# Construction for both self-dual codes and LCD codes

Keita Ishizuka Corresponding author. Research Center for Pure and Applied
Mathematics Graduate School of Information Sciences, Tohoku University, Sendai
980–8579, Japan. email<EMAIL_ADDRESS>Ken Saito Research
Center for Pure and Applied Mathematics Graduate School of Information
Sciences, Tohoku University, Sendai 980–8579, Japan. email:
<EMAIL_ADDRESS>
###### Abstract

From a given $[n,k]$ code $C$, we give a method for constructing many $[n,k]$
codes $C^{\prime}$ such that the hull dimensions of $C$ and $C^{\prime}$ are
identical. This method can be applied to constructions of both self-dual codes
and linear complementary dual codes (LCD codes for short). Using the method,
we construct $661$ new inequivalent extremal doubly even $[56,28,12]$ codes.
Furthermore, constructing LCD codes by the method, we improve some of the
previously known lower bounds on the largest minimum weights of binary LCD
codes of length $n=26,28\leq n\leq 40$.
Keywords. Linear complementary dual code, Self-dual code, Doubly even code,
Hull dimension.
2010 AMS Classification. 94B05

##  1 Introduction

Let $\mathbb{F}_{q}$ be the finite field of order $q$, where $q$ is a prime
power. An $[n,k]$ code $C$ over $\mathbb{F}_{q}$ is said to be a self-dual
code if $C=C^{\perp}$, where $C^{\perp}$ denotes the dual code of $C$. A code
is said to be doubly even if all codewords have weights divisible by four.
Mallows and Sloane [22] proved that the minimum weight $d$ of a binary doubly
even self-dual code of length $n$ is upper bounded by $d\leq 4\lfloor
n/24\rfloor+4$. A binary doubly even self-dual code meeting the bound is
called extremal. An $[n,k]$ code $C$ over $\mathbb{F}_{q}$ is said to be an
LCD code if $C\cap C^{\perp}=\\{\mathbf{0}_{n}\\}$, where $\mathbf{0}_{n}$
denotes the zero vector of length $n$. The concept of LCD codes was invented
by Massey [23]. A binary LCD $[n,k]$ code is said to be optimal if it has the
largest minimum weight among all binary LCD $[n,k]$ codes.

Although the definitions say that self-dual codes and LCD codes are quite
different classes of codes, codes of both classes have similar properties. For
example, it is known that both self-dual codes and LCD codes are characterized
by their generator matrices. Furthermore, self-dual codes are codes with
maximal hull dimension and LCD codes are codes with minimal hull dimension,
where the hull of a code $C$ is defined as $\operatorname{Hull}(C)=C\cap
C^{\perp}$. Recently Harada [16] gave a method for constructing LCD codes
modifying known methods for self-dual codes in [14, Theorem 2.2] and [18,
Theorem 2.2] and constructed $15$ optimal binary LCD $[n,k]$ codes.

In this paper, we give a method for constructing many $[n,k]$ code
$C^{\prime}$ from a given $[n,k]$ code $C$ such that
$\dim(\operatorname{Hull}(C))=\dim(\operatorname{Hull}(C^{\prime}))$. This
method can be applied to constructions of both self-dual codes and LCD codes.
It is shown that the method is a generalized version of [14, Theorem 2.2],
[16, Theorem 3.3] and [18, Theorem 2.2]. As an application, we construct $661$
new inequivalent extremal doubly even $[56,28,12]$ codes. Furthermore,
constructing LCD codes by the method, we improve some of the previously known
lower bounds on the largest minimum weights of binary LCD codes of length
$n=26,28\leq n\leq 40$.

This paper is organized as follows: In Section 2, we recall some basic results
on self-dual codes, LCD codes and hulls of codes. In Section 3, we provide the
construction method. Furthermore, in Section 4, we state basic properties of
the construction method. In Section 5, we construct $661$ new inequivalent
extremal doubly even $[56,28,12]$ from six bordered double circulant doubly
even $[56,28,12]$ codes. In Section 6, we improve some of the largest minimum
weights among all binary LCD $[n,k]$ codes with length $n=26,28\leq n\leq 40$,
which were recently studied by Bouyuklieva [7] and Harada [16]. All
computations in this paper were performed in MAGMA [6].

##  2 Preliminaries

Let $\mathbb{F}_{q}$ be the finite field of order $q$, where $q$ is a prime
power and let $\mathbb{F}_{q}^{n}$ be the vector space of all $n$-tuples over
$\mathbb{F}_{q}$. A $k$-dimensional subspace of $\mathbb{F}_{q}^{n}$ is said
to be an $[n,k]$ code over $\mathbb{F}_{q}$. Especially, codes over
$\mathbb{F}_{2}$ are said to be binary codes. Let $C$ be an $[n,k]$ code over
$\mathbb{F}_{q}$. The parameters $n$, $k$ are said to be the length, the
dimension of $C$ respectively. A vector in $C$ is said to be a codeword. The
weight of $x=(x_{1},x_{2},\dots,x_{n})\in\mathbb{F}_{q}^{n}$ is defined as
$\operatorname{wt}(x)=\\#\\{i\mid x_{i}\neq 0\\}$. The minimum weight of $C$
is defined as $\operatorname{wt}(C)=\min\\{\operatorname{wt}(x)\mid x\in
C,x\neq\mathbf{0}_{n}\\}$. If the minimum weight of $C$ equals to $d$, then
$C$ is said to be an $[n,k,d]$ code over $\mathbb{F}_{q}$. A code $C$ is said
to be an even code if all codewords have even weights. Also, a code is said to
be a doubly even code if all codewords have weights divisible by four. Two
$[n,k]$ codes $C_{1},C_{2}$ over $\mathbb{F}_{q}$ are equivalent if there
exists a monomial matrix $M$ such that $C_{2}=\\{cM\mid c\in C_{1}\\}$. The
equivalence of two codes $C_{1},C_{2}$ is denoted by $C_{1}\simeq C_{2}$. A
generator matrix of a code $C$ is any matrix whose rows form a basis of $C$.

