\in A_{I_{3}}),$

$a.e.\\{P[X_{I_{1}},X_{I_{3}}\in A_{I_{3}}]\\}$ provided
$P[X_{I_{1}},X_{I_{3}}\in A_{I_{3}}]>0$.

######  comment 6.6.

Theorem 6.9 and its corollary are called Fatou’s Theorem of $E$-integral or
Fatou’s Lemma of $E$-integral.

######  Theorem 6.10.

Let $(\Omega,\mathscr{F},P)$ be a given probability space, and let
$(\Omega_{i},\mathscr{F}_{i}),\ i=1,2,3,4$ be measurable spaces. Let
$X_{i}:(\Omega,\mathscr{F})\rightarrow(\Omega_{i},\mathscr{F}_{i}),\
i=1,2,3,4$ be random objects, and $P[X_{i}]$ the probability on
$(\Omega_{i},\mathscr{F}_{i})$ induced by $X_{i}$, $i=1,2,3,4$. Then
$(\Omega_{i},\mathscr{F}_{i},P[X_{i}]),\ i=1,2,3,4$ are probability spaces.
Let $Y$, $Y_{n}(\omega_{2}),n=1,2,3,\cdots,$ be random variables on
$(\Omega_{2},\mathscr{F}_{2},P[X_{2}])$.

1\. If

$\lim_{n\to\infty}Y_{n}=Y,\ \ a.e.\\{P[X_{2}]\\},$

and there exists a nonnegative random variable $g(\omega_{2})$ on
$(\Omega_{2},\mathscr{F}_{2},P[X_{2}])$ with $E_{[X_{2}]}(g)\in\mathbb{R}$
such that

$|Y_{n}|\leq g,\ \ a.e.\\{P[X_{2}]\\},\ \ n=1,2,3,\cdots,$

then for each fixed $A_{i}\in\mathscr{F}_{i},\ i=3,4$, $E_{[X_{2},X_{4}\in
A_{4}]}(Y|X_{3}\in A_{3})\in\mathbb{R}$,

$\lim_{n\to\infty}E_{[X_{2},X_{4}\in A_{4}]}(Y_{n}|X_{3}\in
A_{3})=E_{[X_{2},X_{4}\in A_{4}]}(Y|X_{3}\in A_{3}),$

and

$\lim_{n\to\infty}E_{[X_{2},X_{4}\in A_{4}]}(Y_{n}|X_{1},X_{3}\in
A_{3})=E_{[X_{2},X_{4}\in A_{4}]}(Y|X_{1},X_{3}\in A_{3}),\ \
a.e.\\{P[X_{1},X_{3}\in A_{3}]\\}$

provided $P[X_{1},X_{3}\in A_{3}]>0$.

2\. If

$\lim_{n\to\infty}Y_{n}=Y,\ \ in\ \ P[X_{2}],$

and there exists a nonnegative random variable $g(\omega_{2})$ on
$(\Omega_{2},\mathscr{F}_{2},P[X_{2}])$ with $E_{[X_{2}]}(g)\in\mathbb{R}$
such that

$|Y_{n}|\leq g,\ \ a.e.\\{P[X_{2}]\\},\ \ n=1,2,3,\cdots,$

then for each fixed $A_{i}\in\mathscr{F}_{i},\ i=3,4$, $E_{[X_{2},X_{4}\in
A_{4}]}(Y|X_{3}\in A_{3})\in\mathbb{R}$,

$\lim_{n\to\infty}E_{[X_{2},X_{4}\in A_{4}]}(Y_{n}|X_{3}\in
A_{3})=E_{[X_{2},X_{4}\in A_{4}]}(Y|X_{3}\in A_{3}),$

and

$\lim_{n\to\infty}E_{[X_{2},X_{4}\in A_{4}]}(Y_{n}|X_{1},X_{3}\in
A_{3})=E_{[X_{2},X_{4}\in A_{4}]}(Y|X_{1},X_{3}\in A_{3}),\ \ in\
P[X_{1},X_{3}\in A_{3}]$

provided $P[X_{1},X_{3}\in A_{3}]>0$.

######  Corollary 6.11.

Let $(\Omega,\mathscr{F},P)$ be a given probability space,
$(\Omega_{i},\mathscr{F}_{i}),\ i\in\Lambda$ measurable spaces, where
$\Lambda$ is a nonempty index set. Let
$X_{i}:(\Omega,\mathscr{F})\rightarrow(\Omega_{i},\mathscr{F}_{i}),\
i\in\Lambda$ be random objects. Let $I_{i}\subset\Lambda,\ i=1,2,3,4$, which
are pairwise disjoint. Let $Y(\omega_{I_{2}})$,
$Y_{n}(\omega_{I_{2}}),n=1,2,3,\cdots$ be random variables on
$(\Omega_{I_{2}},\mathscr{F}_{I_{2}},P[X_{I_{2}}])$.

1\. If

$\lim_{n\to\infty}Y_{n}=Y,\ \ a.e.\ P[X_{I_{2}}],$

and there exists a nonnegative random variable $g(\omega_{I_{2}})$ on
$(\Omega_{I_{2}},\mathscr{F}_{I_{2}},P[X_{I_{2}}])$ with
$E_{[X_{I_{2}}]}(g)\in\mathbb{R}$ such that

$|Y_{n}|\leq g,\ \ a.e.\\{P[X_{I_{2}}]\\},\ \ n=1,2,3,\cdots,$

then for each fixed $A_{I_{i}}\in\mathscr{F}_{I_{i}},\ i=3,4$,
$E_{[X_{I_{2}},X_{I_{4}}\in A_{I_{4}}]}(Y|X_{I_{3}}\in
A_{I_{3}})\in\mathbb{R}$,

$\lim_{n\to\infty}E_{[X_{I_{2}},X_{I_{4}}\in A_{I_{4}}]}(Y_{n}|X_{I_{3}}\in
A_{I_{3}})=E_{[X_{I_{2}},X_{I_{4}}\in A_{I_{4}}]}(Y|X_{I_{3}}\in A_{I_{3}}),$

and

$\lim_{n\to\infty}E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(Y_{n}|X_{I_{1}},X_{I_{3}}\in A_{I_{3}})=E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(Y|X_{I_{1}},X_{I_{3}}\in A_{I_{3}}),$

$a.e.\ P[X_{I_{1}},X_{I_{3}}\in A_{I_{3}}]$ provided $P[X_{I_{1}},X_{I_{3}}\in
A_{I_{3}}]>0$.

2\. If

$\lim_{n\to\infty}Y_{n}=Y,\ \ in\ \ P[X_{I_{2}}],$

and there exists a nonnegative random variable $g(\omega_{I_{2}})$ on
$(\Omega_{I_{2}},\mathscr{F}_{I_{2}},P[X_{I_{2}}])$ with
$E_{[X_{I_{2}}]}(g)\in\mathbb{R}$ such that

$|Y_{n}|\leq g,\ \ a.e.\\{P[X_{I_{2}}]\\},\ \ n=1,2,3,\cdots,$

then for each fixed $A_{I_{i}}\in\mathscr{F}_{I_{i}},\ i=3,4$,
$E_{[X_{I_{2}},X_{I_{4}}\in A_{I_{4}}]}(Y|X_{I_{3}}\in
A_{I_{3}})\in\mathbb{R}$,

$\lim_{n\to\infty}E_{[X_{I_{2}},X_{I_{4}}\in A_{I_{4}}]}(Y_{n}|X_{I_{3}}\in
A_{I_{3}})=E_{[X_{I_{2}},X_{I_{4}}\in A_{I_{4}}]}(Y|X_{I_{3}}\in A_{I_{3}}),$

and

$\lim_{n\to\infty}E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(Y_{n}|X_{I_{1}},X_{I_{3}}\in A_{I_{3}})=E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(Y|X_{I_{1}},X_{I_{3}}\in A_{I_{3}}),$

$in\ P[X_{I_{1}},X_{I_{3}}\in A_{I_{3}}]$ provided $P[X_{I_{1}},X_{I_{3}}\in
A_{I_{3}}]>0$.

######  comment 6.7.

Theorem 6.10 and its corollary are called Dominated Convergence Theorem of
$E$-integral.

######  Theorem 6.11.

Let $(\Omega,\mathscr{F},P)$ be a given probability space, and let
$(\Omega_{i},\mathscr{F}_{i})$, $i=1,2,3$ be measurable spaces. Let
$\mathscr{G}$ be a sub $\sigma$-field of $\mathscr{F}_{2}$. Let
$X_{i}:(\Omega,\mathscr{F})\rightarrow(\Omega_{i},\mathscr{F}_{i})$, $i=1,2,3$
be random objects. The random object
$X_{2\mathscr{G}}(\omega):(\Omega,\mathscr{F})\rightarrow(\Omega_{2},\mathscr{G})$
is denoted by $X_{21}$. Let $Y(\omega_{2})$ be a random variable on
$(\Omega_{2},\mathscr{F}_{2},P[X_{2}])$ with $E_{[X_{2}]}(Y)\in\mathbb{R}$,
and $Z(\omega_{2})$ a random variable on $(\Omega_{2},\mathscr{G},P[X_{21}])$
with $E_{[X_{2}]}(ZY)\in\mathbb{R}$. Then for fixed $A_{i}\in\mathscr{F}_{i},\
i=1,3$ with $P[X_{1}\in A_{1},X_{3}\in A_{3}]>0$,

$E_{[X_{2},X_{3}\in A_{3}]}(ZY|X_{21},X_{1}\in A_{1})=ZE_{[X_{2},X_{3}\in
A_{3}]}(Y|X_{21},X_{1}\in A_{1}),\ \ a.e.\\{P[X_{21},X_{1}\in A_{1}]\\}.$

######  comment 6.8.

(i) In Theorem 6.11, let $A_{3}=\Omega_{3}$, then

$E_{[X_{2}]}(ZY|X_{21},X_{1}\in A_{1})=ZE_{[X_{2}]}(Y|X_{21},X_{1}\in A_{1}),\
\ a.e.\\{P[X_{21},X_{1}\in A_{1}]\\}.$

(ii) In Theorem 6.11, let $A_{1}=\Omega_{1}$, then

$E_{[X_{2},X_{3}\in A_{3}]}(ZY|X_{21})=ZE_{[X_{2},X_{3}\in A_{3}]}(Y|X_{21}),\
\ a.e.\\{P[X_{21}]\\}.$

(iii) In Theorem 6.11, let $A_{1}=\Omega_{1}$, $A_{3}=\Omega_{3}$, then

$E_{[X_{2}]}(ZY|X_{21})=ZE_{[X_{2}]}(Y|X_{21}),\ \ a.e.\\{P[X_{21}]\\}.$

(iv) In Theorem 6.11, let $(\Omega_{2},\mathscr{F}_{2})=(\Omega,\mathscr{F})$,
$X_{2}=I_{\mathscr{F}}=I_{0}$, $X_{21}=I_{\mathscr{G}}$, $A_{1}=\Omega_{1}$,
$A_{3}=\Omega_{3}$, then

$E(ZY|\mathscr{G})=E_{[I_{0}]}(ZY|I_{\mathscr{G}})=ZE_{[I_{0}]}(Y|I_{\mathscr{G}})=ZE(Y|\mathscr{G}),\
\ a.e.\\{P\\},$

which is Theorem 7.2.1 (9) in [67] or Theorem 5.5.11 (a) in [4].

Let $(\Omega,\mathscr{F},P)$ be a probability space,
$\mathscr{L}^{p}(\Omega,\mathscr{F},P)$, $(1\leq p<+\infty)$, the set of all
random variables $f:\Omega\rightarrow\mathbb{R}$ on $(\Omega,\mathscr{F})$
such that $E(|f|^{p})\in\mathbb{R}$, and
$\mathscr{L}^{\infty}(\Omega,\mathscr{F},P)$, $(p=+\infty)$, the set of all
random variables $f:\Omega\rightarrow\mathbb{R}$ on $(\Omega,\mathscr{F})$
that are essentially bounded. Then we have H$\ddot{\rm{o}}$lder’s inequality
of $E$-integral below.

######  Theorem 6.12.

Let $(\Omega,\mathscr{F},P)$ be a given probability space, and let
$(\Omega_{i},\mathscr{F}_{i}),\ i=1,2,3,4$ be measurable spaces. Let
$X_{i}:(\Omega,\mathscr{F})\rightarrow(\Omega_{i},\mathscr{F}_{i})$,
$i=1,2,3,4$ be random objects, and $P[X_{i}]$ the probability on
$(\Omega_{i},\mathscr{F}_{i})$ induced by $X_{i}$, $i=1,2,3,4$. Then
$(\Omega_{i},\mathscr{F}_{i},P[X_{i}]),\ i=1,2,3,4$ are probability spaces.
Let $p$ and $q$ satisfy $1\leq p\leq+\infty$, $1\leq q\leq+\infty$, and
$\displaystyle\frac{1}{p}+\frac{1}{q}=1$. Let
$Y(\omega_{2})\in\mathscr{L}^{p}(\Omega_{2},\mathscr{F}_{2},P[X_{2}])$ and
$Z(\omega_{2})\in\mathscr{L}^{q}(\Omega_{2},\mathscr{F}_{2},P[X_{2}])$.

1\. For each fixed $A_{i}\in\mathscr{F}_{i},\ i=3,4$ with $P[X_{3}\in
A_{3},X_{4}\in A_{4}]>0$,

$\begin{array}[]{l}E_{[X_{2},X_{4}\in A_{4}]}(YZ|X_{3}\in
A_{3})\in\mathbb{R};\\\ \\\ E_{[X_{2},X_{4}\in A_{4}]}(|YZ||X_{3}\in A_{3})\\\
\leq(E_{[X_{2},X_{4}\in A_{4}]}(|Y|^{p}|X_{3}\in
A_{3}))^{\frac{1}{p}}\cdot(E_{[X_{2},X_{4}\in A_{4}]}(|Z|^{q}|X_{3}\in
A_{3}))^{\frac{1}{q}},(1<p,q<+\infty);\\\ \\\ E_{[X_{2},X_{4}\in
A_{4}]}(|YZ||X_{3}\in A_{3})\leq\|Y\|_{\infty}\cdot E_{[X_{2},X_{4}\in
A_{4}]}(|Z||X_{3}\in A_{3}),\ \ (p=+\infty,q=1).\end{array}$

2\. For each fixed $A_{i}\in\mathscr{F}_{i},\ i=3,4$ with $P[X_{3}\in
A_{3},X_{4}\in A_{4}]>0$, there exists $E_{[X_{2},X_{4}\in
A_{4}]}(YZ|X_{1},X_{3}\in A_{3})$, and

$\begin{array}[]{l}E_{[X_{2},X_{4}\in A_{4}]}(|YZ||X_{1},X_{3}\in
A_{3})\leq\|Y\|_{\infty}\cdot E_{[X_{2},X_{4}\in A_{4}]}(|Z||X_{1},X_{3}\in
A_{3}),\\\ a.e.\\{P[X_{1},X_{3}\in A_{3}]\\},\ \ (p=+\infty,q=1).\end{array}$

Furthermore if there exists $P[X_{2},X_{4}\in A_{4}|X_{1},X_{3}\in A_{3}]$,
then

$\begin{array}[]{l}E_{[X_{2},X_{4}\in A_{4}]}(|YZ||X_{1},X_{3}\in A_{3})\\\
\leq(E_{[X_{2},X_{4}\in A_{4}]}(|Y|^{p}|X_{1},X_{3}\in
A_{3}))^{\frac{1}{p}}\cdot(E_{[X_{2},X_{4}\in A_{4}]}(|Z|^{q}|X_{1},X_{3}\in
A_{3}))^{\frac{1}{q}},\\\ a.e.\\{P[X_{1},X_{3}\in A_{3}]\\},\ \ (1<p,\
q<+\infty).\end{array}$

######  Corollary 6.12.

Let $(\Omega,\mathscr{F},P)$ be a given probability space,
$(\Omega_{i},\mathscr{F}_{i}),\ i\in\Lambda$ measurable spaces, where
$\Lambda$ is a nonempty index set. Let
$X_{i}:(\Omega,\mathscr{F})\rightarrow(\Omega_{i},\mathscr{F}_{i}),\
i\in\Lambda$ be random objects. Let $I_{i}\subset\Lambda,\ i=1,2,3,4$, which
are pairwise disjoint. Let $p$ and $q$ satisfy $1\leq p\leq+\infty$, $1\leq
q\leq+\infty$, and $\displaystyle\frac{1}{p}+\frac{1}{q}=1$. Let
$Y(\omega_{I_{2}})\in\mathscr{L}^{p}(\Omega_{I_{2}},\mathscr{F}_{I_{2}},P[X_{I_{2}}])$
and
$Z(\omega_{I_{2}})\in\mathscr{L}^{q}(\Omega_{I_{2}},\mathscr{F}_{I_{2}},P[X_{I_{2}}])$.