The dual code $C^{\perp}$ of an $[n,k]$ code $C$ over $\mathbb{F}_{q}$ is
defined as $C^{\perp}=\\{x\in\mathbb{F}_{q}^{n}\mid(x,y)=0\text{ for all }y\in
C\\}$, where $(x,y)$ is the standard inner product. If $C\subset C^{\perp}$,
then $C$ is said to be a self-orthogonal code. If $C=C^{\perp}$, then $C$ is
said to be a self-dual code. A binary self-dual code $C$ is doubly even if and
only if $n\equiv 0\pmod{8}$, where $n$ denotes the length of $C$. Mallows and
Sloane [22] proved that the minimum weight $d$ of a binary doubly even self-
dual code of length $n$ is upper bounded by $d\leq 4\lfloor n/24\rfloor+4$. A
binary doubly even self-dual code meeting the bound is called extremal.

######  Lemma 2.1 ([20, Theorem 1.4.8]).

Let $C$ be a binary code. Then the following holds:

  1. (i)

If $C$ is a self-orthogonal code and has a generator matrix each of whose rows
has weight divisible by four, then $C$ is doubly even.

  2. (ii)

If $C$ is doubly even, then $C$ is a self-orthogonal code.

A pure double circulant code has a generator matrix of the form
$\begin{pmatrix}I_{k}&R\\\ \end{pmatrix}$ and a bordered double circulant code
has a generator matrix of the form

$\begin{pmatrix}\begin{matrix}&&&\\\ &I_{k}&&\\\
&&&\end{matrix}\begin{matrix}0&1&\ldots&1\\\ 1&&&\\\ \vdots&&R&\\\
1&&&\end{matrix}\end{pmatrix},$ (1)

where $I_{k}$ denotes the identity matrix of order $k$ and $R$ is a circulant
matrix. These two families of codes are collectively called double circulant
codes. Harada, Gulliver and Kaneta [17] showed that there exist exactly nine
inequivalent extremal double circulant doubly even $[56,28,12]$ codes and all
of them are bordered double circulant codes. In Section 5, we construct
extremal doubly even self-dual $[56,28,12]$ codes from six inequivalent
extremal double circulant doubly even $[56,28,12]$ codes
$D11,C_{56,1},\dots,C_{56,5}$. Generator matrices of
$D11,C_{56,1},\ldots,C_{56,5}$ are of the form (1) with first rows

$\displaystyle(000101011011111000111111111),(000000000000110010101111011),$
$\displaystyle(000000001011011111110010111),(000000010011100111101110111),$
$\displaystyle(000000011011001001111101111),(000000101001111101011101011),$

respectively.

An $[n,k]$ code $C$ over $\mathbb{F}_{q}$ is said to be an LCD code if $C\cap
C^{\perp}=\\{\mathbf{0}_{n}\\}$. The concept of LCD codes was invented by
Massey [23]. LCD codes have been applied in data storage, communication
systems and cryptography. For example, it is known that binary LCD codes can
be used against side-channel attacks and fault injection attacks [9]. A binary
LCD $[n,k]$ code is said to be optimal if it has the largest minimum weight
among all binary LCD $[n,k]$ codes. Massey [23] gave the following
characterization of LCD codes.

######  Theorem 2.2 (Massey [23]).

Let $C$ be an $[n,k]$ code over $\mathbb{F}_{q}$ and let $G$ be a generator
matrix of $C$. Then $C$ is an LCD code if and only if the $k\times k$ matrix
$GG^{T}$ is nonsingular.

The hull of a code $C$ is defined as $\operatorname{Hull}(C)=C\cap C^{\perp}$.
By definition, it follows that self-dual codes are codes with maximal hull
dimension and LCD codes are codes with minimal hull dimension.

######  Lemma 2.3 ([13, Proposition 3.1]).

Let $C$ be an $[n,k]$ code over $\mathbb{F}_{q}$ with generator matrix $G$.
Then

$\operatorname{rank}(GG^{T})=k-\dim(\operatorname{Hull}(C)).$

##  3 Construction method

Let $C$ be an $[n,k]$ code over $\mathbb{F}_{q}$ with generator matrix
$\begin{pmatrix}I_{k}&A\\\ \end{pmatrix}$ and let
$x,y\in\mathbb{F}_{q}^{n-k}$. We denote by $r_{i}$ the $i$-th row of $A$.
Define an $n\times(n-k)$ matrix $A(x,y)$, where the $i$-th row
$r^{\prime}_{i}$ is defined as follows:

$r^{\prime}_{i}=r_{i}+(r_{i},y)x-(r_{i},x)y.$

We denote by $C(A(x,y))$ the code with generator matrix
$\begin{pmatrix}I_{k}&A(x,y)\\\ \end{pmatrix}$.

######  Remark 3.1.

With the above notation, suppose that $x=\mathbf{0}_{n-k}$ or
$y=\mathbf{0}_{n-k}$. Then it holds that $A(x,y)=A$. Hereafter, we assume that
$x\neq\mathbf{0}_{n-k}$ and $y\neq\mathbf{0}_{n-k}$.

######  Theorem 3.2.

Let $C$ be an $[n,k]$ code over $\mathbb{F}_{q}$ with generator matrix
$G=\begin{pmatrix}I_{k}&A\\\ \end{pmatrix}$ and let
$x,y\in\mathbb{F}_{q}^{n-k}$. Suppose that $(x,x)=(y,y)=(x,y)=0$. Then
$\dim(\operatorname{Hull}(C(A(x,y))))=\dim(\operatorname{Hull}(C))$.

###### Proof.

We denote by $r_{i},r^{\prime}_{i}$ the $i$-th rows of $A,A(x,y)$
respectively. It holds that

$\displaystyle(r_{i}^{\prime},r_{j}^{\prime})$
$\displaystyle=(r_{i}+(r_{i},y)x-(r_{i},x)y,\ r_{j}+(r_{j},y)x-(r_{j},x)y)$
$\displaystyle=(r_{i},r_{j})+(r_{j},y)(r_{i},x)-(r_{j},x)(r_{i},y)+(r_{i},y)(x,r_{j})-(r_{i},x)(y,r_{j})$
$\displaystyle=(r_{i},r_{j}).$

Therefore it follows that

$\begin{split}\begin{pmatrix}I_{k}&A(x,y)\\\
\end{pmatrix}\begin{pmatrix}I_{k}&A(x,y)\\\
\end{pmatrix}^{T}&=I_{k}+A(x,y)A(x,y)^{T}\\\ &=I_{k}+AA^{T}\\\
&=\begin{pmatrix}I_{k}&A\\\ \end{pmatrix}\begin{pmatrix}I_{k}&A\\\
\end{pmatrix}^{T}.\end{split}$

By Lemma 2.3, the result follows. ∎

######  Corollary 3.3.

Let $C$ be an $[n,k]$ code over $\mathbb{F}_{q}$ with generator matrix
$G=\begin{pmatrix}I_{k}&A\\\ \end{pmatrix}$ and let
$x,y\in\mathbb{F}_{q}^{n-k}$. Suppose that $(x,x)=(y,y)=(x,y)=0$. Then
$C(A(x,y))$ is a self-orthogonal code if and only if $C$ is a self-orthogonal
code.

###### Proof.

It holds that $C$ is a self-orthogonal code if and only if
$\dim(\operatorname{Hull}(C))=k$. The result follows from Theorem 3.2. ∎

######  Remark 3.4.

With the notation of Corollary 3.3, suppose that $C$ is a binary self-dual
code with length $n\equiv 0\pmod{4}$. Then it follows that
$(x,x)=(x,\mathbf{1}_{n-k})=(\mathbf{1}_{n-k},\mathbf{1}_{n-k})=0$ if $x$ has
an even weight. Let $r_{i},r^{\prime}_{i}$ be the $i$-th rows of
$A,A(x,\mathbf{1}_{n-k})$ respectively. Since $C$ is an even code,
$\operatorname{wt}(r_{i})\equiv 1\pmod{2}$ for all $1\leq i\leq n$. Therefore
we obtain the following:

$\begin{split}r_{i}^{\prime}&=r_{i}+(r_{i},\mathbf{1}_{n-k})x+(r_{i},x)\mathbf{1}_{n-k}\\\
&=r_{i}+x+(r_{i},x)\mathbf{1}_{n-k},\end{split}$

which shows that $A(x,\mathbf{1}_{n-k})$ is identical to
$B^{\prime}_{\Gamma^{\prime}}$, the first case of [14, Theorem 2.2]. Therefore
Corollary 3.3 is a generalized version of the first case of [14, Theorem 2.2].

######  Corollary 3.5.

Let $C$ be an $[n,k]$ code over $\mathbb{F}_{q}$ with generator matrix
$G=\begin{pmatrix}I_{k}&A\\\ \end{pmatrix}$ and let
$x,y\in\mathbb{F}_{q}^{n-k}$. Suppose that $(x,x)=(y,y)=(x,y)=0$. Then
$C(A(x,y))$ is an LCD code if and only if $C$ is an LCD code.

###### Proof.

It holds that $C$ is an LCD code if and only if
$\dim(\operatorname{Hull}(C))=0$. The result follows from Theorem 3.2. ∎

######  Remark 3.6.

With the notation of Corollary 3.5, suppose that $C$ is a binary even LCD code
and $n-k$ is even. Then it follows that
$(x,x)=(x,\mathbf{1}_{n-k})=(\mathbf{1}_{n-k},\mathbf{1}_{n-k})=0$ if $x$ has
an even weight. Let $r_{i},r^{\prime}_{i}$ be the $i$-th rows of
$A,A(x,\mathbf{1}_{n-k})$ respectively. Since $C$ is an even code,
$\operatorname{wt}(r_{i})\equiv 1\pmod{2}$ for all $1\leq i\leq n$. Therefore
we obtain the following:

$\begin{split}r_{i}^{\prime}&=r_{i}+(r_{i},\mathbf{1}_{n-k})x+(r_{i},x)\mathbf{1}_{n-k}\\\
&=r_{i}+x+(r_{i},x)\mathbf{1}_{n-k},\end{split}$

which shows that $A(x,\mathbf{1}_{n-k})$ is identical to $A(x)$ in [16,
Theorem 3.3]. Therefore Corollary 3.5 is a generalized version of [16, Theorem
3.3].

######  Lemma 3.7 ([20, Theorem 1.4.3]).

Let
$x=(x_{1},x_{2},\ldots,x_{n}),y=(y_{1},y_{2},\ldots,y_{n})\in\mathbb{F}_{2}^{n}$.
Then
$\operatorname{wt}(x+y)=\operatorname{wt}(x)+\operatorname{wt}(y)-2\operatorname{wt}(x*y)$,
where $x*y=(x_{1}y_{1},x_{2}y_{2},\ldots,x_{n}y_{n})$.