1\. For each fixed $A_{I_{i}}\in\mathscr{F}_{I_{i}},\ i=3,4$ with
$P[X_{I_{3}}\in A_{I_{3}},X_{I_{4}}\in A_{I_{4}}]>0$,

$\begin{array}[]{l}E_{[X_{I_{2}},X_{I_{4}}\in A_{I_{4}}]}(YZ|X_{I_{3}}\in
A_{I_{3}})\in\mathbb{R};\\\ \\\ E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(|YZ||X_{I_{3}}\in A_{I_{3}})\\\ \leq(E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(|Y|^{p}|X_{I_{3}}\in
A_{I_{3}}))^{\frac{1}{p}}\cdot(E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(|Z|^{q}|X_{I_{3}}\in A_{I_{3}}))^{\frac{1}{q}},(1<p,q<+\infty);\\\
\\\ E_{[X_{I_{2}},X_{I_{4}}\in A_{I_{4}}]}(|YZ||X_{I_{3}}\in
A_{I_{3}})\leq\|Y\|_{\infty}\cdot E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(|Z||X_{I_{3}}\in A_{I_{3}}),\ \ (p=+\infty,q=1).\end{array}$

2\. For each fixed $A_{I_{i}}\in\mathscr{F}_{I_{i}},\ i=3,4$ with
$P[X_{I_{3}}\in A_{I_{3}},X_{I_{4}}\in A_{I_{4}}]>0$, there exists
$E_{[X_{I_{2}},X_{I_{4}}\in A_{I_{4}}]}(YZ|X_{I_{1}},X_{I_{3}}\in A_{I_{3}})$,
and

$\begin{array}[]{l}E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(|YZ||X_{I_{1}},X_{I_{3}}\in A_{I_{3}})\leq\|Y\|_{\infty}\cdot
E_{[X_{I_{2}},X_{I_{4}}\in A_{I_{4}}]}(|Z||X_{I_{1}},X_{I_{3}}\in
A_{I_{3}}),\\\ a.e.\\{P[X_{I_{1}},X_{I_{3}}\in A_{I_{3}}]\\},\ \ (p=+\infty,\
q=1).\end{array}$

Furthermore if there exists $P[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}|X_{I_{1}},X_{I_{3}}\in A_{I_{3}}]$, then

$\begin{array}[]{l}E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(|YZ||X_{I_{1}},X_{I_{3}}\in A_{I_{3}})\\\
\leq(E_{[X_{I_{2}},X_{I_{4}}\in A_{I_{4}}]}(|Y|^{p}|X_{I_{1}},X_{I_{3}}\in
A_{I_{3}}))^{\frac{1}{p}}\cdot(E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(|Z|^{q}|X_{I_{1}},X_{I_{3}}\in A_{I_{3}}))^{\frac{1}{q}},\\\
a.e.\\{P[X_{I_{1}},X_{I_{3}}\in A_{I_{3}}]\\},\ \ (1<p,\
q<+\infty).\end{array}$

By Corollary 6.12 and Theorem 2.12 we have the following corollary.

######  Corollary 6.13.

Let $(\Omega,\mathscr{F},P)$ be a given probability space, and
$X_{i}(\omega),\ i\in\mathbb{N}$ random variables on $(\Omega,\mathscr{F},P)$.
Let $I_{i}\subset\mathbb{N},\ i=1,2,3,4$, which are pairwise disjoint. Let $p$
and $q$ satisfy $1\leq p\leq+\infty$, $1\leq q\leq+\infty$, and
$\displaystyle\frac{1}{p}+\frac{1}{q}=1$. Let
$Y(x_{I_{2}})\in\mathscr{L}^{p}(\mathbb{R}^{I_{2}},\mathscr{B}(\mathbb{R}^{I_{2}}),P[X_{I_{2}}])$
and
$Z(x_{I_{2}})\in\mathscr{L}^{q}(\mathbb{R}^{I_{2}},\mathscr{B}(\mathbb{R}^{I_{2}}),P[X_{I_{2}}])$.

1\. For each fixed $A_{i}\in\mathscr{B}(\mathbb{R}),\ i\in I_{3}+I_{4}$ with
$P[X_{i}\in A_{i},\ i\in I_{3}+I_{4}]>0$,

$\begin{array}[]{l}E_{[X_{i},i\in I_{2};X_{i}\in A_{i},i\in
I_{4}]}(YZ|X_{i}\in A_{i},i\in I_{3})\in\mathbb{R};\\\ \\\ E_{[X_{i},i\in
I_{2};X_{i}\in A_{i},i\in I_{4}}(|YZ||X_{i}\in A_{i},i\in I_{3})\\\
\leq(E_{[X_{i},i\in I_{2};X_{i}\in A_{i},i\in I_{4}]}(|Y|^{p}|X_{i}\in
A_{i},i\in I_{3}))^{\frac{1}{p}}\cdot(E_{[X_{i},i\in I_{2};X_{i}\in A_{i},i\in
I_{4}]}(|Z|^{q}|X_{i}\in A_{i},i\in I_{3}))^{\frac{1}{q}},\\\ \ \ \
(1<p,q<+\infty);\\\ \\\ E_{[X_{i},i\in I_{2};X_{i}\in A_{i},i\in
I_{4}]}(|YZ||X_{i}\in A_{i},i\in I_{3})\leq\|Y\|_{\infty}\cdot E_{[X_{i},i\in
I_{2};X_{i}\in A_{i},i\in I_{4}]}(|Z||X_{i}\in A_{i},i\in I_{3}),\\\ \ \ \
(p=+\infty,q=1).\end{array}$

2\. For each fixed $A_{i}\in\mathscr{B}(\mathbb{R}),\ i\in I_{3}+I_{4}$ with
$P[X_{i}\in A_{i},\ i\in I_{3}+I_{4}]>0$, there exists $E_{[X_{i},i\in
I_{2};X_{i}\in A_{i},i\in I_{4}]}(YZ|X_{i},i\in I_{1};X_{i}\in A_{i},i\in
I_{3})$, and

$\begin{array}[]{l}E_{[X_{i},i\in I_{2};X_{i}\in A_{i},i\in
I_{4}]}(|YZ||X_{i},i\in I_{1};X_{i}\in A_{i},i\in I_{3})\\\
\leq(E_{[X_{i},i\in I_{2};X_{i}\in A_{i},i\in I_{4}]}(|Y|^{p}|X_{i},i\in
I_{1};X_{i}\in A_{i},i\in I_{3}))^{\frac{1}{p}}\\\ (E_{[X_{i},i\in
I_{2};X_{i}\in A_{i},i\in I_{4}]}(|Z|^{q}|X_{i},i\in I_{1};X_{i}\in A_{i},i\in
I_{3}))^{\frac{1}{q}},\ \ a.e.\\{P[X_{i},i\in I_{1};X_{i}\in A_{i},i\in
I_{3}]\\},\\\ (1<p,\ q<+\infty);\\\ \\\ E_{[X_{i},i\in I_{2};X_{i}\in
A_{i},i\in I_{4}]}(|YZ||X_{i},i\in I_{1};X_{i}\in A_{i},i\in I_{3})\\\
\leq\|Y\|_{\infty}E_{[X_{i},i\in I_{2};X_{i}\in A_{i},i\in
I_{4}]}(|Z||X_{i},i\in I_{1};X_{i}\in A_{i},i\in I_{3}),\ \
a.e.\\{P[X_{i},i\in I_{1};X_{i}\in A_{i},i\in I_{3}]\\},\\\
(p=+\infty,q=1).\end{array}$

The following Theorem is Minkowski’s inequality of $E$-integral.

######  Theorem 6.13.

Let $(\Omega,\mathscr{F},P)$ be a given probability space, and let
$(\Omega_{i},\mathscr{F}_{i}),\ i=1,2,3,4$ be measurable spaces. Let
$X_{i}:(\Omega,\mathscr{F})\rightarrow(\Omega_{i},\mathscr{F}_{i})$,
$i=1,2,3,4$ be random objects, and $P[X_{i}]$ the probability on
$(\Omega_{i},\mathscr{F}_{i})$ induced by $X_{i}$, $i=1,2,3,4$. Then
$(\Omega_{i},\mathscr{F}_{i},P[X_{i}]),\ i=1,2,3,4$ are probability spaces.
Let $p$ satisfy $1\leq p<+\infty$, and
$Y(\omega_{2}),Z(\omega_{2})\in\mathscr{L}^{p}(\Omega_{2},\mathscr{F}_{2},P[X_{2}])$.

1\. For each fixed $A_{i}\in\mathscr{F}_{i},\ i=3,4$ with $P[X_{3}\in
A_{3},X_{4}\in A_{4}]>0$,

$\begin{array}[]{l}\\{E_{[X_{2},X_{4}\in A_{4}]}(|Y+Z|^{p}|X_{3}\in
A_{3})\\}^{\frac{1}{p}}\\\ \leq\\{E_{[X_{2},X_{4}\in A_{4}]}(|Y|^{p}|X_{3}\in
A_{3})\\}^{\frac{1}{p}}+\\{E_{[X_{2},X_{4}\in A_{4}]}(|Z|^{p}|X_{3}\in
A_{3})\\}^{\frac{1}{p}}.\end{array}$

2\. For each fixed $A_{i}\in\mathscr{F}_{i},\ i=3,4$ with $P[X_{3}\in
A_{3},X_{4}\in A_{4}]>0$, if there exists $P[X_{2},X_{4}\in
A_{4}|X_{1},X_{3}\in A_{3}]$, then

$\begin{array}[]{l}\\{E_{[X_{2},X_{4}\in A_{4}]}(|Y+Z|^{p}|X_{1},X_{3}\in
A_{3})\\}^{\frac{1}{p}}\\\ \leq\\{E_{[X_{2},X_{4}\in
A_{4}]}(|Y|^{p}|X_{1},X_{3}\in A_{3})\\}^{\frac{1}{p}}+\\{E_{[X_{2},X_{4}\in
A_{4}]}(|Z|^{p}|X_{1},X_{3}\in A_{3})\\}^{\frac{1}{p}},\\\
a.e.\\{P[X_{1},X_{3}\in A_{3}]\\}.\end{array}$

######  Corollary 6.14.

Let $(\Omega,\mathscr{F},P)$ be a given probability space,
$(\Omega_{i},\mathscr{F}_{i}),\ i\in\Lambda$ measurable spaces, where
$\Lambda$ is a nonempty index set. Let
$X_{i}:(\Omega,\mathscr{F})\rightarrow(\Omega_{i},\mathscr{F}_{i}),\
i\in\Lambda$ be random objects. Let $I_{i}\subset\Lambda,\ i=1,2,3,4$, which
are pairwise disjoint. Let $p$ satisfy $1\leq p<+\infty$, and
$Y(\omega_{I_{2}}),Z(\omega_{I_{2}})\in\mathscr{L}^{p}(\Omega_{I_{2}},\mathscr{F}_{I_{2}},P[X_{I_{2}}])$.

1\. For each fixed $A_{I_{i}}\in\mathscr{F}_{I_{i}},\ i=3,4$ with
$P[X_{I_{3}}\in A_{I_{3}},X_{I_{4}}\in A_{I_{4}}]>0$,

$\begin{array}[]{l}\\{E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(|Y+Z|^{p}|X_{I_{3}}\in A_{I_{3}})\\}^{\frac{1}{p}}\\\
\leq\\{E_{[X_{I_{2}},X_{I_{4}}\in A_{I_{4}}]}(|Y|^{p}|X_{I_{3}}\in
A_{I_{3}})\\}^{\frac{1}{p}}+\\{E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(|Z|^{p}|X_{I_{3}}\in A_{I_{3}})\\}^{\frac{1}{p}}.\end{array}$

2\. For each fixed $A_{I_{i}}\in\mathscr{F}_{I_{i}},\ i=3,4$ with
$P[X_{I_{3}}\in A_{I_{3}},X_{I_{4}}\in A_{I_{4}}]>0$, if there exists
$P[X_{I_{2}},X_{I_{4}}\in A_{I_{4}}|X_{I_{1}},X_{I_{3}}\in A_{I_{3}}]$, then

$\begin{array}[]{l}\\{E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(|Y+Z|^{p}|X_{I_{1}},X_{I_{3}}\in A_{I_{3}})\\}^{\frac{1}{p}}\\\
\leq\\{E_{[X_{I_{2}},X_{I_{4}}\in A_{I_{4}}]}(|Y|^{p}|X_{I_{1}},X_{I_{3}}\in
A_{I_{3}})\\}^{\frac{1}{p}}+\\{E_{[X_{I_{2}},X_{I_{4}}\in
A_{I_{4}}]}(|Z|^{p}|X_{I_{1}},X_{I_{3}}\in A_{I_{3}})\\}^{\frac{1}{p}},\\\
a.e.\\{P[X_{I_{1}},X_{I_{3}}\in A_{I_{3}}]\\}.\end{array}$

By Corollary 6.14 and Theorem 2.12 we have the following corollary.

######  Corollary 6.15.

Let $(\Omega,\mathscr{F},P)$ be a given probability space, and
$X_{i}(\omega),\ i\in\mathbb{N}$ random variables on $(\Omega,\mathscr{F},P)$.
Let $I_{i}\subset\mathbb{N},\ i=1,2,3,4$, which are pairwise disjoint. Let $p$
satisfy $1\leq p<+\infty$, and
$Y(x_{I_{2}}),Z(x_{I_{2}})\in\mathscr{L}^{p}(\mathbb{R}^{I_{2}},\mathscr{B}(\mathbb{R}^{I_{2}}),P[X_{I_{2}}])$.
Then for each fixed $A_{i}\in\mathscr{B}(\mathbb{R}),\ i\in I_{3}+I_{4}$ with
$P[X_{i}\in A_{i},\ i\in I_{3}+I_{4}]>0$,

$\begin{array}[]{l}\\{E_{[X_{i},\ i\in I_{2};X_{i}\in A_{i},\ i\in
I_{4}]}(|Y+Z|^{p}|X_{i}\in A_{i},\ i\in I_{3})\\}^{\frac{1}{p}}\\\
\leq\\{E_{[X_{i},\ i\in I_{2};X_{i}\in A_{i},\ i\in I_{4}]}(|Y|^{p}|X_{i}\in
A_{i},\ i\in I_{3})\\}^{\frac{1}{p}}\\\ +\\{E_{[X_{i},\ i\in I_{2};X_{i}\in
A_{i},\ i\in I_{4}]}(|Z|^{p}|X_{i}\in A_{i},\ i\in
I_{3})\\}^{\frac{1}{p}},\end{array}$

and

$\begin{array}[]{l}\\{E_{[X_{i},\ i\in I_{2};X_{i}\in A_{i},\ i\in
I_{4}]}(|Y+Z|^{p}|X_{i},\ i\in I_{1};X_{i}\in A_{i},\ i\in
I_{3})\\}^{\frac{1}{p}}\\\ \leq\\{E_{[X_{i},\ i\in I_{2};X_{i}\in A_{i},\ i\in
I_{4}]}(|Y|^{p}|X_{i},\ i\in I_{1};X_{i}\in A_{i},\ i\in
I_{3})\\}^{\frac{1}{p}}\\\ +\\{E_{[X_{i},\ i\in I_{2};X_{i}\in A_{i},\ i\in
I_{4}]}(|Z|^{p}|X_{i},\ i\in I_{1};X_{i}\in A_{i},\ i\in
I_{3})\\}^{\frac{1}{p}},\\\ a.e.\\{P[X_{i},\ i\in I_{1};X_{i}\in A_{i},\ i\in
I_{3}]\\}.\end{array}$

The following Theorem is Jensen’s inequality of $E$-integral.