######  Lemma 3.8.

Let $C$ be a binary $[n,k]$ code with generator matrix
$\begin{pmatrix}I_{k}&A\\\ \end{pmatrix}$ and let
$x,y\in\mathbb{F}_{q}^{n-k}$. Suppose that
$\operatorname{wt}(x)\equiv\operatorname{wt}(y)\equiv 0\pmod{4}$ and
$(x,y)=0$. Then
$\operatorname{wt}(r_{i}^{\prime})\equiv\operatorname{wt}(r_{i})\pmod{4}$,
where $r_{i},r^{\prime}_{i}$ denote the $i$-th rows of $A,A(x,y)$
respectively.

###### Proof.

It holds that

$\displaystyle\operatorname{wt}(r_{i}*((r_{i},y)x-(r_{i},x)y))$
$\displaystyle=(r_{i},\ (r_{i},y)x-(r_{i},x)y)$
$\displaystyle=(r_{i},y)(r_{i},x)-(r_{i},x)(r_{i},y)$ $\displaystyle=0,$
$\displaystyle\operatorname{wt}((r_{i},y)x*(r_{i},x)y)$
$\displaystyle=((r_{i},y)x,(r_{i},x)y)$
$\displaystyle=(r_{i},y)(r_{i},x)(x,y)$ $\displaystyle\equiv 0\pmod{2},$

where we regard $r_{i},x,y$ as vectors in $\mathbb{Z}^{n-k}$. Therefore, by
Lemma 3.7, it follows that

$\displaystyle\operatorname{wt}(r_{i}^{\prime})$
$\displaystyle=\operatorname{wt}(r_{i}+(r_{i},y)x-(r_{i},x)y)$
$\displaystyle=\operatorname{wt}(r_{i})+\operatorname{wt}((r_{i},y)x-(r_{i},x)y)-2\operatorname{wt}(r_{i}*((r_{i},y)x-(r_{i},x)y))$
$\displaystyle=\operatorname{wt}(r_{i})+\operatorname{wt}((r_{i},y)x)+\operatorname{wt}((r_{i},x)y)-2\operatorname{wt}((r_{i},y)x*(r_{i},x)y)$
$\displaystyle\equiv\operatorname{wt}(r_{i})\pmod{4}.$

This completes the proof. ∎

######  Theorem 3.9.

Let $C$ be a binary $[n,k]$ code with generator matrix
$\begin{pmatrix}I_{k}&A\\\ \end{pmatrix}$ and let
$x,y\in\mathbb{F}_{2}^{n-k}$. Suppose that
$\operatorname{wt}(x)\equiv\operatorname{wt}(y)\equiv 0\pmod{4}$ and
$(x,y)=0$. Then $C(A(x,y))$ is a doubly even code if and only if $C$ is a
doubly even code.

###### Proof.

We denote by $r_{i},r^{\prime}_{i}$ the $i$-th rows of $A,A(x,y)$
respectively. Suppose that $C$ is a doubly even code. Then, by Corollary 3.3
and the second part of Lemma 2.1, $C(A(x,y))$ is a self-orthogonal code.
Furthermore, by Lemma 3.8,
$\operatorname{wt}(r^{\prime}_{i})\equiv\operatorname{wt}(r_{i})\equiv
3\pmod{4}$ follows for all $1\leq i\leq n$. Therefore it holds that
$C(A(x,y))$ is a doubly even code by the first part of Lemma 2.1 By the same
argument, the converse holds. ∎

######  Remark 3.10.

With the notation of Theorem 3.9, suppose that $C$ is a binary doubly even
self-dual code. Then it follows that
$\operatorname{wt}(x)\equiv\operatorname{wt}(\mathbf{1}_{n-k})\equiv
0\pmod{4}$ and $(x,\mathbf{1}_{n-k})=0$ if $x$ has a weight divisible by four.
Let $r_{i},r^{\prime}_{i}$ be the $i$-th rows of $A,A(x,\mathbf{1}_{n-k})$
respectively. Since $C$ is doubly even, $\operatorname{wt}(r_{i})\equiv
3\pmod{4}$ for all $1\leq i\leq n$. Therefore we obtain the following:

$\begin{split}r_{i}^{\prime}&=r_{i}+(r_{i},\mathbf{1}_{n-k})x+(r_{i},x)\mathbf{1}_{n-k}\\\
&=r_{i}+x+(r_{i},x)\mathbf{1}_{n-k},\end{split}$

which shows that $A(x,\mathbf{1}_{n-k})$ is identical to $A^{\prime}_{\Gamma}$
in [18, Theorem 2.2], where $m$ is even. Therefore Theorem 3.9 is a
generalized version of [18, Theorem 2.2], where $m$ is even.

##  4 Basic properties

In this section, we state basic properties of the construction method given in
Section 3. Let $x,y\in\mathbb{F}_{q}^{m}$. Define the following $m\times m$
matrix $M(x,y)$:

$M(x,y)=I_{m}+y^{T}x-x^{T}y.$

######  Lemma 4.1.

Let $C$ be an $[n,k]$ code over $\mathbb{F}_{q}$ with generator matrix
$\begin{pmatrix}I_{k}&A\\\ \end{pmatrix}$ and let
$x,y\in\mathbb{F}_{q}^{n-k}$. Then it holds that

$A(x,y)=AM(x,y).$

###### Proof.

For a matrix $M$, we denote by $M_{i,j}$ the $(i,j)$-entry of $M$. By
definition it follows that