######  Theorem 6.14.

Let $(\Omega,\mathscr{F},P)$ be a given probability space, and let
$(\Omega_{i},\mathscr{F}_{i}),\ i=1,2,3$ be measurable spaces. Let
$X_{i}:(\Omega,\mathscr{F})\rightarrow(\Omega_{i},\mathscr{F}_{i})$, $i=1,2,3$
be random objects, and $P[X_{i}]$ the probability on
$(\Omega_{i},\mathscr{F}_{i})$ induced by $X_{i}$, $i=1,2,3$. Then
$(\Omega_{i},\mathscr{F}_{i},P[X_{i}]),\ i=1,2,3$ are probability spaces.
Suppose that $\varphi:\mathbb{R}\rightarrow\mathbb{R}$ is convex, in the sense
that $\varphi(tx+(1-t)y)\leq t\varphi(x)+(1-t)\varphi(y)$ holds for $\forall
x,y\in\mathbb{R}$ and $\forall t\in[0,1]$. Let $Y$ be a random variable on
$(\Omega_{2},\mathscr{F}_{2},P[X_{2}])$ with $E_{[X_{2}]}(Y)\in\mathbb{R}$,
and $A_{3}\in\mathscr{F}_{3}$ with $P[X_{3}\in A_{3}]>0$. If there exists
$P[X_{2}|X_{1},X_{3}\in A_{3}]$, then

$\varphi(E_{[X_{2}]}(Y|X_{1},X_{3}\in A_{3}))\leq
E_{[X_{2}]}(\varphi(Y)|X_{1},X_{3}\in A_{3}),\ \ a.e.\\{P[X_{1},X_{3}\in
A_{3}]\\}.$

######  Corollary 6.16.

Let $(\Omega,\mathscr{F},P)$ be a given probability space,
$(\Omega_{i},\mathscr{F}_{i}),\ i\in\Lambda$ measurable spaces, where
$\Lambda$ is a nonempty index set. Let
$X_{i}:(\Omega,\mathscr{F})\rightarrow(\Omega_{i},\mathscr{F}_{i}),\
i\in\Lambda$ be random objects. Let $I_{i}\subset\Lambda,\ i=1,2,3$, which are
pairwise disjoint. Suppose that $\varphi:\mathbb{R}\rightarrow\mathbb{R}$ is
convex, in the sense that $\varphi(tx+(1-t)y)\leq t\varphi(x)+(1-t)\varphi(y)$
holds for $\forall x,y\in\mathbb{R}$ and $\forall t\in[0,1]$. Let $Y$ be a
random variable on $(\Omega_{I_{2}},\mathscr{F}_{I_{2}},P[X_{I_{2}}])$ with
$E_{[X_{I_{2}}]}(Y)\in\mathbb{R}$, and $A_{I_{3}}\in\mathscr{F}_{I_{3}}$ with
$P[X_{I_{3}}\in A_{I_{3}}]>0$. If there exists
$P[X_{I_{2}}|X_{I_{1}},X_{I_{3}}\in A_{I_{3}}]$, then

$\varphi(E_{[X_{I_{2}}]}(Y|X_{I_{1}},X_{I_{3}}\in A_{I_{3}}))\leq
E_{[X_{I_{2}}]}(\varphi(Y)|X_{I_{1}},X_{I_{3}}\in A_{I_{3}}),\ \
a.e.\\{P[X_{I_{1}},X_{I_{3}}\in A_{I_{3}}]\\}.$

By Corollary 6.16 and Theorem 2.12 we have the following corollary.

######  Corollary 6.17.

Let $(\Omega,\mathscr{F},P)$ be a given probability space, and
$X_{i}(\omega),\ i\in\mathbb{N}$ random variables on $(\Omega,\mathscr{F},P)$.
Let $I_{i}\subset\mathbb{N},\ i=1,2,3$, which are pairwise disjoint. Suppose
that $\varphi:\mathbb{R}\rightarrow\mathbb{R}$ is convex, in the sense that
$\varphi(tx+(1-t)y)\leq t\varphi(x)+(1-t)\varphi(y)$ holds for $\forall
x,y\in\mathbb{R}$ and $\forall t\in[0,1]$. Let $Y$ be a random variable on
$(\mathbb{R}^{I_{2}},\mathscr{B}(\mathbb{R}^{I_{2}}),P[X_{I_{2}}])$ with
$E_{[X_{I_{2}}]}(Y)\in\mathbb{R}$, and $A_{i}\in\mathscr{B}(\mathbb{R}),i\in
I_{3}$ with $P[X_{i}\in A_{i},i\in I_{3}]>0$. Then

$\begin{array}[]{l}\varphi(E_{[X_{i},i\in I_{2}]}(Y|X_{i},i\in I_{1};X_{i}\in
A_{i},i\in I_{3}))\leq E_{[X_{i},i\in I_{2}]}(\varphi(Y)|X_{i},i\in
I_{1};X_{i}\in A_{i},i\in I_{3}),\\\ \\\ a.e.\\{P[X_{i},i\in I_{1};X_{i}\in
A_{i},i\in I_{3}]\\}.\end{array}$

Funding: This research did not receive any specific grant from funding
agencies in the public, commercial, or not-for-profit sectors.

Acknowledgements:

Conflicts of Interest: The authors declare no conflict of interest.

## References

  * [1] Albert, R. and Barabási, A. L. (2000). Topology of evolving networks: Local events and universality. Physical Review Letters, 85, 5234-5237.
  * [2] Agborsangaya, C. B., Lau, D., Lahtinen, M., Cooke, T. and Johnson, J. A. (2012). Multimorbidity prevalence and patterns across socioeconomic determinants: a cross-sectional survey. BMC Public Health, 12:201.
  * [3] Anselin, L. (2000). The Alchemy of Statistics, or Creating Data Where No Data Exist. Annals of The Association of American Geographers, 90(3), 586-592.
  * [4] Ash, Robert B. (2000). Probability and Measure Theory, 2nd ed. Academic Press.
  * [5] Baldi, P. and Brunak, S. (2001). Bioinformatics: The Machine Learning Approach, 2nd ed. MIT Press, Cambridge.
  * [6] Barnett, K., Mercer, S. W., Norbury, M., Watt, G., Wyke, S. and Guthrie, B. (2012). Epidemiology of multimorbidity and implications for health care, research, and medical education: a cross-sectional study. The Lancet, 380:37-43.
  * [7] Begun, Janet M., Hall, W.J., Huang, Wei-Min and Wellner, Jon A. (1983). Information and Asymptotic Efficiency In Parametric-Nonparametric Models, The Annals of Statistics, 11(2), 432-452.
  * [8] Billingsley, Patrick (1995). Probability and Measure, 3rd ed. Wiley-Interscience.
  * [9] Barnard, K., Duygulu, P., de Freitas, N., Forsyth, D., Blei, D., and Jordan, M. I. (1995). Matching Words and Pictures, Journal of Machine Learning Research, 3, 1107-1135.
  * [10] Bogachev, V.I. (2007). Measure Theory, Vol. I and II. Springer-Verlag Berlin Heidelberg.
  * [11] Bollen, K. A. (1989). Structural Equations with Latent Variables. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Wiley, New York. MR0996025. https://doi.org/10.1002/9781118619179.
  * [12] Bongers, S., Forré, P., Peters, J. and Mooij, J. M. (2021) Foundations of structural causal models with cycles and latent variables. Ann. Statist. 49(5): 2885-2915.
  * [13] Bühlmann, P., Peters, J. and Ernest, J. (2014). CAM: Causal additive models, high-dimensional order search and penalized regression. Ann. Statist. 42 2526-2556. MR3277670. https://doi.org/10.1214/14-AOS1260.
  * [14] Brezger, A., Fahrmeir, L. and Hennerfeind, A. (2007). Adaptive Gaussian Markov random fields with applications in human brain mapping. Journal of the Royal Statistical Society, Series C (Applied Statistics), 56, 327-345.
  * [15] Chung, K. L. (2001). A Course in Probability Theory, 3rd ed. Academic Press.
  * [16] Cohn, D. L. (1980). Measure Theory. Birkh$\ddot{\rm a}$user Boston, USA.
  * [17] Derin, H., Elliott, H., Cristi, R. and Geman, D. (1984). Bayes Smoothing Algorithms for Segmentation of Binary Images Modeled by Markov Random Fields. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 707-720.
  * [18] Duncan, O. D. (1975). Introduction to Structural Equation Models: Studies in Population. Academic Press, New York. MR0398558.
  * [19] Durrett, R. (1995) Probability (Theory and Examples), 5th ed. Cambridge University Press.
  * [20] Feng, W., Jia, J. and Liu, Z. Q. (2010). Self-Validated Labeling of Markov Random Fields for Image Segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 32(10), 1871-1887.
  * [21] Fortin, M., Bravo, G., Hudon, C., Vanasse, A. and Lapointe, L. (2005). Prevalence of multimorbidity among adults seen in family practice. The Annals of Family Medicine, May 2005, 3(3), 223-228.
  * [22] Gao, G. S. (2000). Topological Space Theory. Science Press, China.
  * [23] Geman, S. and Geman, D. (1984). Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721-741.
  * [24] Giné, E. and Nickl, R. (2016). Mathematical Foundations of Infinite-Dimensional Statistical Models. Cambridge University Press.
  * [25] Glynn, L. G., Valderas, J. M., Healy, P., Burke, E., Newell, J., Gillespie, P., et al. (2011). The prevalence of multimorbidity in primary care and its effect on health care utilization and cost. Family Practice, 28(5), 516-523.
  * [26] Goldberger, A. S. and Duncan, O. D. (1973). Structural Equation Models in the Social Sciences. Seminar Press, New York.
  * [27] Gravier, G., Sigelle, M. and Chollet, G. (2000). A markov random field model for automatic speech recognition. In: Proceedings of the International Conference on Pattern Recognition, pp. 3258-3261.
  * [28] Haavelmo, T. (1943). The statistical implications of a system of simultaneous equations. Econometrica, 11, 1-12. MR0007954 https://doi.org/10.2307/1905714.
  * [29] Hedström, P. and Ylikoski (2010). Causal Mechanisms in the Social Sciences. Annual Review of Sociology, 36, 49-67.
  * [30] Held, K., Kops, E. R., Krause, B. J., Wells, W. M. I., Kikinis, R. and Müller-Gärtner, H.W. (1997). Markov random field segmentation of brain MR images. IEEE Transactions on Medical Imaging, 16(6), 878-886.
  * [31] Hofmann, T. and Puzicha, J. (1998). Statistical models for co-occurrence data. A.I. Memo 1625. Massachusetts Institute of Technology.
  * [32] King, G. (1997). A Solution to The Ecological Inference Problem: Reconstructing Individual Behavior From Aggregate Data. Princeton University Press.
  * [33] Llorente, A., Manmatha, R. and Rüger, S. M.(2010). Image retrieval using Markov Random Fields and global image features. In: Proceedings of the ACM International Conference on Image and Video Retrieval, pp. 243-250.
  * [34] Maathuis, M. H., Kalisch, M. and Bühlmann, P. (2009). Estimating high-dimensional intervention effects from observational data. Ann. Statist. 37, 3133-3164. MR2549555, https://doi.org/10.1214/09-AOS685.
  * [35] Maglogiannis, I., Vouyioukas, D. and Aggelopoulos, C. (2009). Face Detection and Recognition of Natural Human Emotion Using Markov Random Fields, Personal and Ubiquitous Computing, 13(1), 95-101.
  * [36] Mardia, K. V. (1988). Multi-dimensional multivariate Gaussian Markov random fields with application to image processing. Journal of Multivariate Analysis, 24, 265-284.
  * [37] Marengoni, A., Winblad, B., Karp, A. and Fratiglioni, L. (2008). Prevalence of chronic diseases and multimorbidity among the elderly population in Sweden.  American Journal of Public Health, 98, no. 7 (July 1, 2008): 1198-1200.
  * [38] Marengoni, A., Angleman, S., Melis, R., Mangialasche, F., Karp, A., Garmen, A., et al. (2011). Aging with multimorbidity: a systematic review of the literature. Ageing Research Reviews, 10(4), September 2011, 430-439.
  * [39] Mason, S.J. (1953). Feedback theory-Some properties of signal flow graphs. In Proceedings of the IRE 41, 1144-1156. IEEE.
  * [40] Mason, S.J. (1956). Feedback theory-Further properties of signal flow graphs. In Proceedings of the IRE 44, 920-926. IEEE.
  * [41] McPhail, S. M. (2016). Multimorbidity in chronic disease: impact on health care resources and costs. Risk Management and Healthcare Policy, Volume 2016:9, 143-156.
  * [42] Mignotte, M., Collet, C., Pérez, P. and Bouthemy, P. (2000). Markov Random Field and Fuzzy Logic Modeling in Sonar Imagery: Application to the Classification of Underwater Floor. Computer Vision and Image Understanding, 79(1), 4-24.
  * [43] Mooij, J. M. and Heskes, T.(2013). Cyclic causal discovery from continuous equilibrium data. In Proceedings of the 29th Conference on Uncertainty in Artificial Intelligence (UAI-13). (A. Nicholson and P. Smyth, eds.) 431-439. AUAI Press, Corvallis, OR, USA.
  * [44] Mooij, J. M., Peters, J., Janzing, D., Zscheischler, J. and Schölkope, B. (2016). Distinguishing cause from effect using observational data: Methods and benchmarks. J. Mach. Learn. Res. 17 Paper No. 32, 102. MR3491126.
  * [45] Mori, Y., Takahashi, H., and Oka, R. (1999). Image-to-word transformation based on dividing and vector quantizing images with words. In First International Workshop on Multimedia Intelligent Storage and Retrieval Management (in conjunction with ACM Multimedia Conference 1999), Orlando, Florida.
  * [46] Oakes, D. (1981). Survival times: aspects of partial likelihood. Internal.Statist. Rev. 49, 235-264.
  * [47] Oliver, D. (2012). 21st century health services for an ageing population: 10 challenges for general practice. British Journal of General Practice, 62(601), 396-397.
  * [48] Openshaw, S. (1983). The modifiable areal unit problem. CATMOG. 38 (Norwich, UK: Geo Books).
  * [49] Pearl, J. (2009). Causality: Models, Reasoning, and Inference, 2nd ed. Cambridge Univ. Press, Cambridge. MR2548166. https://doi.org/10.1017/CBO9780511803161.
  * [50] Peters, J., Bühlmann, P. and Meinshausen, N. (2016). Causal inference by using invariant prediction: identification and confidence intervals. J. R. Statist. Soc. B 78, Part 5, 947-1012.
  * [51] Peters, J., Janzing, D. and Schölkopf, B. (2017). Elements of Causal Inference: Foundations and Learning Algorithms. Adaptive Computation and Machine Learning. MIT Press, Cambridge, MA. https://doi.org/10.1017/CBO9780511803161. MR3822088.
  * [52] Peters, J., Mooij, J. M., Janzing, D. and Schölkope, B. (2014). Causal discovery with continuous additive noise models. J. Mach. Learn. Res. 15, 2009-2053. MR3231600.
  * [53] Pérez, P. (1998). Markov Random Fields and Images. CWI Quarterly, 11, 413-437.
  * [54] Pollard, D. (2001). A User’s Guide to Mesure Theoretic Probability. Cambridge University Press.
  * [55] Robinson, W. S. (1950). Ecological correlations and the behavior of individuals. American Sociological Review, 15(601), 351-357.
  * [56] Segev, E. (2021). Semantic Network Analysis in Social Sciences. London: Routledge.
  * [57] Spirtes, P., Glymour, C. and Scheines, R. (2000). Causation, Prediction, and Search, 2nd ed. Adaptive Computation and Machine Learning. MIT Press, Cambridge, MA. MR1815675.
  * [58] Stainvas, I. and Lowe, D. (2003). A Generative Model for Separating Illumination and Reflectance from Images. Journal of Machine Learning Research, 4, 1499-1519.
  * [59] Tate, N. and Atkinson, P. M. (2001). Modelling Scale in Geographical Information Science, 1st ed. Wiley.
  * [60] Torrione, P. A. and Collins, L. (2008). Application of Markov random fields to landmine detection in ground penetrating radar data. In: Harmon, R.S., Holloway, J.H.J., Broach, J.T. (eds.) Detection and Sensing of Mines, Explosive Objects, and Obscured Targets XIII, vol. 6953.
  * [61] Trianni, G. and Gamba, P. (2009). Fast damage mapping in case of earthquakes using multitemporal SAR data. Journal of Real-Time Image Processing, 4, 195-203.
  * [62] van der vaart, A. W. and Wellner, J. A. (2000). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer.
  * [63] Wang, J. Y. and Yu, H. (2022). Supplement to “Probability of Co-occurrence and E-Integral”.
  * [64] Whittle, P. (1992). Probability via Expectation, 3th edition. Springer-Verlag, New York. First edition 1970, under the tilte “Probability”.
  * [65] Whittle, P. (2000). Probability via Expectation, 4th edition. Springer.
  * [66] Wright, S. (1921). Correlation and causation. J. Agric. Res. 20, 557-585.
  * [67] Yan, J. A. (2004). Measure Theory Handout. Science Press, China.
  * [68] Zheng, H., Daoudi, M. and Jedynak, B. (2004). Blocking adult images based on statistical skin detection. Electronic Letters on Computer Vision and Image Analysis, 4(2), 1-14.
  * [69] Zheng, Y., Li, H. and Doermann, D.(2004). Machine Printed Text and Handwriting Identification in Noisy Document Images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26, 337-353.
  * [70] Zhu, J., Nie, Z., Zhang, B. and Wen, J. R. (2008). Dynamic Hierarchical Markov Random Fields for Integrated Web Data Extraction, Journal of Machine Learning Research, 9, 1583-1614.