$(AM(x,y))_{i,j}=\sum_{l=1}^{n-k}A_{i,l}M(x,y)_{l,j}.$

On the other hand, it holds that

$\displaystyle r^{\prime}_{i}$ $\displaystyle=r_{i}+(r_{i},y)x-(r_{i},x)y$
$\displaystyle=r_{i}(I_{n-k}+y^{T}x-x^{T}y)$
$\displaystyle=(A_{i,1},A_{i,2},\ldots,A_{i,n})M(x,y)$
$\displaystyle=(\sum_{l=1}^{n-k}A_{i,l}M(x,y)_{l,1},\sum_{l=1}^{n-k}A_{i,l}M(x,y)_{l,2},\dots,\sum_{l=1}^{n-k}A_{i,l}M(x,y)_{l,n}),$

where $r_{i},r^{\prime}_{i}$ denote the $i$-th rows of $A,A(x,y)$
respectively. Therefore it holds that

$A(x,y)_{i,j}=\sum_{l=1}^{n-k}A_{i,l}M(x,y)_{l,j}.$

This completes the proof. ∎

######  Theorem 4.2.

Let $C$ be an $[n,k]$ code over $\mathbb{F}_{q}$ with generator matrix
$\begin{pmatrix}I_{k}&A\\\ \end{pmatrix}$ and let
$x,y\in\mathbb{F}_{q}^{n-k}$. Suppose that $(x,x)=(y,y)=(x,y)=0$. Then the
following holds:

$\displaystyle C(A(x,y))=C(A(-x,-y)),$ (2) $\displaystyle
C(A(y,x))=C(A(x,-y))=C(A(-x,y)).$ (3)

###### Proof.

For (2), it holds that

$\displaystyle M(-x,-y)$ $\displaystyle=I_{m}+(-y)^{T}(-x)-(-x)^{T}(-y)$
$\displaystyle=I_{m}+y^{T}x-x^{T}y$ $\displaystyle=M(x,y).$

For (3), the following holds:

$\displaystyle M(x,-y)$ $\displaystyle=I_{m}-y^{T}x+x^{T}y$
$\displaystyle=I_{m}+x^{T}y-y^{T}x$ $\displaystyle=M(y,x),$ $\displaystyle
M(-x,y)$ $\displaystyle=I_{m}-y^{T}x+x^{T}y$
$\displaystyle=I_{m}+x^{T}y-y^{T}x$ $\displaystyle=M(y,x).$

This completes the proof. ∎

In the following sections, we apply the construction method only to binary
codes. However, for codes over $\mathbb{F}_{q}\ (q\geq 3)$, Theorem 4.2
reduces computations.

##  5 Extremal binary doubly even $[56,28,12]$ codes

In this section we are concerned only with binary codes. Therefore we omit the
term “binary”. Bhargava, Young and Bhargava [5] constructed an extremal doubly
even $[56,28,12]$ code. Yorgov [26] proved that there exist exactly $16$
inequivalent extremal doubly even $[56,28,12]$ codes with automorphisms of
order $13$. As stated in [15], one of the $16$ codes in [26] is equivalent to
the code in [5]. Bussemarker and Tonchev [8] constructed $6$ extremal doubly
even $[56,28,12]$ codes. As stated in [15], the first code among the six codes
in [8] is equivalent to the code in [5]. Moreover, Kimura [21] showed that
$5$th and $6$th codes in [8] are equivalent. Harada [14] constructed $137$
inequivalent extremal doubly even $[56,28,12]$ codes. Harada, Gulliver and
Kaneta [17] showed that there exist exactly nine inequivalent extremal double
circulant doubly even $[56,28,12]$ codes. Harada [15] constructed $1122$
inequivalent extremal doubly even $[56,28,12]$ codes. This result is a
generalization of [14] and for any code $C$ in [14] there exists a code
$C^{\prime}$ in [15] such that $C\simeq C^{\prime}$. Yankov and Russeva [25]
proved that there exist exactly $4202$ inequivalent extremal doubly even
$[56,28,12]$ codes having automorphisms of order $7$. As stated in [25], one
of the $4202$ codes in [25] is equivalent to a code in [14]. Yankov and Lee
[24] proved that there exist exactly $3763$ inequivalent extremal $[56,28,12]$
codes having automorphisms of order $5$. Therefore the number of previously
known extremal doubly even $[56,28,12]$ codes is $9115$, as stated in [24,
Proposition 7].

In [15], Harada applied [18, Theorem 2.2] to six inequivalent extremal double
circulant doubly even $[56,28,12]$ codes $D11,C_{56,1},\ldots,C_{56,5}$. As
stated earlier, Theorem 3.9 is a generalized version of [18, Theorem 2.2]. In
this section, we apply Theorem 3.9 to $D11,C_{56,1},\dots,C_{56,5}$ and
construct $661$ new inequivalent extremal doubly even $[56,28,12]$ codes. This
illustrates the effectiveness of Theorem 3.9.

In order to illustrate our method, we consider the code $D11$ as an example.
Let $y_{i}$ denote the vector of length $28$ such that
$y_{i}=(0,\dots,0,1,\dots,1),\operatorname{wt}(y_{i})=i$. Applying Theorem 3.9
to $y_{4}$ and all $x\in\mathbb{F}_{2}^{28}$ such that $x\neq\mathbf{0}_{28}$
and $(x,x)=(x,y_{4})=0$, we constructed $45$ inequivalent extremal doubly even
$[56,28,12]$ codes. Furthermore, applying Theorem 3.9 to $y_{i}\
(i=8,12,16,20,24)$ and all $x\in\mathbb{F}_{2}^{28}$ such that
$x\neq\mathbf{0}_{28}$ and $(x,x)=(x,y_{i})=0$, we constructed
$45,19,15,2,33,4$ inequivalent extremal doubly even $[56,28,12]$ codes
respectively.

By the following method, we verified that the above codes are all
inequivalent. For an extremal doubly even $[56,28,12]$ code $C$, we define
$M=(m_{i,j})$ to be an $8196\times 56$ matrix whose rows composed of codewords
of $C$ with weight $12$. Furthermore, for a positive integer $t$, we define

$N_{t}=\\#\left\\{\\{j_{1},j_{2},j_{3},j_{4}\\}\in{\displaystyle{56\choose
4}}\mid\sum_{i=1}^{8196}m_{i,j_{1}}m_{i,j_{2}}m_{i,j_{3}}m_{i,j_{4}}=t\right\\}.$

Harada [15] showed that two extremal doubly even $[56,28,12]$ codes
$C_{1},C_{2}$ are inequivalent if the sequences $(N_{1},N_{2},\dots)$
constructed from $C_{1},C_{2}$ are distinct. According to this result, we
compared the sequence $(N_{1},N_{2},\dots,N_{56})$ for the classification.
Consequently we found no pair of codes whose sequences are identical.
Therefore we verified that the number of inequivalent codes constructed from
$D11$ is $118$. By the same method, we constructed inequivalent extremal
doubly even $[56,28,12]$ codes from $C_{56,i}\ (i=1,2,\dots,5)$. In Table 1 we
show the number of inequivalent codes constructed by this method. We denote
the inequivalent codes constructed from $D11,C_{56,1},\dots,C_{56,5}$ by
$D_{i}\ (i=1,2,\dots,118),E_{i}\ (i=1,2,\dots,56),F_{i}\
(i=1,2,\dots,105),G_{i}\ (i=1,2,\dots,59),H_{i}\ (i=1,2,\dots,212),K_{i}\
(i=1,2,\dots,115)$ respectively. The $x,y$ in Corollary 3.9 for all codes we
constructed can be obtained electronically from
https://www.math.is.tohoku.ac.jp/~mharada/Ishizuka/56.txt.

Table 1: Inequivalent extremal doubly even $[56,28,12]$ codes | $y_{4}$ | $y_{8}$ | $y_{12}$ | $y_{16}$ | $y_{20}$ | $y_{24}$ | total
---|---|---|---|---|---|---|---
$D11$ | $45$ | $19$ | $15$ | $2$ | $33$ | $4$ | $118$
$C_{56,1}$ | $16$ | $3$ | $1$ | $0$ | $10$ | $26$ | $56$
$C_{56,2}$ | $34$ | $27$ | $26$ | $1$ | $3$ | $14$ | $105$
$C_{56,3}$ | $10$ | $0$ | $23$ | $2$ | $0$ | $24$ | $59$
$C_{56,4}$ | $10$ | $109$ | $17$ | $58$ | $2$ | $16$ | $212$
$C_{56,5}$ | $17$ | $53$ | $25$ | $11$ | $5$ | $4$ | $115$

Comparing sequences $(N_{1},N_{2},\dots,N_{56})$, we found that there exist
four pairs of codes
$(D_{115},K_{112}),(D_{116},K_{113}),(D_{117},K_{114}),(D_{118},K_{115})$
whose sequences are identical. By the Magma function IsIsomorphic, we verified
that two codes of all the four pairs are equivalent. Therefore the number of
inequivlent extremal doubly even $[56,28,12]$ codes constructed as above is
$661$.

Finally, we verified that the $661$ codes are inequivalent to any of the
previously known extremal doubly even $[56,28,12]$ codes as follows: By the
Magma function AutomorphismGroup, we verified that the $661$ codes have
automorphism groups of order $1$. Consequently it follows that the $661$ codes
are inequivalent to any of the codes in [24], [25], [26]. Furthermore we
verified that all the codes except $D_{118}$ have sequences
$(N_{1},N_{2},\ldots,N_{56})$ different from that of any code in [5], [8],
[15], [17], [26]. The sequence $(N_{1},N_{2},\dots,N_{56})$ of $D_{118}$ is
identical to that of the $25$th code constructed from $C_{56,2}$ in [15].
However we verified by the Magma function IsIsomorphic that the two codes are
inequivalent. Consequently it follows that the $661$ codes are inequivalent to
any of the codes in [5], [8], [15], [17], [26]. As stated in the beginning of
this section, the number of the previously known inequivalent doubly even
$[56,28,12]$ codes is $9115$. Therefore we have Proposition 5.1.

######  Proposition 5.1.

There exist at least $9776$ inequivalent extremal doubly even $[56,28,12]$
codes.

##  6 Optimal binary LCD codes of length $n=26,28\leq n\leq 40$

In this section we are concerned only with binary codes. Therefore we omit the
term “binary”. Let $d_{LCD}(n,k)$ denote the largest minimum weight among all
LCD $[n,k]$ code. Galvez, Kim, Lee, Roe and Won [11], Harada and Saito [19],
Araya and Harada [1] determined the exact value of $d_{LCD}(n,k)$ for $n\leq
12,13\leq n\leq 16,17\leq n\leq 24$ respectively. Bouyuklieva [7] determined
the exact value of $d_{LCD}(n,k)$ for $n=25,27$ and gave $d_{LCD}(n,k)$ for
$n=26,28\leq n\leq 40$. Galvez, Kim, Lee, Roe and Won [11], Harada and Saito
[19], Araya and Harada [1], Araya, Harada and Saito [3] determined the exact
value of $d_{LCD}(n,k)$ for $k=2,3,4,5$ respectively. Also, Dougherty, Kim,
Ozkaya, Sok and Solé [10], Araya and Harada [2], Araya, Harada and Saito [4]
determined the exact value of $d_{LCD}(n,k)$ for $k=n-1$,
$k\in\\{n-2,n-3,n-4\\}$, $k=n-5$ respectively. For all $n=26,28\leq n\leq 40$,
Bouyuklieva [7] determined the exact value of $d_{LCD}(n,k)$ for $5\leq k\leq
8$.