# Algebraic structure of $\mathbb{F}_{q}$-linear conjucyclic codes over finite
field $\mathbb{F}_{q^{2}}$

Jingjie Lv, Ruihu Li∗ Department of Basic Sciences, Air Force Engineering
University
Xi’an, Shaanxi 710051, P. R. China<EMAIL_ADDRESS>(J. Lv<EMAIL_ADDRESS>(R. Li)

###### Abstract

Recently, Abualrub et al. [1] illustrated the algebraic structure of additive
conjucyclic codes over $\mathbb{F}_{4}$ (Finite Fields Appl. 65 (2020)
101678). In this paper, our main objective is to generalize their theory. Via
an isomorphic map, we give a canonical bijective correspondence between
$\mathbb{F}_{q}$-linear additive conjucyclic codes of length $n$ over
$\mathbb{F}_{q^{2}}$ and $q$-ary linear cyclic codes of length $2n$. By
defining the alternating inner product, our proposed isomorphic map preserving
the orthogonality can also be proved. From the factorization of the polynomial
$x^{2n}-1$ over $\mathbb{F}_{q}$, the enumeration of $\mathbb{F}_{q}$-linear
additive conjucyclic codes of length $n$ over $\mathbb{F}_{q^{2}}$ will be
obtained. Moreover, we provide the generator and parity-check matrices of
these $q^{2}$-ary additive conjucyclic codes of length $n$.

###### keywords:

Additive codes, Conjucyclic codes, Generator and parity-check matrices

††journal: Finite Fields and Their Applications

##  1 Introduction

Additive codes are important and special in coding theory. In 1973, the
additive code was defined by Delsarte et al. as the subgroup of the underling
abelian group on translation schemes [8]. In the year 1998, Calderbank et al.
[6] built a direct relationship between a quaternary additive code and a
binary quantum error-correcting code. The latter one can effectively reduce
the influences of environmental and operational noise (decoherence) of quantum
information, which plays an important role in both quantum communication and
quantum computation. Thereafter, the researches on additive codes have been a
hot topic, and there were many papers exploring the algebraic structure and
other applications of additive codes over finite fields or finite rings [3,
11, 16, 17, 18, 19].

Additive cyclic codes, as a special class of additive codes, have a pretty
good property, i.e., they are closed with respect to the cyclic shifts (see
Definition 2.1). The algebraic structure of quaternary additive cyclic codes
was explored by Huffman [15] in 2007, who found a canonical decomposition for
those codes of odd length. Later, the generalized theory of additive cyclic
codes over general fields was developed in [5]. At present, additive cyclic
codes have received a lot of attention by scholars [9, 13, 20, 21].

Similar to the cyclic codes, conjucyclic codes are closed for the conjugated
cyclic shift (see Definition 2.2). However, compared with the wide studies of
additive cyclic codes over finite fields or finite rings, there was little
work on conjucyclic codes. The major cause is that these conjucyclic codes can
not be represented canonically by polynomials, which leads to some
difficulties in analyzing these codes. Until 2020, the algebraic structure of
quaternary additive conjucyclic codes has been first researched by Abualrub et
al. [1]. Using an isomorphic map, they have built a connection between
quaternary additive conjucyclic codes of length $n$ and binary cyclic codes of
length $2n$. Inspired by the work, this paper will generalize the theory of
$\mathbb{F}_{q}$-linear additive conjucyclic codes over finite field
$\mathbb{F}_{q^{2}}$.

This paper is organized as follows. Sect. 2 discusses some definitions and
preliminary properties of cyclic codes, conjucyclic codes and some inner
products. In Sect. 3, we set up a bijective correspondence between
$\mathbb{F}_{q}$-linear conjucyclic codes of length $n$ over
$\mathbb{F}_{q^{2}}$ and $q$-ary linear cyclic codes of length $2n$. Moreover,
the algebraic structure of these conjucyclic codes is explored. In Sect. 4, we
illustrate the factorization of the polynomial $x^{2n}-1$ over
$\mathbb{F}_{q}$ and provide generator and parity-check matrices of
$q^{2}$-ary conjucyclic codes of length $n$.

##  2 Preliminaries

Let $\mathbb{F}_{q^{2}}$ be a finite field with $q^{2}$ elements, where $q$ is
a power of prime $p$. It is obvious that the characteristic of
$\mathbb{F}_{q^{2}}$ denoted by ${\rm char}(\mathbb{F}_{q^{2}})$ is $p$. For
any $\alpha\in\mathbb{F}_{q^{2}}$, let $\bar{\alpha}$ denote its conjugation
$\alpha^{q}$. Recall that a linear code $\mathscr{C}$ with parameters
$[n,k]_{q^{2}}$ is defined as a linear subspace of $\mathbb{F}_{q^{2}}^{n}$
with dimension $k$. An $(n,M)_{q^{2}}$ additive code over $\mathbb{F}_{q^{2}}$
is a subgroup of $\mathbb{F}_{q^{2}}^{n}$ with $M$ elements for the additive
operation. This means that the scalar multiples of codewords in an additive
code may no longer belong to it. In particular, if a code is a subspace over
field $\mathbb{F}_{q}$, then we call the code an $\mathbb{F}_{q}$-linear
additive code. For simplicity, we use additive codes to indicate them unless
otherwise stated.

Let vector $\vec{u}=(u_{0},u_{1},\ldots,u_{n-1})\in\mathbb{F}_{q}^{n}$. Denote
$w_{h}(\vec{u})={\rm card}\\{i|u_{i}\neq 0\\}$ for $0\leq i\leq n-1$. For a
$q$-ary code $\mathscr{C}$ of length $n$, its minimum Hamming weight is
defined as $w_{h}(\mathscr{C})={\rm
min}\\{w_{h}(\vec{u})|\vec{u}\in\mathscr{C}\\}$. If $n=2m$, we write
$w_{s}(\vec{u})={\rm card}\\{j|(u_{j},u_{m+j})\neq(0,0)\\}$ for $0\leq j\leq
m-1.$ Then $w_{s}(\mathscr{C})={\rm
min}\\{w_{s}(\vec{u})|\vec{u}\in\mathscr{C}\\}$ is the minimum symplectic
weight of $\mathscr{C}$.

Now we introduce the definition of cyclic codes as follows:

######  Definition 2.1.

Let $\mathscr{C}$ be a linear or additive code of length $n$ over
$\mathbb{F}_{q}$. For any codeword
$c=(c_{0},c_{1},\ldots,c_{n-1})\in\mathscr{C}$, if vector
$\sigma(c)=(c_{n-1},c_{0},\ldots,c_{n-2})$ also belongs to $\mathscr{C}$,
where $\sigma$ is a right cyclic shift operator, then the code $\mathscr{C}$
is called a $q$-ary cyclic code of length $n$.

Let $\mathbb{R}=\mathbb{F}_{q}[x]/\langle x^{n}-1\rangle$ be a quotient ring.
Define an $\mathbb{F}_{q}$-module isomorphic map $\varphi$ from vector space
$\mathbb{F}_{q}^{n}$ to $\mathbb{R}$ as

$\varphi(c_{0},c_{1},\ldots,c_{n-1})=c_{0}+c_{1}x+\cdots+c_{n-1}x^{n-1}.$

It is easy to see that $\mathscr{C}$ is a $q$-ary linear cyclic code of length
$n$ if and only if $\varphi(\mathscr{C})$ is an ideal of ring $\mathbb{R}$.
Since $\mathbb{R}$ is a principal ideal ring, one can prove that there exists
a one-to-one correspondence between a monic divisor of $x^{n}-1$ in
$\mathbb{F}_{q}[x]$ and a $q$-ary linear cyclic code of length $n$. As such,
the factorization of $x^{n}-1$ is crucial to classify linear cyclic codes of
length $n$. If a linear cyclic code $\mathscr{C}$ is generated by a monic
divisor $g(x)$ of $x^{n}-1$, $g(x)$ is called the generator polynomial of
$\mathscr{C}$. For convenience, we denote the linear cyclic code $\mathscr{C}$
by $\langle g(x)\rangle$ in the rest of the paper.

The definition of conjucyclic codes can be given similarly.

######  Definition 2.2.

Let $\mathscr{C}$ be a $q^{2}$-ary linear or additive code of length $n$. For
each codeword $c=(c_{0},c_{1},\ldots,c_{n-1})\in\mathscr{C}$, the code
$\mathscr{C}$ is said to be conjucyclic if
$T(c)=(\bar{c}_{n-1},c_{0},\ldots,c_{n-2})\in\mathscr{C}$, where $T$ is a
right conjucyclic shift operator.

Next, we propose some inner products and corresponding dual codes of
$\mathscr{C}$ that will be used in this paper.

Given two vectors $\vec{u}=(u_{0},u_{1}\ldots,u_{n-1})$ and
$\vec{v}=(v_{0},v_{1}\ldots,v_{n-1})$ in vector space $\mathbb{F}_{q}^{n}$,
define their Euclidean inner product as

$\langle\vec{u},\vec{v}\rangle_{e}=\sum_{i=0}^{n-1}u_{i}v_{i}.$

If $n=2m$, their symplectic inner product is defined as

$\langle\vec{u},\vec{v}\rangle_{s}=\sum_{i=0}^{m-1}(u_{i}v_{m+i}-u_{m+i}v_{i}).$

The Euclidean and symplectic dual codes of code $\mathscr{C}$ are respectively
denoted as

$\mathscr{C}^{\perp_{e}}=\\{\vec{v}\in\mathbb{F}_{q}^{n}|\langle\vec{u},\vec{v}\rangle_{e}=0,~{}\forall\vec{u}\in\mathscr{C}\\},~{}~{}\mathscr{C}^{\perp_{s}}=\\{\vec{v}\in\mathbb{F}_{q}^{2m}|\langle\vec{u},\vec{v}\rangle_{s}=0,~{}\forall\vec{u}\in\mathscr{C}\\}.$

Note that the dual codes $\mathscr{C}^{\perp_{e}}$ and
$\mathscr{C}^{\perp_{s}}$ must be $\mathbb{F}_{q^{2}}$-linear whether the code
$\mathscr{C}$ is $\mathbb{F}_{q^{2}}$-linear or not. Since we are generally
interested in additive codes over $\mathbb{F}_{q^{2}}$, then these usual inner
products are not the most useful to study the dual structure of additive codes
anymore.

In the following, we give an alternating inner product on vector space
$\mathbb{F}_{q^{2}}^{n}$. Let $\beta$ be a primitive element of field
$\mathbb{F}_{q^{2}}$. For $\vec{u}=(u_{0},u_{1},\ldots,u_{n-1})$,
$\vec{v}=(v_{0},v_{1},\ldots,v_{n-1})\in\mathbb{F}_{q^{2}}^{n}$, the
alternating inner product is defined as

$\langle
u,v\rangle_{a}=(\bar{\beta}^{2}-\beta^{2})\sum_{i=1}^{n}(u_{i}\bar{v}_{i}-\bar{u}_{i}v_{i}).$

Since $\langle u,v\rangle_{a}^{q}=\langle u,v\rangle_{a}$, we have $\langle
u,v\rangle_{a}\in\mathbb{F}_{q}$. In addition, the alternating inner product
$\langle\cdot~{},~{}\cdot\rangle_{a}$ is $\mathbb{F}_{q}$-bilinear.
Particularly, when $q=2$, $\langle
u,v\rangle_{a}=\sum_{i=1}^{n}(u_{i}\bar{v}_{i}+\bar{u}_{i}v_{i})={\rm
Tr}\left(\sum_{i=1}^{n}u_{i}\bar{v}_{i}\right),$ where ${\rm Tr}$ is the trace
function of $\mathbb{F}_{q^{2}}$ down $\mathbb{F}_{q}$ that is defined as
${\rm Tr}(x)=x+\bar{x}$ for all $x\in\mathbb{F}_{q^{2}}$. Therefore, the
alternating inner product is a generalization of the trace-Hermitian inner
product in vector space $\mathbb{F}_{4}^{n}$ appeared in [6].

Similarly, the alternating dual codes of $\mathscr{C}$ can be presented as
follows:

$\mathscr{C}^{\perp_{a}}=\\{\vec{v}\in\mathbb{F}_{q^{2}}^{n}|\langle\vec{u},\vec{v}\rangle_{a}=0,~{}\forall\vec{u}\in\mathscr{C}\\}.$

##  3 The algebraic structure of additive conjucyclic codes

In [1], Abualrub et al. have shown that quaternary conjucyclic codes of length
$n$ are related to binary cyclic codes of length $2n$. In this section, by
virtue of an isomorphic map, we indicate that the bijective correspondence
between $q^{2}$-ary conjucyclic codes of length $n$ and $q$-ary cyclic codes
of length $2n$ also exists.

For any element $\alpha\in\mathbb{F}_{q^{2}}$, we define a map $\phi$ from
$\mathbb{F}_{q^{2}}$ to $\mathbb{F}_{q}^{2}$ as follows:

$\begin{split}\phi:~{}~{}\mathbb{F}_{q^{2}}&\longrightarrow\mathbb{F}_{q}^{2}\\\
\alpha&\longmapsto\left({\rm Tr}(\beta\alpha),{\rm
Tr}(\bar{\beta}\alpha)\right),\\\ \end{split}$

where $\beta$ is a primitive element of field $\mathbb{F}_{q^{2}}$ and ${\rm
Tr}$ is the trace function of $\mathbb{F}_{q^{2}}$ down $\mathbb{F}_{q}$.

######  Proposition 3.1.

The map $\phi$ is an $\mathbb{F}_{q}$-linear isomorphism.

  * Proof.

For any $k_{1}$, $k_{2}$ $\in\mathbb{F}_{q}$ and $\alpha_{1}$, $\alpha_{2}$
$\in\mathbb{F}_{q^{2}}$, we have

$\begin{split}\phi(k_{1}\alpha_{1}+k_{2}\alpha_{2})&=\left({\rm
Tr}(\beta(k_{1}\alpha_{1}+k_{2}\alpha_{2})\right),{\rm
Tr}(\bar{\beta}(k_{1}\alpha_{1}+k_{2}\alpha_{2})))\\\ &=\left(k_{1}{\rm
Tr}(\beta\alpha_{1})+k_{2}{\rm Tr}(\beta\alpha_{2}),k_{1}{\rm
Tr}(\bar{\beta}\alpha_{1})+k_{2}{\rm Tr}(\bar{\beta}\alpha_{2})\right)\\\
&=k_{1}({\rm Tr}(\beta\alpha_{1}),{\rm Tr}(\bar{\beta}\alpha_{1}))+k_{2}({\rm
Tr}(\beta\alpha_{2}),{\rm Tr}(\bar{\beta}\alpha_{2}))\\\
&=k_{1}\phi(\alpha_{1})+k_{2}\phi(\alpha_{2}),\end{split}$

which implies that the map $\phi$ is $\mathbb{F}_{q}$-linear.

Next, we prove that $\phi$ is an injective map. Assume
$\phi(\alpha_{1})=\phi(\alpha_{2})$, then $\left({\rm
Tr}(\beta\alpha_{1}),{\rm Tr}(\bar{\beta}\alpha_{1})\right)-\left({\rm
Tr}(\beta\alpha_{2}),{\rm Tr}(\bar{\beta}\alpha_{2})\right)=\left({\rm
Tr}(\beta(\alpha_{1}-\alpha_{2})),{\rm
Tr}(\bar{\beta}(\alpha_{1}-\alpha_{2})\right)=(0,0)$, i.e.,

$\displaystyle\begin{cases}\beta(\alpha_{1}-\alpha_{2})+\bar{\beta}(\alpha_{1}-\alpha_{2})^{q}=0\\\
\bar{\beta}(\alpha_{1}-\alpha_{2})+\beta(\alpha_{1}-\alpha_{2})^{q}=0.\end{cases}$
(1)

If ${\rm char}(\mathbb{F}_{q^{2}})=2$, by Equation (1), we have

$\displaystyle\begin{cases}\beta(\alpha_{1}+\alpha_{2})=\bar{\beta}(\alpha_{1}+\alpha_{2})^{q}\\\
\bar{\beta}(\alpha_{1}+\alpha_{2})=\beta(\alpha_{1}+\alpha_{2})^{q},\end{cases}$

which means that $\beta(\alpha_{1}+\alpha_{2})$,
$\bar{\beta}(\alpha_{1}+\alpha_{2})\in\mathbb{F}_{q}$. Since $\beta$ is a
primitive element of $\mathbb{F}_{q^{2}}$ and $\alpha_{1},\alpha_{2}$ are
arbitrary elements in $\mathbb{F}_{q^{2}}$, then $\alpha_{1}=\alpha_{2}$ can
be derived directly.