Recently Harada [16] constructed $15$ optimal LCD codes by [16, Theorem 3.3].
As stated earlier, Corollary 3.5 is a generalized version of [16, Theorem
3.3]. In this section, we apply Corollary 3.5 in order to improve some of the
previously known lower bounds on $d_{LCD}(n,k)$ for $n=26,28\leq n\leq 40$ and
$9\leq k\leq n-6$. For a code $C$, we denote by $C^{T},C_{T}$ the punctured,
the shortened codes of $C$ on a set of coordinates $T$ respectively. In this
section, shortened codes, punctured codes were constructed by the Magma
functions ShortenCode, PunctureCode respectively.

In order to obtain lower bounds, we use the following method: First, by the
Magma function BestKnownLinearCode, we obtained a $[49,32,7]$ code
$C_{49,32,7}$, a $[42,14,13]$ code $C_{42,14,13}$ and a $[51,28,9]$ code
$C_{51,28,9}$. Generator matrices of these codes can be obtained
electronically from
https://www.math.is.tohoku.ac.jp/~mharada/Ishizuka/generator.txt. Then we
verified that $((C_{49,32,7})_{S_{1}})^{P_{1}}$,
$((C_{42,14,13})_{S_{2}})^{P_{2}}$, $((C_{51,28,9})_{S_{3}})^{P_{3}}$ are an
LCD $[37,22,5]$ code, an LCD $[38,13,11]$ code, an LCD $[40,22,6]$ code
respectively, where the set of coordinates $P_{i},\ S_{i}\ (i=1,2,3)$ are
given in Table 2. Define $A_{37,22,5}$, $A_{38,13,10}$, $A_{40,22,6}$ as in
Figure 1. Then $\begin{pmatrix}I_{22}&A_{37,22,5}\\\ \end{pmatrix}$,
$\begin{pmatrix}I_{13}&A_{38,13,10}\\\ \end{pmatrix}$,
$\begin{pmatrix}I_{22}&A_{40,22,6}\\\ \end{pmatrix}$ are generator matrices of
the LCD $[37,22,5]$, the LCD $[38,13,10]$, the LCD $[40,22,6]$ codes
respectively. Applying Corollary 3.5 to $\begin{pmatrix}I_{22}&A_{37,22,5}\\\
\end{pmatrix}$, $\begin{pmatrix}I_{13}&A_{38,13,10}\\\ \end{pmatrix}$,
$\begin{pmatrix}I_{22}&A_{40,22,6}\\\ \end{pmatrix}$, we found an LCD
$[37,22,6]$ code $C_{37,22,6}$, an LCD $[38,13,11]$ code $C_{38,13,11}$, an
LCD $[40,22,7]$ code $C_{40,22,7}$ respectively. The vectors $x,y$ in
Corollary 3.5 are listed in Table 3. Therefore we obtain Proposition 6.1.

######  Proposition 6.1.

  1. (i)

There exists an LCD $[37,22,6]$ code.

  2. (ii)

There exists an LCD $[38,13,11]$ code.

  3. (iii)

There exists an LCD $[40,22,7]$ code.

Table 2: $P_{i},S_{i}$ for $i=1,2,3$ $i$ | $P_{i}$ | $S_{i}$
---|---|---
$1$ | $\\{2,5\\}$ | $\\{1,2,3,4,5,6,7,8,9,11\\}$
$2$ | $\\{1,2,4\\}$ | $\\{3\\}$
$3$ | $\\{1,4,5,6,7\\}$ | $\\{1,2,3,4,5,7\\}$
Table 3: $C_{n,k,d}$ with $x,y$ $C_{n,k,d}$ | $x$ | $y$
---|---|---
$C_{37,22,6}$ | $(010110011011111)$ | $(110010110000001)$
$C_{38,13,11}$ | $(0010011100110001011000100)$ | $(1110100001110101110001101)$
$C_{40,22,7}$ | $(101111011011010011)$ | $(001011100011100001)$

From the previously known results of $d_{LCD}(n,k)$ described in the beginning
of this section, we are concerned only with $d_{LCD}(n,k)$ for $n=26,28\leq
n\leq 40$ and $9\leq k\leq n-6$. Let $d_{K}(n,k)$ denote the largest minimum
weight among currently known $[n,k]$ codes. By the Magma function
BestKnownLinearCode, one can construct an $[n,k,d_{K}(n,k)]$ code for all
$n=26,28\leq n\leq 40$ and $9\leq k\leq n-6$. In addition, by considering
shortened codes and punctured codes of $[n,k,d_{K}(n,k)]$ codes, we found LCD
$[n,k,d_{K}(n,k)]$ codes for

$\begin{split}(n,k,d_{K}(n,k))=&(29,11,9),(30,12,9),(31,11,10),(31,12,10),(31,13,9),\\\
&(31,21,5),(32,12,10),(32,22,5),(33,23,5),(34,9,13),\\\
&(34,13,10),(34,14,10),(35,22,6),(35,24,5),(36,15,10),\\\
&(36,16,10),(36,22,6),(36,24,6),(36,25,5),(37,23,6),\\\
&(37,24,6),(37,26,5),(38,24,6),(38,25,6),(38,26,6),\\\
&(38,27,5),(39,18,10),(39,25,6),(39,26,6),(39,28,5),\\\
&(40,26,6),(40,28,6),(40,29,5).\end{split}$ (4)

Consequently we obtain Proposition 6.2.

######  Proposition 6.2.

There exists an optimal LCD $[n,k,d]$ code for $(n,k,d)$ listed in (4).