Now we suppose ${\rm char}(\mathbb{F}_{q^{2}})\neq 2$, then Equation (1) gives

$\displaystyle\begin{cases}(\beta+\bar{\beta})\left((\alpha_{1}-\alpha_{2})+(\alpha_{1}-\alpha_{2})^{q}\right)=0\\\
(\beta-\bar{\beta})\left((\alpha_{1}-\alpha_{2})-(\alpha_{1}-\alpha_{2})^{q}\right)=0.\end{cases}$

Obviously, $(\beta+\bar{\beta})\neq 0$ and $(\beta-\bar{\beta})\neq 0$. If
not, $\beta^{2q-2}=1$ or $\beta^{q-1}=1$. There exists a contradiction because
the order of $\beta$ is $q^{2}-1$ and $q-1<2q-2<q^{2}-1$. So the above
equations can be transformed into

$\displaystyle\begin{cases}(\alpha_{1}-\alpha_{2})+(\alpha_{1}-\alpha_{2})^{q}=0\\\
(\alpha_{1}-\alpha_{2})-(\alpha_{1}-\alpha_{2})^{q}=0\end{cases}\Rightarrow\begin{cases}2(\alpha_{1}-\alpha_{2})=0\\\
2(\alpha_{1}-\alpha_{2})^{q}=0\end{cases}\Rightarrow\alpha_{1}=\alpha_{2}.$

Hence, $\phi$ is an injective map. Moreover, since
$|\mathbb{F}_{q^{2}}|=|\mathbb{F}_{q}^{2}|=q^{2}$, $\phi$ must be surjective.
In conclusion, $\phi$ is an $\mathbb{F}_{q}$-linear isomorphism.  $\square$

######  Remark 3.2.

Notice that the map $\phi$ is $\mathbb{F}_{q}$-linear isomorphic. Then any
element $\alpha\in\mathbb{F}_{q^{2}}$ is determined by $({\rm
Tr}(\beta\alpha),{\rm Tr}(\bar{\beta}\alpha))\in\mathbb{F}_{q}^{2}$. In fact,

$\alpha=\frac{1}{\beta-\beta^{2q-1}}{\rm
Tr}(\beta\alpha)-\frac{\beta^{q-1}}{\beta-\beta^{2q-1}}{\rm
Tr}(\bar{\beta}\alpha).$ (2)

######  Example 3.3.

Let $q=3$ and $\beta$ be a primitive element of field $\mathbb{F}_{9}$. Then
the map $\phi$ from $\mathbb{F}_{9}$ to $\mathbb{F}_{3}$ can be given as
follows:

$\begin{split}\phi:~{}~{}\mathbb{F}_{9}&\longrightarrow\mathbb{F}_{3}^{2}\\\
0\longmapsto(0,0)&~{}~{}~{}~{}~{}~{}~{}\beta^{4}\longmapsto(2,2)\\\
1\longmapsto(1,1)&~{}~{}~{}~{}~{}~{}~{}\beta^{5}\longmapsto(0,2)\\\
\beta\longmapsto(0,1)&~{}~{}~{}~{}~{}~{}~{}\beta^{6}\longmapsto(2,1)\\\
\beta^{2}\longmapsto(1,2)&~{}~{}~{}~{}~{}~{}~{}\beta^{7}\longmapsto(2,0)~{}.\\\
\beta^{3}\longmapsto(1,0)&~{}~{}~{}~{}~{}~{}~{}\end{split}$

One can check that the map $\phi$ is an $\mathbb{F}_{3}$-linear isomorphism.
According to Equation (2), we have $\frac{1}{\beta-\beta^{5}}=\beta^{3}$,
$\frac{\beta^{2}}{\beta-\beta^{5}}=\beta^{5}$ and

$\begin{split}0=\beta^{3}\times 0-\beta^{5}\times
0&~{}~{}~{}~{}~{}~{}~{}\beta^{4}=\beta^{3}\times 2-\beta^{5}\times 2\\\
1=\beta^{3}\times 1-\beta^{5}\times
1&~{}~{}~{}~{}~{}~{}~{}\beta^{5}=\beta^{3}\times 0-\beta^{5}\times 2\\\
\beta=\beta^{3}\times 0-\beta^{5}\times
1&~{}~{}~{}~{}~{}~{}~{}\beta^{6}=\beta^{3}\times 2-\beta^{5}\times 1\\\
\beta^{2}=\beta^{3}\times 1-\beta^{5}\times
2&~{}~{}~{}~{}~{}~{}~{}\beta^{7}=\beta^{3}\times 2-\beta^{5}\times 0.\\\
\beta^{3}=\beta^{3}\times 1-\beta^{5}\times 0&\end{split}$

∎

Naturally, the map $\phi$ can be generalized to vector space
$\mathbb{F}_{q^{2}}^{n}$. For vector
$\vec{\alpha}=(\alpha_{0},\ldots,\alpha_{n-1})\in\mathbb{F}_{q^{2}}^{n}$, the
extended map denoted by $\Phi$ is

$\begin{split}\Phi(\vec{\alpha}=(\alpha_{0},\ldots,\alpha_{n-1}))=\left({\rm
Tr}(\beta\alpha_{0}),{\rm Tr}(\bar{\beta}\alpha_{0}),\ldots,{\rm
Tr}(\beta\alpha_{n-1}),{\rm Tr}(\bar{\beta}\alpha_{n-1})\right).\\\
\end{split}$

If we define a permutation $\varsigma$ on vector
$\vec{v}=(v_{0},v_{1},\ldots,v_{2n-1})\in\mathbb{F}_{q}^{2n}$ as

$\varsigma(\vec{v}=(v_{0},v_{1},\ldots,v_{2n-1}))=(v_{0},v_{2},\ldots,v_{2n-2},v_{1},v_{3},\ldots,v_{2n-1}),$

then the following proposition can be gained directly.

######  Proposition 3.4.

$\Psi=\varsigma\cdot\Phi$ is an $\mathbb{F}_{q}$-linear isomorphism, and for
all vector
$\vec{\alpha}=(\alpha_{0},\ldots,\alpha_{n-1})\in\mathbb{F}_{q^{2}}^{n}$, we
have

$\Psi(\alpha_{0},\ldots,\alpha_{n-1})=\left({\rm
Tr}(\beta\alpha_{0}),\ldots,{\rm Tr}(\beta\alpha_{n-1}),{\rm
Tr}(\bar{\beta}\alpha_{0}),{\rm Tr}(\bar{\beta}\alpha_{n-1})\right).$

Now we can build a relationship between the conjucyclic shift on vectors in
$\mathbb{F}_{q^{2}}^{n}$ and the cyclic shift on vectors in
$\mathbb{F}_{q}^{2n}$.

######  Proposition 3.5.

For any vector $\vec{\alpha}\in\mathbb{F}_{q^{2}}^{n}$, we have
$\Psi(T(\vec{\alpha}))=\sigma(\Psi(\vec{\alpha}))$, where $\sigma$ and $T$ are
cyclic and conjucyclic shift operators, respectively. To be a more intuitive
understanding, a commutative diagram is exhibited as follows:

$\begin{CD}\mathbb{F}_{q^{2}}^{n}@>{T}>{}>\mathbb{F}_{q^{2}}^{n}\\\
@V{\Psi}V{}V@V{}V{\Psi}V\\\
<EMAIL_ADDRESS>
  * Proof.

For any
$\vec{\alpha}=(\alpha_{0},\ldots,\alpha_{n-1})\in\mathbb{F}_{q^{2}}^{n}$, we
have

$\begin{split}\Psi(T(\vec{\alpha}))=&\Psi(\bar{\alpha}_{n-1},\alpha_{0},\ldots,\alpha_{n-2})=({\rm
Tr}(\beta\bar{\alpha}_{n-1}),{\rm Tr}(\beta\alpha_{0}),\ldots,{\rm
Tr}(\beta\alpha_{n-2}),\\\ &{\rm Tr}(\bar{\beta}\bar{\alpha}_{n-1}),{\rm
Tr}(\bar{\beta}\alpha_{0}),\ldots,{\rm
Tr}(\bar{\beta}\alpha_{n-2})),\end{split}$
$\begin{split}\sigma(\Psi(\vec{\alpha}))=&\sigma({\rm
Tr}(\beta\alpha_{0}),\ldots,{\rm Tr}(\beta\alpha_{n-1}),{\rm
Tr}(\bar{\beta}\alpha_{0}),\ldots,{\rm Tr}(\bar{\beta}\alpha_{n-1}))=({\rm
Tr}(\bar{\beta}\alpha_{n-1}),\\\ &{\rm Tr}(\beta\alpha_{0}),\ldots,{\rm
Tr}(\beta\alpha_{n-2}),{\rm Tr}(\beta\alpha_{n-1}),{\rm
Tr}(\bar{\beta}\alpha_{0}),\ldots,{\rm
Tr}(\bar{\beta}\alpha_{n-2})).\end{split}$

Since ${\rm Tr}(\beta\bar{\alpha}_{n-1})={\rm Tr}(\bar{\beta}\alpha_{n-1})$,
${\rm Tr}(\bar{\beta}\bar{\alpha}_{n-1})={\rm Tr}(\beta\alpha_{n-1})$, the
proposition holds obviously.  $\square$

Then a bijective correspondence between additive conjucyclic codes and linear
cyclic codes can be obtained.

######  Theorem 3.6.

$\mathscr{C}$ is an additive conjucyclic code over $\mathbb{F}_{q^{2}}$ of
length $n$ if and only if there exists a $q$-ary linear cyclic code
$\mathscr{D}=\Psi(\mathscr{C})$ of length $2n$. Moreover, the minimum Hamming
weight of $\mathscr{C}$ is equal to the minimum symplectic weight of
$\mathscr{C}$, i.e., $w_{h}(\mathscr{C})=w_{s}(\mathscr{D})$.

  * Proof.

If $\mathscr{C}$ is an additive conjucyclic code over $\mathbb{F}_{q^{2}}$ of
length $n$, the map $\Psi$ gives that $\mathscr{D}$ is a vector subspace of
$\mathbb{F}_{q}^{2n}$. It is equivalent to say, $\mathscr{D}$ is an
$\mathbb{F}_{q}$-linear code of length $2n$. Via Proposition 3.5, we have
$\sigma(\Psi(\mathscr{C}))=\Psi(T(\mathscr{C}))=\Psi(\mathscr{C})$. So
$\mathscr{D}$ is a $q$-ary linear cyclic code of length $2n$.

Conversely, assume that $\mathscr{D}$ is a $q$-ary linear cyclic code of
length $2n$. Since $\Psi$ is an $\mathbb{F}_{q}$-linear isomorphism, then
$\mathscr{C}=\Psi^{-1}(\mathscr{D})$ is an additive subgroup of
$\mathbb{F}_{q^{2}}^{n}$. In addition, by
$\Psi(T(\mathscr{C}))=\sigma(\Psi(\mathscr{C}))=\Psi(\mathscr{C})$, we have
$T(\mathscr{C})=\mathscr{C}$. Hence, $\mathscr{C}$ is an additive conjucyclic
code.

For each codeword $\vec{c}=(c_{0},c_{1},\ldots,c_{n-1})\in\mathscr{C}$,
according to Equation (2), it is easy to see that $c_{i}=0$ if and only if
$\left({\rm Tr}(\beta c_{i}),{\rm Tr}(\bar{\beta}c_{i})\right)=0$ for $0\leq
i\leq n-1$, which implies $w_{h}(\mathscr{C})=w_{s}(\mathscr{D})$.  $\square$

######  Remark 3.7.

By the theory of cyclic codes [14], these binary cyclic codes from quaternary
additive conjucyclic codes in [1] must be repeated-root since ${\rm
gcd}(2n,2)=2$. However, [7] has shown that repeated-root cyclic codes cannot
be asymptotically better than simple-root cyclic codes. In this work, our
cyclic codes from additive conjucyclic codes can be simple-root.

In the following result, we will exhibit that the map $\Psi$ also preserves
the orthogonality.

######  Proposition 3.8.

Let $\vec{u}$ and $\vec{v}$ be two vectors in $\mathbb{F}_{q^{2}}^{n}$. Then

$\langle\Psi(\vec{u}),\Psi(\vec{v})\rangle_{s}=\langle\vec{u},\vec{v}\rangle_{a},$

where $\langle\cdot~{},~{}\cdot\rangle_{s}$ and
$\langle\cdot~{},~{}\cdot\rangle_{a}$ denote the symplectic and alternating
inner product, respectively.

  * Proof.

Select $\vec{u}=(u_{0},\ldots,u_{n-1})$ and
$\vec{v}=(v_{0},\ldots,v_{n-1})\in\mathbb{F}_{q^{2}}^{n}$, we have

$\Psi(\vec{u}=(u_{0},\ldots,u_{n-1}))=\left({\rm Tr}(\beta u_{0}),\ldots,{\rm
Tr}(\beta u_{n-1}),{\rm Tr}(\bar{\beta}u_{0}),\ldots,{\rm
Tr}(\bar{\beta}u_{n-1})\right),$
$\Psi(\vec{v}=(v_{0},\ldots,v_{n-1}))=\left({\rm Tr}(\beta v_{0}),\ldots,{\rm
Tr}(\beta v_{n-1}),{\rm Tr}(\bar{\beta}v_{0}),\ldots,{\rm
Tr}(\bar{\beta}v_{n-1})\right).$
$\begin{split}\text{Then}~{}\langle\Psi(\vec{u}),\Psi(\vec{v})\rangle_{s}=&\sum_{i=0}^{n-1}\left({\rm
Tr}(\bar{\beta}u_{i}){\rm Tr}(\beta v_{i})-{\rm Tr}(\beta u_{i}){\rm
Tr}(\bar{\beta}v_{i})\right)\\\
=&\sum_{i=0}^{n-1}\left((\bar{\beta}u_{i}+\beta\bar{u}_{i})(\beta
v_{i}+\bar{\beta}\bar{v}_{i})-(\beta
u_{i}+\bar{\beta}\bar{u}_{i})(\bar{\beta}v_{i}+\beta\bar{v}_{i})\right)\\\
=&\sum_{i=0}^{n-1}(\beta^{q+1}u_{i}v_{i}+\bar{\beta}^{2}u_{i}\bar{v}_{i}+\beta^{2}\bar{u}_{i}v_{i}+\beta^{q+1}\bar{u}_{i}\bar{v}_{i}-\beta^{q+1}u_{i}v_{i}\\\
&~{}~{}~{}~{}~{}-\beta^{2}u_{i}\bar{v}_{i}-\bar{\beta}^{2}\bar{u}_{i}v_{i}-\beta^{q+1}\bar{u}_{i}\bar{v}_{i})\\\
=&\sum_{i=0}^{n-1}(\bar{\beta}^{2}u_{i}\bar{v}_{i}+\beta^{2}\bar{u}_{i}v_{i}-\beta^{2}u_{i}\bar{v}_{i}-\bar{\beta}^{2}\bar{u}_{i}v_{i})\\\
=&\sum_{i=0}^{n-1}(\bar{\beta}^{2}(u_{i}\bar{v}_{i}-\bar{u}_{i}v_{i})+\beta^{2}(\bar{u}_{i}v_{i}-u_{i}\bar{v}_{i}))\\\
=&(\bar{\beta}^{2}-\beta^{2})\sum_{i=0}^{n-1}(u_{i}\bar{v}_{i}-\bar{u}_{i}v_{i})=\langle\vec{u},\vec{v}\rangle_{a}.\\\
\end{split}$

$\square$

######  Theorem 3.9.

Let $\mathscr{C}$ be an additive conjucyclic code over $\mathbb{F}_{q^{2}}$ of
length $n$ and $\mathscr{D}=\Psi(\mathscr{C})$ be a $q$-ary linear cyclic code
of length $2n$. Then we have

$\mathscr{C}^{\perp_{a}}=\Psi^{-1}(\mathscr{D}^{\perp_{s}}).$

  * Proof.

For any $\vec{c}\in\mathscr{C}$ and $\vec{d}\in\mathscr{C}^{\bot_{a}}$,
according to Proposition 3.8, we have
$\langle\vec{c},\vec{d}\rangle_{a}=\langle\Psi(\vec{c}),\Psi(\vec{d})\rangle_{s}=0$.
By the arbitrariness of codeword $\vec{c}\in\mathscr{C}$, we gain
$\Psi(\vec{d})\in\mathscr{D}^{\perp_{s}}$ and
$\Psi(\mathscr{C}^{\bot_{a}})\subseteq\mathscr{D}^{\perp_{s}}$, then
$\mathscr{C}^{\bot_{a}}\subseteq\Psi^{-1}(\mathscr{D}^{\perp_{s}})$.
Analogously, we can also prove
$\Psi^{-1}(\mathscr{D}^{\perp_{s}})\subseteq\mathscr{C}^{\bot_{a}}$. This
concludes the result.  $\square$

######  Corollary 3.10.

Let $\mathscr{C}\subseteq\mathbb{F}_{q^{2}}^{n}$ be an additive conjucyclic
code and $\mathscr{D}=\Psi(\mathscr{C})\subseteq\mathbb{F}_{q}^{2n}$ be a
linear cyclic code. Then $\mathscr{C}$ is alternating dual-containing if and
only if $\mathscr{D}$ is symplectic dual-containing, that is,
$\mathscr{C}^{\perp_{a}}\subseteq\mathscr{C}$ if and only if
$\mathscr{D}^{\perp_{s}}\subseteq\mathscr{D}.$

In 2001, Ashikhmin et al. [2] expounded a vitally important connection between
quantum stabilizer codes and classical codes with respect to the symplectic
inner product.

######  Lemma 3.11.

[2] If $\mathscr{C}$ is a symplectic dual-containing linear code with
parameters $[2n,k]_{q}$, then there exists an $[[n,k-n,\geq
w_{s}(\mathscr{C})]]_{q}$ quantum stabilizer code that is pure to
$w_{s}(\mathscr{C}$).

Utilizing the alternating dual-containing additive conjucyclic codes, a
construction of quantum stabilizer codes can also be proposed.

######  Theorem 3.12.

If $\mathscr{C}$ is an $\left(n,M=q^{k},w_{h}(\mathscr{C})\right)_{q^{2}}$
additive conjucyclic code such that
$\mathscr{C}^{\perp_{a}}\subseteq\mathscr{C}$, then a pure $[[n,k-n,\geq
w_{h}(\mathscr{C})]]_{q}$ stabilizer code can be provided.

  * Proof.

From Theorem 3.6, Corollary 3.10 and Lemma 3.11, the construction can be
derived immediately.  $\square$

In [1], $\mathcal{H}(\mathscr{C})$ was defined by Abualrub et al. as the
largest binary cyclic code contained in a conjucyclic code $\mathscr{C}$ over
$\mathbb{F}_{4}$. Some algebraic properties of $\mathcal{H}(\mathscr{C})$ have
been further discussed. Now we define $\mathcal{H}(\mathscr{C})$ as the
largest $q$-ary cyclic code in a $q^{2}$-ary conjucyclic code $\mathscr{C}$
and some generalized conclusions are also available.

######  Theorem 3.13.

Let $\mathscr{C}$ be an additive conjucyclic code and
$\mathscr{D}=\Psi(\mathscr{C})$ be a linear cyclic code. Then
$\Psi(\mathcal{H}(\mathscr{C}))$ is a subcode of $\mathscr{D}$ and cosets of
$\Psi(\mathcal{H}(\mathscr{C}))$ are contained in $\mathscr{D}$.

  * Proof.

Since $\mathcal{H}(\mathscr{C})$ is the largest $q$-ary cyclic code in
$\mathscr{C}$, we have that $\mathcal{H}(\mathscr{C})$ is an
$\mathbb{F}_{q}$-linear subcode of $\mathscr{C}$, which gives that
$\Psi(\mathcal{H}(\mathscr{C}))$ is a subcode of $\mathscr{D}$ and
$\mathscr{D}$ contains cosets of $\Psi(\mathcal{H}(\mathscr{C}))$.  $\square$

######  Theorem 3.14.

Let $\mathscr{C}$ be a $q^{2}$-ary conjucyclic code of length $n$ and
$\mathscr{D}=\Psi(\mathscr{C})$ be a $q$-ary linear cyclic code of length
$2n$. For any codeword $\vec{d}=(d_{0},d_{1},\ldots,d_{2n-1})\in\mathscr{D}$,
$\vec{d}\in\Psi(\mathcal{H}(\mathscr{C}))$ if and only if $d_{i}=d_{n+i}$ for
$0\leq i\leq n-1$.

  * Proof.

If $\vec{d}=(d_{0},d_{1},\ldots,d_{2n-1})\in\Psi(\mathcal{H}(\mathscr{C}))$,
then $\vec{c}=\Psi^{-1}(\vec{d})=(c_{0},c_{1},\ldots,\\\
c_{n-1})\in\mathcal{H}(\mathscr{C})$. Since $c_{i}\in\mathbb{F}_{q}$ for
$0\leq i\leq n-1$, we have $\vec{d}=\Psi(\vec{d})=({\rm Tr}(\beta
c_{0}),\ldots,{\rm Tr}(\beta c_{n-1}),{\rm Tr}(\beta^{q}c_{0}),\ldots,{\rm
Tr}(\beta^{q}c_{n-1}))=(c_{0}{\rm Tr}(\beta),\ldots,c_{n-1}{\rm Tr}(\beta),\\\
c_{0}{\rm Tr}(\beta^{q}),\ldots,c_{n-1}{\rm Tr}(\beta^{q}))$. By ${\rm
Tr}(\beta)={\rm Tr}(\beta^{q})$, $d_{i}=d_{n+i}$ can be deduced.

On the contrary, if $d_{i}=d_{n+i}$, by Equation (2), we have
$c_{i}=\frac{1}{\beta-\beta^{2q-1}}d_{i}-\frac{\beta^{q-1}}{\beta-\beta^{2q-1}}d_{n+i}=\frac{1-\beta^{q-1}}{\beta-\beta^{2q-1}}d_{i}=\frac{1}{\beta+\beta^{q}}d_{i}\in\mathbb{F}_{q}$
for $0\leq i\leq n-1$. Since $\mathcal{H}(\mathscr{C})$ is the largest $q$-ary
cyclic code in $\mathscr{C}$, one can get that
$\vec{c}\in\mathcal{H}(\mathscr{C})$ and
$\vec{d}=\Psi(c)\in\Psi(\mathcal{H}(\mathscr{C}))$.  $\square$

######  Corollary 3.15.

Let $\mathscr{C}$ be a conjucyclic code and $\mathscr{D}=\Psi(\mathscr{C})$ be
a linear cyclic code. Then $\Psi(\mathcal{H}(\mathscr{C}))$ is a subcode of
$\mathscr{D}$ consisting of all codewords with $d_{i}=d_{n+i}$ for $0\leq
i\leq n-1$.

######  Example 3.16.

Choose $\beta$ as a primitive element of $\mathbb{F}_{9}$ and the 9-ary
conjucyclic additive code $\mathscr{C}$ of length 3 is generated by

$\\{(2,1,0),(0,2,1),(\beta^{2},\beta^{6},\beta^{2})\\}.$

The code $\mathscr{C}$ has cardinality 27, and its codewords are listed as
follows:

$(0,0,0)$ | $(\beta,\beta^{3},\beta)$ | $(\beta^{3},\beta,\beta^{3})$ | $(1,2,0)$ | $(\beta^{2},\beta^{5},\beta)$ | $(\beta^{6},\beta^{7},\beta^{3})$
---|---|---|---|---|---
$(0,1,2)$ | $(\beta,\beta^{6},\beta^{7})$ | $(\beta^{3},\beta^{2},\beta^{5})$ | $(2,0,1)$ | $(\beta^{7},\beta^{3},\beta^{2})$ | $(\beta^{5},\beta,\beta^{6})$
$(0,2,1)$ | $(\beta,\beta^{5},\beta^{2})$ | $(\beta^{3},\beta^{7},\beta^{6})$ | $(2,1,0)$ | $(\beta^{7},\beta^{6},\beta)$ | $(\beta^{5},\beta^{2},\beta^{3})$
$(1,0,2)$ | $(\beta^{2},\beta^{3},\beta^{7})$ | $(\beta^{6},\beta,\beta^{5})$ | $(2,2,2)$ | $(\beta^{7},\beta^{5},\beta^{7})$ |  $(\beta^{5},\beta^{7},\beta^{5})$.
$(1,1,1)$ | $(\beta^{2},\beta^{6},\beta^{2})$ | $(\beta^{6},\beta^{2},\beta^{6})$ |  |  |

Then $\mathscr{D}=\Psi(\mathscr{C})$ is a ternary cyclic code of length 6,
whose codewords can be written as

$(0,0,0,0,0,0)$ | $(0,1,0,1,0,1)$ | $(1,0,1,0,1,0)$
---|---|---
$(0,1,2,0,1,2)$ | $(0,2,2,1,1,0)$ | $(1,1,0,0,2,2)$
$(0,2,1,0,2,1)$ | $(0,0,1,1,2,2)$ | $(1,2,2,0,0,1)$
$(1,0,2,1,0,2)$ | $(1,1,2,2,0,0)$ | $(2,0,0,1,1,2)$
$(1,1,1,1,1,1)$ | $(1,2,1,2,1,2)$ | $(2,1,2,1,2,1)$
$(1,2,0,1,2,0)$ | $(1,0,0,2,2,1)$ | $(2,2,1,1,0,0)$
$(2,0,1,2,0,1)$ | $(2,1,1,0,0,2)$ | $(0,0,2,2,1,1)$
$(2,1,0,2,1,0)$ | $(2,2,0,0,1,1)$ | $(0,1,1,2,2,0)$
$(2,2,2,2,2,2)$ | $(2,0,2,0,2,0)$ |  $(0,2,0,2,0,2)$.

By our definition,
$\mathcal{H}(\mathscr{C})=\\{(0,0,0),(0,1,2),(0,2,1),(1,0,2),(1,1,1),(1,2,0),\\\
(2,0,1),(2,1,0),(2,2,2)\\}$. From
$\Psi(\mathcal{H}(\mathscr{C}))=\\{(0,0,0,0,0,0),(0,1,2,0,1,2),\\\
(0,2,1,0,2,1),(1,0,2,1,0,2),(1,1,1,1,1,1),(1,2,0,1,2,0),(2,0,1,2,0,1),\\\
(2,1,0,2,1,0),(2,2,2,2,2,2)\\}$, we can see that
$\Psi(\mathcal{H}(\mathscr{C}))$ is a subcode of $\mathscr{D}$ consisting of
all codewords with $d_{i}=d_{3+i}$ for $0\leq i\leq 2$. ∎

######  Example 3.17.

Assume ${\rm char}(\mathbb{F}_{q^{2}})=2$. Let
$\mathscr{D}=\Psi(\mathscr{C})=\langle x+1\rangle$ be a $q$-ary linear cyclic
code of length $2n$. By the theory of cyclic codes [14], we have its Euclidean
dual code $\mathscr{D}^{\bot_{e}}=\langle 1+x+\cdots+x^{2n-1}\rangle$.
Obviously, vector $\vec{v}=(1,0,\ldots 0\mid 1,0,\ldots 0)\in\mathscr{D}$,
which deduces
$\Psi^{-1}(\vec{v})=(\frac{1}{\beta+\beta^{q}},0,\ldots,0)\in\mathbb{F}_{q}^{n}$.
Therefore, the coordinate vectors
$e_{1}=(1,0,\ldots,0),\ldots,e_{n}=(0,\ldots,0,1)\in\mathscr{C}$ and
$\mathcal{H}(\mathscr{C})=\mathbb{F}_{q}^{n}$. ∎

######  Example 3.18.

Now we let $\mathscr{D}=\Psi(\mathscr{C})=\langle 1+x+\cdots+x^{2n-1}\rangle$
be a $q$-ary linear cyclic code of length $2n$. Then $\mathscr{D}$ has
cardinality $q$ and consists of all $\gamma$ vectors for
$\gamma\in\mathbb{F}_{q}$. Hence,
$\mathscr{C}=\Psi^{-1}(\mathscr{D})=\left\\{\left(\frac{\gamma}{\beta+\beta^{q}},\ldots,\frac{\gamma}{\beta+\beta^{q}}\right)|\gamma\in\mathbb{F}_{q}\right\\}\subseteq\mathbb{F}_{q}^{n}$.
In this case, we have $\mathcal{H}(\mathscr{C})=\mathscr{C}$. ∎

##  4 Generator and parity-check matrices for additive conjucyclic codes

In this section, we start with the factorization of the polynomial $x^{2n}-1$
over $\mathbb{F}_{q}$ to classify $q$-ary linear cyclic codes of length $2n$.

Let $2n=p^{\ell}n_{0}$, where $p$ is the characteristic of field
$\mathbb{F}_{q}$ and $\ell\geq 0$, $n_{0}>0$ are integers such that ${\rm
gcd}(n_{0},p)=1$. Then we have $x^{n_{0}}-1=g_{1}(x)g_{2}(x)\ldots g_{t}(x)$,
where $g_{1}(x)$, $g_{2}(x)$,…,$g_{t}(x)$ are distinct irreducible polynomials
in $\mathbb{F}_{q}[x]$. It means that

$x^{2n}-1=(x^{n_{0}}-1)^{p^{\ell}}=g_{1}(x)^{p^{\ell}}g_{2}(x)^{p^{\ell}}\ldots
g_{t}(x)^{p^{\ell}}.$ (3)

The set of divisors of $x^{2n}-1$ over $\mathbb{F}_{q}$ can be written as

${\rm
Div}_{\mathbb{F}_{q}}(x^{2n}-1)=\\{g_{1}(x)^{s_{1}}g_{2}(x)^{s_{2}}\cdots
g_{t}(x)^{s_{t}}~{}|~{}0\leq s_{1},s_{2},\ldots,s_{t}\leq p^{\ell}\\}.$

Obviously, the polynomial $x^{2n}-1$ has $(p^{\ell}+1)^{t}$ distinct divisors
over $\mathbb{F}_{q}$. Therefore, according to Theorem 3.6, the enumeration of
$q^{2}$-ary conjucyclic codes of length $n$ can be obtained naturally.

######  Theorem 4.1.

Assume that the polynomial $x^{2n}-1$ is factorized as Equation (3), then
there exist $(p^{\ell}+1)^{t}$ distinct $q^{2}$-ary additive conjucyclic codes
of length $n$.

Let $g(x)=g_{1}(x)^{s_{1}}g_{2}(x)^{s_{2}}\cdots g_{t}(x)^{s_{t}}\in{\rm
Div}_{\mathbb{F}_{q}}(x^{2n}-1)$. Suppose that ${\rm Degree}(g_{i}(x))=d_{i}$
for $1\leq i\leq t$, then the dimension of the $q$-ary cyclic code
$\mathscr{D}_{g(x)}$ is

${\rm dim}_{\mathbb{F}_{q}}(\mathscr{D}_{g(x)})=2n-{\rm
deg}(g(x))=2n-\sum_{i=1}^{t}s_{i}d_{i}.$

This means that $|\mathscr{D}_{g(x)}|=q^{2n-{\rm deg}(g(x))}$. Let $g(x)\neq
x^{2n}-1$ and $k={\rm deg}(g(x))=\sum_{i=1}^{t}s_{i}d_{i}.$ We have
$g(x)=g_{0}+g_{1}x+\cdots+g_{k}x^{k}$ and $0\leq k\leq 2n-1$, where
$g_{0},g_{1},\ldots,g_{k}\in\mathbb{F}_{q}$. Define
$h(x)=\frac{x^{2n}-1}{g(x)}=h_{0}+h_{1}x+\cdots+h_{2n-k}x^{2n-k}$ and
$h^{*}(x)=\frac{1}{h_{2n-k}}x^{2n-k}h(\frac{1}{x})$. For simplicity, attached
to polynomial
$f(x)=f_{0}+f_{1}x+\cdots+f_{2n-1}x^{2n-1}\in{\mathbb{F}_{q}}[x]/\langle
x^{2n}-1\rangle$, we denote
$V_{f(x)}=(f_{0},f_{1},\ldots,f_{2n-1})\in\mathbb{F}_{q}^{2n}$ and
$W_{f(x)}=\Psi^{-1}(V_{f(x)})=\Psi^{-1}(f_{0},f_{1},\ldots,f_{n-1})\in\mathbb{F}_{q^{2}}^{n}$
in the rest of the paper.

######  Definition 4.2.

A generator matrix of a $q^{2}$-ary linear code $\mathscr{C}$ is defined as a
matrix whose rows form an $\mathbb{F}_{q^{2}}$-basis for $\mathscr{C}$. If
$\mathscr{C}$ is an additive code over $\mathbb{F}_{q^{2}}$, these rows of its
additive generator matrix form an $\mathbb{F}_{q}$-basis.

Now we produce their additive generator matrices of $q^{2}$-ary additive
conjucyclic codes.

######  Theorem 4.3.

Let $\mathscr{C}$ be an additive conjucyclic code over $\mathbb{F}_{q^{2}}$ of
length $n$ and $\mathscr{D}=\Psi(\mathscr{C})$ be a $q$-ary linear cyclic code
of length $2n$, where $\mathscr{D}=\langle g(x)\rangle$ and $g(x)\in{\rm
Div}_{\mathbb{F}_{q}}(x^{2n}-1)$. Then an additive generator matrix of
$\mathscr{C}$ can be given as follows:

$\small\widehat{G}=\left(\begin{array}[]{c}W_{g(x)}\\\
T\left(W_{g(x)}\right)\\\ \vdots\\\ T^{2n-k-1}\left(W_{g(x)}\right)\\\
\end{array}\right).$

  * Proof.

We first prove that each codeword $\vec{c}\in\mathscr{C}$ can be generated by
the rows of matrix $\widehat{G}$. Since $\mathscr{D}=\langle g(x)\rangle$, by
the theory of cyclic codes [14], its generator matrix is

$\small G=\left(\begin{array}[]{c}V_{g(x)}\\\ \sigma\left(V_{g(x)}\right)\\\
\vdots\\\ \sigma^{2n-k-1}\left(V_{g(x)}\right)\\\ \end{array}\right).$

Hence, there exist elements $k_{0},k_{1},\ldots,k_{2n-k-1}\in\mathbb{F}_{q}$
such that

$\Psi(\vec{c})=k_{0}V_{g(x)}+k_{1}\sigma\left(V_{g(x)}\right)+\cdots+k_{2n-k-1}\sigma^{2n-k-1}\left(V_{g(x)}\right).$

Since $\Psi$ is an $\mathbb{F}_{q}$-linear isomorphism, then

$\vec{c}=k_{0}\Psi^{-1}\left(V_{g(x)}\right)+k_{1}\Psi^{-1}\left(\sigma\left(V_{g(x)}\right)\right)+\cdots+k_{2n-k-1}\Psi^{-1}\left(\sigma^{2n-k-1}\left(V_{g(x)}\right)\right).$

According to Proposition 3.5, we can deduce
$\Psi^{-1}\left(\sigma^{i}\left(V_{g(x)}\right)\right)=T^{i}(W_{g(x)}),$ where
$W_{g(x)}=\Psi^{-1}(V_{g(x)})$ and $0\leq i\leq 2n-k-1$. Therefore,

$\vec{c}=k_{0}W_{g(x)}+k_{1}T(W_{g(x)})+\cdots+k_{2n-k-1}T^{2n-k-1}(W_{g(x)}).$

In addition, from $|\mathscr{C}|=|\mathscr{D}|=q^{2n-k}$, it is obvious that
there are no subsets of the rows of $\widehat{G}$ can produce $\mathscr{C}$.
As a consequence, the rows of matrix $\widehat{G}$ form an
$\mathbb{F}_{q}$-basis. In other words, matrix $\widehat{G}$ is an additive
generator matrix of the conjucyclic code $\mathscr{C}$.  $\square$

In order to provide parity-check matrices of $q^{2}$-ary additive conjucyclic
codes, we need to define the following linear transformation $\tau$:

$\tau(\vec{v})=(v_{0},v_{1},\ldots,v_{2n-1})\left(\begin{array}[]{cc}0&I_{n}\\\
-I_{n}&0\\\
\end{array}\right)=(-v_{n},\ldots,-v_{2n-1},v_{0},\ldots,v_{n-1}),$

where $\vec{v}=(v_{0},v_{1},\ldots,v_{2n-1})\in\mathbb{F}_{q}^{2n}$ and
$I_{n}$ denotes $n\times n$ identity matrix.

######  Lemma 4.4.

Let $\mathscr{D}$ be a $q$-ary linear cyclic code of length $2n$, where
$\mathscr{D}=\langle g(x)\rangle$ and $g(x)\in{\rm
Div}_{\mathbb{F}_{q}}(x^{2n}-1)$. Then a generator matrix of the symplectic
dual code $\mathscr{D}^{\perp_{s}}$ has the following form,

$\small H_{s}=\left(\begin{array}[]{c}\tau\left(V_{h^{*}(x)}\right)\\\
\tau\left(\sigma\left(V_{h^{*}(x)}\right)\right)\\\ \vdots\\\
\tau\left(\sigma^{k-1}\left(V_{h^{*}(x)}\right)\right)\\\ \end{array}\right).$

  * Proof.

By the theory of cyclic codes [14], we know that a generator matrix of the
Euclidean dual code $\mathscr{D}^{\perp_{e}}$ can be given as

$\small H_{e}=\left(\begin{array}[]{c}V_{h^{*}(x)}\\\
\sigma\left(V_{h^{*}(x)}\right)\\\ \vdots\\\
\sigma^{k-1}\left(V_{h^{*}(x)}\right)\\\ \end{array}\right).$

For each codeword $\vec{d}\in\mathscr{D}$ and row vector
$\tau\left(\sigma^{i}\left(V_{h^{*}(x)}\right)\right)$ of $H_{s}$, we have
$\langle
d,\tau\left(\sigma^{i}\left(V_{h^{*}(x)}\right)\right)\rangle_{s}=\langle
d,\sigma^{i}\left(V_{h^{*}(x)}\right)\rangle_{e}=0$ for $0\leq i\leq k-1$.
Hence, the code generated by $H_{s}$ is contained in the code
$\mathscr{D}^{\perp_{s}}$. In addition, from
$H_{s}=H_{e}\left(\begin{array}[]{cc}0&I_{n}\\\ -I_{n}&0\\\
\end{array}\right)$ , we have ${\rm Rank}(H_{s})={\rm Rank}(H_{e})=k$.
Therefore, $H_{s}$ is a generator matrix of $\mathscr{D}^{\perp_{s}}$
obviously.  $\square$

######  Theorem 4.5.

Let $\mathscr{C}$ be a $q^{2}$-ary additive conjucyclic code of length $n$
given by the $q$-ary linear cyclic code $\mathscr{D}=\langle g(x)\rangle$,
where $g(x)\in{\rm Div}_{\mathbb{F}_{q}}(x^{2n}-1)$. Then the additive
generator matrix of $\mathscr{C}^{\perp_{a}}$ can be written as

$\small\widehat{H}_{a}=\left(\begin{array}[]{c}\Psi^{-1}\left(\tau\left(V_{h^{*}(x)}\right)\right)\\\
\Psi^{-1}\left(\tau\left(\sigma\left(V_{h^{*}(x)}\right)\right)\right)\\\
\vdots\\\
\Psi^{-1}\left(\tau\left(\sigma^{k-1}\left(V_{h^{*}(x)}\right)\right)\right)\\\
\end{array}\right).$

  * Proof.

Since $\Psi$ is an $\mathbb{F}_{q}$-linear isomorphism, from Theorem 3.9 and
Lemma 4.4, this result can be obtained directly.  $\square$

Moreover, the additive generator matrix $\widehat{H}_{a}$ for code
$\mathscr{C}^{\perp_{a}}$ is a parity-check matrix of code $\mathscr{C}$. We
have

$\mathscr{C}=\left\\{(c_{0},c_{1},\ldots,c_{2n-1})\in\mathbb{F}_{q^{2}}^{n}~{}|~{}\langle\widehat{H}_{a},(c_{0},c_{1},\ldots,c_{2n-1})^{t}\rangle_{a}=0\right\\},$

where $t$ is the transpose transform and
$\langle\widehat{H}_{a},(c_{0},c_{1},\ldots,c_{2n-1})^{t}\rangle_{a}$ denotes
the matrix multiplication of $\widehat{H}_{a}$ and
$(c_{0},c_{1},\ldots,c_{2n-1})^{t}$ for the alternating inner product.

######  Theorem 4.6.

With the above notations, we let $W_{h^{\epsilon}(x)}=\tau(W_{h^{*}(x)})$. If
${\rm char}(\mathbb{F}_{q^{2}})=2$, then the additive generator matrix of
$\mathscr{C}^{\perp_{a}}$ can be also denoted as

$\small\widehat{H}_{a}=\left(\begin{array}[]{c}W_{h^{\epsilon}(x)}\\\
T\left(W_{h^{\epsilon}(x)}\right)\\\ \vdots\\\
T^{k-1}\left(W_{h^{\epsilon}(x)}\right)\\\ \end{array}\right).$

  * Proof.

If ${\rm char}(\mathbb{F}_{q^{2}})=2$, then for any vector
$\vec{v}=(v_{0},v_{1},\ldots,v_{2n-1})\in\mathbb{F}_{q}^{2n}$, one can check
that

$\tau(\sigma(\vec{v}))=(v_{n-1},v_{n},\ldots,v_{2n-2},v_{2n-1},v_{0},\ldots,v_{n-2})=\sigma(\tau(\vec{v})).$

Hence, for $0\leq i\leq k-1$,
$\tau(\sigma^{i}(\vec{v}))=\sigma(\tau(\sigma^{i-1}(\vec{v})))=\cdots=\sigma^{i}(\tau(\vec{v}))$.
In addition, according to Proposition 3.5, we have
$\Psi^{-1}(\sigma^{i}(\vec{v}))=T^{i}(\Psi^{-1}(\vec{v}))$. Then the additive
generator matrix of $\mathscr{C}^{\perp_{a}}$ can be rewritten as

$\small\widehat{H}_{a}=\left(\begin{array}[]{c}\Psi^{-1}\left(\tau\left(V_{h^{*}(x)}\right)\right)\\\
\Psi^{-1}\left(\tau\left(\sigma\left(V_{h^{*}(x)}\right)\right)\right)\\\
\vdots\\\
\Psi^{-1}\left(\tau\left(\sigma^{k-1}\left(V_{h^{*}(x)}\right)\right)\right)\\\
\end{array}\right)=\left(\begin{array}[]{c}\Psi^{-1}\left(\tau\left(V_{h^{*}(x)}\right)\right)\\\
\Psi^{-1}\left(\sigma\left(\tau\left(V_{h^{*}(x)}\right)\right)\right)\\\
\vdots\\\
\Psi^{-1}\left(\sigma^{k-1}\left(\tau\left(V_{h^{*}(x)}\right)\right)\right)\\\
\end{array}\right)$
$\small=\left(\begin{array}[]{c}\Psi^{-1}\left(V_{h^{\epsilon}(x)}\right)\\\
T\left(\Psi^{-1}\left(V_{h^{\epsilon}(x)}\right)\right)\\\ \vdots\\\
T^{k-1}\left(\Psi^{-1}\left(V_{h^{\epsilon}(x)}\right)\right)\\\
\end{array}\right)=\left(\begin{array}[]{c}W_{h^{\epsilon}(x)}\\\
T\left(W_{h^{\epsilon}(x)}\right)\\\ \vdots\\\
T^{k-1}\left(W_{h^{\epsilon}(x)}\right)\\\
\end{array}\right).~{}~{}~{}~{}~{}~{}~{}$

$\square$

######  Corollary 4.7.

Let $\mathscr{C}$ be a $q^{2}$-ary additive conjucyclic code. If ${\rm
char}(\mathbb{F}_{q^{2}})=2$, then the alternating dual code
$\mathscr{C}^{\perp_{a}}$ is also additive conjucyclic.

  * Proof.

By observing the form of matrix $\widehat{H}_{a}$, the corollary follows
obviously.  $\square$

######  Remark 4.8.

Theorem 9 in [1] has claimed that if $\mathscr{C}$ is a quaternary additive
conjucyclic code, then its trace dual code $\mathscr{C}^{Tr}$ is also a
quaternary additive conjucyclic code, where
$\mathscr{C}^{Tr}=\\{\vec{v}\in\mathbb{F}_{q^{2}}^{n}|{\rm
Tr}\langle\vec{u},\vec{v}\rangle_{e}=0,~{}\forall\vec{u}\in\mathscr{C}\\}$.
Now we can give another explanation. When $q=2$, by Corollary 4.7, we know
that $\mathscr{C}^{\perp_{a}}$ is an additive conjucyclic code. This leads
$\mathscr{C}^{Tr}$ to be also additive conjucyclic based on the fact
$\mathscr{C}^{Tr}=(\mathscr{C}^{\perp_{a}})^{2}$, where
$(\mathscr{C}^{\perp_{a}})^{2}=\\{(c_{0}^{2},c_{1}^{2},\ldots,c_{n-1}^{2})\mid(c_{0},c_{1},\ldots,c_{n-1})\in\mathscr{C}^{\perp_{a}}\\}.$

In [1], Abualrub et al. have constructed an additive conjucyclic code when
$q=2$. In the following, we provide two examples to show the construction
method of additive conjucyclic codes when $q=3$ or $4$.

######  Example 4.9.

Let $q=3$ and $n=11$. The polynomial $x^{22}-1$ in $\mathbb{F}_{3}[x]$ can be
decomposed as

$\begin{split}x^{22}-1=&(1+x)(2+x)(2+2x+x^{2}+2x^{3}+x^{5})(1+2x+2x^{2}+2x^{3}+x^{5})\\\
&(1+2x^{2}+2x^{3}+2x^{4}+x^{5})(2+x^{2}+2x^{3}+x^{4}+x^{5}).\end{split}$

Obviously, there are $2^{6}=64$ factors of $x^{22}-1$, which correspond to
$64$ additive conjucyclic codes over $\mathbb{F}_{9}$ of length $11$. Let
$\mathscr{C}$ be a 9-ary additive conjucyclic code of length $11$ given by

$\begin{split}g(x)=&(1+2x+2x^{2}+2x^{3}+x^{5})(2+x^{2}+2x^{3}+x^{4}+x^{5})\\\
=&2+x+2x^{2}+2x^{3}+x^{4}+2x^{5}+2x^{6}+2x^{7}+x^{8}+x^{9}+x^{10}.\end{split}$

Then
$V_{g(x)}=(2,1,2,2,1,2,2,2,1,1,1,0,0,0,0,0,0,0,0,0,0,0)\in\mathbb{F}_{3}^{22}$
and
$W_{g(x)}=\Psi^{-1}(V_{g(x)})=(\beta^{7},\beta^{3},\beta^{7},\beta^{7},\beta^{3},\beta^{7},\beta^{7},\beta^{7},\beta^{3},\beta^{3},\beta^{3})\in\mathbb{F}_{9}^{11},$
where $\beta$ is a primitive element of $\mathbb{F}_{9}$. According to Theorem
4.3, an additive generator matrix of $\mathscr{C}$ can be given by

$\footnotesize\widehat{G}=\left(\begin{array}[]{ccccc cccc
cc}\beta^{7}&\beta^{3}&\beta^{7}&\beta^{7}&\beta^{3}&\beta^{7}&\beta^{7}&\beta^{7}&\beta^{3}&\beta^{3}&\beta^{3}\\\
\beta&\beta^{7}&\beta^{3}&\beta^{7}&\beta^{7}&\beta^{3}&\beta^{7}&\beta^{7}&\beta^{7}&\beta^{3}&\beta^{3}\\\
\beta&\beta&\beta^{7}&\beta^{3}&\beta^{7}&\beta^{7}&\beta^{3}&\beta^{7}&\beta^{7}&\beta^{7}&\beta^{3}\\\
\beta&\beta&\beta&\beta^{7}&\beta^{3}&\beta^{7}&\beta^{7}&\beta^{3}&\beta^{7}&\beta^{7}&\beta^{7}\\\
\beta^{5}&\beta&\beta&\beta&\beta^{7}&\beta^{3}&\beta^{7}&\beta^{7}&\beta^{3}&\beta^{7}&\beta^{7}\\\
\beta^{5}&\beta^{5}&\beta&\beta&\beta&\beta^{7}&\beta^{3}&\beta^{7}&\beta^{7}&\beta^{3}&\beta^{7}\\\
\beta^{5}&\beta^{5}&\beta^{5}&\beta&\beta&\beta&\beta^{7}&\beta^{3}&\beta^{7}&\beta^{7}&\beta^{3}\\\
\beta&\beta^{5}&\beta^{5}&\beta^{5}&\beta&\beta&\beta&\beta^{7}&\beta^{3}&\beta^{7}&\beta^{7}\\\
\beta^{5}&\beta&\beta^{5}&\beta^{5}&\beta^{5}&\beta&\beta&\beta&\beta^{7}&\beta^{3}&\beta^{7}\\\
\beta^{5}&\beta^{5}&\beta&\beta^{5}&\beta^{5}&\beta^{5}&\beta&\beta&\beta&\beta^{7}&\beta^{3}\\\
\beta&\beta^{5}&\beta^{5}&\beta&\beta^{5}&\beta^{5}&\beta^{5}&\beta&\beta&\beta&\beta^{7}\\\
\beta^{5}&\beta&\beta^{5}&\beta^{5}&\beta&\beta^{5}&\beta^{5}&\beta^{5}&\beta&\beta&\beta\\\
\end{array}\right).$

Hence, we have ${\rm dim}_{\mathbb{F}_{3}}(\mathscr{C})={\rm
dim}_{\mathbb{F}_{3}}(\mathscr{D})=22-10=12$ and
$\mathscr{C}=\\{(a_{1},a_{2},a_{3},a_{4},\\\
a_{5},a_{6},a_{7},a_{8},a_{9},a_{10},a_{11})\widehat{G}~{}|~{}(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{7},a_{8},a_{9},a_{10},a_{11})\in\mathbb{F}_{3}^{11}\\}$.
Using Magma [4], we calculate $w_{h}(\mathscr{C})=5$. Therefore, we can
construct a $(11,3^{12},5)_{9}$ additive conjucyclic code, whose weight
distribution is
$1+264z^{5}+2112z^{6}+11880z^{7}+46200z^{8}+125840z^{9}+199584z^{10}+199584z^{11}$.

On the other hand, since
$h(x)=\frac{x^{22}-1}{g(x)}=1+x+x^{3}+x^{4}+2x^{6}+x^{8}+2x^{9}+2x^{11}+x^{12}$
and $h^{*}(x)=1+2x+2x^{3}+x^{4}+2x^{6}+x^{8}+x^{9}+x^{11}+x^{12}$, then
$\tau(V_{h^{*}(x)})=(2,2,0,0,0,0,0,0,0,0,0,1,2,0,2,1,0,2,0,1,1,0),\ldots,\tau(\sigma^{9}(V_{h^{*}(x)}))\\\
=(0,1,2,0,1,0,2,2,0,2,2,0,0,0,0,0,0,0,0,0,1,2)\in\mathbb{F}_{3}^{22}$ and
$\Psi^{-1}(\tau(V_{h^{*}(x)}))\\\
=(\beta^{6},2,0,\beta^{5},\beta,0,\beta^{5},0,\beta,\beta,0),\ldots,\Psi^{-1}(\tau(\sigma^{9}(V_{h^{*}(x)})))=(0,\beta^{3},\beta^{7},0,\beta^{3},0,\\\
\beta^{7},\beta^{7},0,\beta^{6},2)\in\mathbb{F}_{9}^{11}$. Applying Theorem
4.5, we gain the following additive generator matrix of the code
$\mathscr{C}^{\bot_{a}}$,

$\footnotesize\widehat{H}_{a}=\left(\begin{array}[]{ccccc cccc
cc}\beta^{6}&2&0&\beta^{5}&\beta&0&\beta^{5}&0&\beta&\beta&0\\\
0&\beta^{6}&2&0&\beta^{5}&\beta&0&\beta^{5}&0&\beta&\beta\\\
\beta^{7}&0&\beta^{6}&2&0&\beta^{5}&\beta&0&\beta^{5}&0&\beta\\\
\beta^{7}&\beta^{7}&0&\beta^{6}&2&0&\beta^{5}&\beta&0&\beta^{5}&0\\\
0&\beta^{7}&\beta^{7}&0&\beta^{6}&2&0&\beta^{5}&\beta&0&\beta^{5}\\\
\beta^{3}&0&\beta^{7}&\beta^{7}&0&\beta^{6}&2&0&\beta^{5}&\beta&0\\\
0&\beta^{3}&0&\beta^{7}&\beta^{7}&0&\beta^{6}&2&0&\beta^{5}&\beta\\\
\beta^{7}&0&\beta^{3}&0&\beta^{7}&\beta^{7}&0&\beta^{6}&2&0&\beta^{5}\\\
\beta^{3}&\beta^{7}&0&\beta^{3}&0&\beta^{7}&\beta^{7}&0&\beta^{6}&2&0\\\
0&\beta^{3}&\beta^{7}&0&\beta^{3}&0&\beta^{7}&\beta^{7}&0&\beta^{6}&2\\\
\end{array}\right).$

Then
$\mathscr{C}_{g(x)}^{\bot_{a}}=\\{(b_{1},b_{2},b_{3},b_{4},b_{5},b_{6},b_{7},b_{8},b_{9},b_{10},b_{11})\widehat{H}_{a}~{}|~{}(b_{1},b_{2},b_{3},b_{4},b_{5},b_{6},b_{7},\\\
b_{8},b_{9},b_{10},b_{11})\in\mathbb{F}_{3}^{11}\\}$.

######  Example 4.10.

Now we consider the case $q=4$. Over $\mathbb{F}_{4}$, the factorization of
$x^{22}-1$ is

$\begin{split}(1+x)^{2}(1+\omega^{2}x+x^{2}+x^{3}+\omega
x^{4}+x^{5})^{2}(1+\omega
x+x^{2}+x^{3}+\omega^{2}x^{4}+x^{5})^{2},\end{split}$

where $\omega$ is a primitive element in $\mathbb{F}_{4}$. Then the factors of
$x^{22}-1$ can be described as $(1+x)^{i_{1}}(1+\omega^{2}x+x^{2}+x^{3}+\omega
x^{4}+x^{5})^{i_{2}}(1+\omega x+x^{2}+x^{3}+\omega^{2}x^{4}+x^{5})^{i_{3}}$
for $0\leq i_{1},i_{2},i_{3}\leq 2.$ By Theorem 4.1, there are $3^{3}=27$
additive conjucyclic codes over $\mathbb{F}_{16}$ of length $11$. Let
$\mathscr{C}$ be a 16-ary additive conjucyclic code of length $11$ determined
by

$\begin{split}g(x)&=(1+\omega^{2}x+x^{2}+x^{3}+\omega
x^{4}+x^{5})^{2}=1+\omega
x^{2}+x^{4}+x^{6}+\omega^{2}x^{8}+x^{10}.\end{split}$

So
$V_{g(x)}=(1,0,\omega,0,1,0,1,0,\omega^{2},0,1,0,0,0,0,0,0,0,0,0,0,0)\in\mathbb{F}_{4}^{22}$.
Choose $\beta$ as a primitive element in $\mathbb{F}_{16}$, then

$W_{g(x)}=\Psi^{-1}(V_{g(x)})=(\beta,0,\beta^{6},0,\beta,0,\beta,0,\beta^{11},0,\beta)\in\mathbb{F}_{16}^{11}.$

An additive generator matrix of $\mathscr{C}$ can be written as

$\footnotesize\widehat{G}_{a}=\left(\begin{array}[]{ccccc cccc
cc}\beta&0&\beta^{6}&0&\beta&0&\beta&0&\beta^{11}&0&\beta\\\
\beta^{4}&\beta&0&\beta^{6}&0&\beta&0&\beta&0&\beta^{11}&0\\\
0&\beta^{4}&\beta&0&\beta^{6}&0&\beta&0&\beta&0&\beta^{11}\\\
\beta^{14}&0&\beta^{4}&\beta&0&\beta^{6}&0&\beta&0&\beta&0\\\
0&\beta^{14}&0&\beta^{4}&\beta&0&\beta^{6}&0&\beta&0&\beta\\\
\beta^{4}&0&\beta^{14}&0&\beta^{4}&\beta&0&\beta^{6}&0&\beta&0\\\
0&\beta^{4}&0&\beta^{14}&0&\beta^{4}&\beta&0&\beta^{6}&0&\beta\\\
\beta^{4}&0&\beta^{4}&0&\beta^{14}&0&\beta^{4}&\beta&0&\beta^{6}&0\\\
0&\beta^{4}&0&\beta^{4}&0&\beta^{14}&0&\beta^{4}&\beta&0&\beta^{6}\\\
\beta^{9}&0&\beta^{4}&0&\beta^{4}&0&\beta^{14}&0&\beta^{4}&\beta&0\\\
0&\beta^{9}&0&\beta^{4}&0&\beta^{4}&0&\beta^{14}&0&\beta^{4}&\beta\\\
\beta^{4}&0&\beta^{9}&0&\beta^{4}&0&\beta^{4}&0&\beta^{14}&0&\beta^{4}\\\
\end{array}\right).$

By calculation, the code $\mathscr{C}$ has parameters $(11,4^{12},5)_{16}$
with a weight distribution
$1+825z^{5}+1980z^{6}+61875z^{7}+391875z^{8}+2025375z^{9}+6045600z^{10}+8249685z^{11}$.

On the other side, from $h(x)=\frac{x^{22}-1}{g(x)}=1+\omega x^{2}+\omega
x^{4}+\omega^{2}x^{8}+\omega^{2}x^{10}+x^{12}$, we have
$h^{*}(x)=1+\omega^{2}x^{2}+\omega^{2}x^{4}+\omega x^{8}+\omega x^{10}+x^{12}$
and
$V_{h^{*}(x)}=(1,0,\omega^{2},0,\omega^{2},0,0,0,\omega,0,\omega,0,1,0,0,0,0,0,0,0,0,0)\in\mathbb{F}_{4}^{22}$.
Then
$V_{h^{\epsilon}(x)}=(0,1,0,0,0,0,0,0,0,0,0,1,0,\omega^{2},0,\omega^{2},0,0,0,\omega,0,\omega)\in\mathbb{F}_{4}^{22}$
and $W_{h^{\epsilon}(x)}=(\beta^{4},\\\
\beta,\beta^{14},0,\beta^{14},0,0,0,\beta^{9},0,\beta^{9})\in\mathbb{F}_{16}^{11}$.
According to Theorem 4.6, a parity-check matrix of $\mathscr{C}$ for the
alternating inner product can be given as

$\footnotesize\widehat{H}_{a}=\left(\begin{array}[]{ccccc cccc
cc}\beta^{4}&\beta&\beta^{14}&0&\beta^{14}&0&0&0&\beta^{9}&0&\beta^{9}\\\
\beta^{6}&\beta^{4}&\beta&\beta^{14}&0&\beta^{14}&0&0&0&\beta^{9}&0\\\
0&\beta^{6}&\beta^{4}&\beta&\beta^{14}&0&\beta^{14}&0&0&0&\beta^{9}\\\
\beta^{6}&0&\beta^{6}&\beta^{4}&\beta&\beta^{14}&0&\beta^{14}&0&0&0\\\
0&\beta^{6}&0&\beta^{6}&\beta^{4}&\beta&\beta^{14}&0&\beta^{14}&0&0\\\
0&0&\beta^{6}&0&\beta^{6}&\beta^{4}&\beta&\beta^{14}&0&\beta^{14}&0\\\
0&0&0&\beta^{6}&0&\beta^{6}&\beta^{4}&\beta&\beta^{14}&0&\beta^{14}\\\
\beta^{11}&0&0&0&\beta^{6}&0&\beta^{6}&\beta^{4}&\beta&\beta^{14}&0\\\
0&\beta^{11}&0&0&0&\beta^{6}&0&\beta^{6}&\beta^{4}&\beta&\beta^{14}\\\
\beta^{11}&0&\beta^{11}&0&0&0&\beta^{6}&0&\beta^{6}&\beta^{4}&\beta\\\
\end{array}\right).$

By the matrix $\widehat{H}_{a}$, we can easily see that the alternating dual
code $\mathscr{C}^{\perp_{a}}$ is also an additive conjucyclic code.

Moreover, one can check that the additive conjucyclic code $\mathscr{C}$ is
dual-containing for the alternating inner product. Therefore, according to
Theorem 3.12, a stabilizer code with parameters $[[11,1,5]]_{4}$ can be
constructed, whose weight distribution is
$825z^{5}+1155z^{6}+61050z^{7}+361350+91904925z^{9}+5664615z^{10}+7734720z^{11}$.
Note that this code exceeds the the quantum Gilbert-Varshamov bound [12] and
betters than $[[13,1,3]]_{4}$ and $[[14,0,4]]_{4}$ stabilizer codes available
in [10].

## Acknowledgments

This work is supported by National Natural Science Foundation of China (Nos.
11801564, 11901579).

## References

  * [1] T. Abualrub, Y. Cao, S.T. Dougherty, Algebraic structure of additive conjucyclic codes over $\mathbb{F}_{4}$, Finite Fields Appl. 65 (2020) 101678\.
  * [2] A. Ashikhmin, E. Knill, Nonbinary quantum stabilizer codes, IEEE Trans. Inf. Theory 47 (2001) 3065-3072.
  * [3] N. Benbelkacem, J. Borges, S.T. Dougherty, C. Fernández-Córdoba, On $\mathbb{Z}_{2}\mathbb{Z}_{4}$-additive complementary dual codes and related LCD codes, Finite Fields Appl. 62 (2020) 101622.
  * [4] W. Bosma, J. Cannon, C. Playoust, The MAGMA algebra system I: the user language, J. Symb. Comput. 24 (1997) 235-265.
  * [5] J. Bierbrauer, Cyclic additive codes, J. Algebra 372 (2012) 661-672.
  * [6] A.R. Calderbank, E.M. Rains, P.W. Shor, N.J.A. Sloane, Quantum error correction via GF(4), IEEE Trans. Inf. Theory 44 (1998) 1369-1387.
  * [7] G. Castagnoli, J.L. Massey, P.A. Schoeller, N. Seeman, On repeated-root cyclic codes, IEEE Trans. Inf. Theory 37 (1991) 337-342.
  * [8] P. Delsarte, V. Levenshtein, An algebraic approach to the association schemes of coding theory, Philips Res. Repts. Suppl., 10 (1973).
  * [9] L. Diao, J. Gao, J. Lu, Some results on $\mathbb{Z}_{p}\mathbb{Z}_{p}[v]$-additive cyclic codes, Adv. Math. Commun. doi: 10.3934/amc.2020029 (Online).
  * [10] Y.Edel, Table of quantum twisted codes, Online available at http://
www.mathi.uni-heidelberg.de/$\thicksim$ yves/matritzen/QTBCH/QTBCHIndex.
html. (Accessed 8 July 2020).

  * [11] M.F. Ezerman, S. Ling, P. Solé, Additive asymmetric quantum codes, IEEE Trans. Inf. Theory 57 (8) (2011) 5536-5550.
  * [12] K. Feng, Z. Ma, A finite Gilbert-Varshamov bound for pure stabilizer quantum codes, IEEE Trans. Inf. Theory 50 (2004) 3323-3325.
  * [13] C. Güneri, F. Özbudak, F. ÖzdemIr, On complementary dual additive cyclic codes, Adv. Math. Commun. 11 (2) (2017) 353-357.
  * [14] W.C. Huffman, V. Pless, Fundamentals of error-correcting codes, Cambridge, U.K.: Cambridge University Press, 2003.
  * [15] W.C. Huffman, Additive cyclic codes over $\mathbb{F}_{4}$, Adv. Math. Commun. 1 (4) (2007) 427-459.
  * [16] W.C. Huffman, On the theory of $\mathbb{F}_{q}$-linear $\mathbb{F}_{q^{t}}$ codes, Adv. Math. Commun. 7 (3) (2013) 349-378.
  * [17] J.-L. Kim, Designs in additive codes over GF(4), Designs, Codes Cryptogr., 30 (2003) 187-199.
  * [18] J.-L. Kim, N. Lee, Secret sharing schemes based on additive codes over $\mathbb{F}_{4}$, Appl. Algebra Eng. Commun. Comput. 28 (1) (2017) 79-97.
  * [19] S. Mahmoudi, K. Samei, Additive codes over Galois rings, Finite Fields Appl. 56 (2019) 332-350.
  * [20] M. Shi, R. Wu, P. Solé, Asymptotically good additive cyclic codes exist, IEEE Commun. Lett. 22 (10) (2018) 1980-1983.
  * [21] T. Yao, S. Zhu, X. Kai, Asymptotically good $\mathbb{Z}_{p^{r}}\mathbb{Z}_{p^{s}}$-additive cyclic codes, Finite Fields Appl. 63 (2020) 101633.