By a method similar to that given in the above, we found LCD
$[n,k,d_{K}(n,k)-1]$ codes and LCD $[n,k,d_{K}(n,k)-2]$ codes for

$\begin{split}(n,k,d_{K}(n,k)-1)=&(30,11,9),(31,15,7),(32,13,9),(32,15,7),(32,16,7),\\\
&(32,21,5),(33,14,9),(33,15,8),(33,16,7),(33,21,5),\\\
&(34,15,9),(34,16,8),(34,17,7),(34,23,5),(35,9,13),\\\
&(35,16,9),(35,17,7),(35,18,7),(35,21,5),(35,23,5),\\\
&(36,17,8),(36,18,7),(36,19,7),(36,21,6),(37,17,9),\\\
&(37,18,8),(37,19,7),(37,20,7),(37,25,5),(38,10,13),\\\
&(38,17,9),(38,18,9),(38,19,8),(38,20,7),(38,21,7),\\\
&(38,23,6),(39,11,13),(39,14,11),(39,17,10),(39,19,9),\\\
&(39,20,8),(39,21,7),(39,22,7),(39,24,6),(39,27,5),\\\
&(39,33,2),(40,9,15),(40,12,13),(40,18,10),(40,19,9),\\\
&(40,20,9),(40,23,7),(40,25,6),(40,27,5),(40,33,3),\\\ &(40,34,2).\end{split}$
(5)
$\begin{split}(n,k,d_{K}(n,k)-2)=&(37,21,6),(38,22,6),(39,13,11),(39,23,6),(40,11,13),\\\
&(40,13,12),(40,14,11),(40,17,10),(40,21,7),(40,24,6).\end{split}$ (6)

Consequently we obtain Proposition 6.3.

######  Proposition 6.3.

There exists an LCD $[n,k,d]$ code for $(n,k,d)$ listed in (5) and (6).

In Tables 4 through 6, we give $d_{LCD}(n,k)$ for $n=26,28\leq n\leq 40$ and
$9\leq k\leq n-6$. In order to obtain upper bounds, we use the following:

$d_{LCD}(n,k)\leq d(n,k),$

where $d(n,k)$ denotes the largest minimum weight among all $[n,k]$ codes. The
values of $d(n,k)$ are given in [12]. For the parameters listed in Proposition
6.1, we mark $d_{LCD}(n,k)$ by $*$ in Tables 4 through 6. Furthermore, for the
parameters given in Propositions 6.1 through 6.3, we give $d_{LCD}(n,k)$ in
boldface. For each of the parameters, an LCD code can be obtained
electronically from
https://www.math.is.tohoku.ac.jp/~mharada/Ishizuka/LCD.txt.

Table 4: $d_{LCD}(n,k)$, where $26\leq n\leq 40,9\leq k\leq 17$
---
$n\backslash k$ | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17
26 | 9 | 8 | 8 | 8 | 7 | 6 | 5–6 | 5 | 4
27 | 9 | 9 | 8 | 8 | 7 | 6 | 6 | 6 | 5
28 | 10 | 10 | 8 | 8 | 8 | 7 | 6 | 6 | 5–6
29 | 10 | 10 | 9 | 8 | 8 | 8 | 6 | 6 | 6
30 | 11 | 10 |  9–10 | 9 | 8 | 8 | 6–7 | 6 | 6
31 | 11 | 10 | 10 | 10 | 9 | 8 |  7–8 | 6–7 | 6
32 | 12 | 11 | 10 | 10 |  9–10 | 8–9 |  7–8 |  7–8 | 6–7
33 | 12 | 12 | 10–11 | 10 | 9–10 |  9–10 |  8–9 |  7–8 | 6–8
34 | 13 | 12 | 11–12 | 10–12 | 10 | 10 |  9–10 |  8–9 |  7–8
35 |  13–14 | 12–13 | 12 | 10–12 | 10–11 | 10 | 9–10 |  9–10 |  7–8
36 | 13–14 | 12–14 | 12–13 | 11–12 | 10–12 | 10–11 | 10 | 10 |  8–9
37 | 13–15 | 12–14 | 12–14 | 12–13 | 10–12 | 10–12 | 10–11 | 10 |  9–10
38 | 14–16 |  13–14 | 12–14 | 12–14 |  11*–12 | 10–12 | 10–12 | 10–11 |  9–10
39 | 14–16 | 14–15 |  13–14 | 12–14 |  11–13 |  11–12 | 10–12 | 10–12 |  10–11
40 |  15–16 | 14–16 |  13–15 |  13–14 |  12–14 |  11–13 | 10–12 | 10–12 |  10–12
Table 5: $d_{LCD}(n,k)$, where $26\leq n\leq 40,18\leq k\leq 26$
---
$n\backslash k$ | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26
26 | 4 | 4 | 4 |  |  |  |  |  |
27 | 4 | 4 | 4 | 3 |  |  |  |  |
28 | 5 | 4 | 4 | 4 | 3 |  |  |  |
29 | 6 | 5 | 4 | 4 | 4 | 3 |  |  |
30 | 6 | 5 | 5 | 4 | 4 | 4 | 3 |  |
31 | 6 | 6 | 6 | 5 | 4 | 4 | 4 | 3 |
32 | 6 | 6 | 6 |  5–6 | 5 | 4 | 4 | 3–4 | 3
33 | 6–7 | 6 | 6 |  5–6 | 6 | 5 | 4 | 4 | 4
34 | 6–8 | 6–7 | 6 | 6 | 6 |  5–6 | 4 | 4 | 4
35 |  7–8 | 6–8 | 6–7 |  5–6 | 6 |  5–6 | 5 | 4 | 4
36 |  7–8 |  7–8 | 6–8 |  6–7 | 6 | 6 | 6 | 5 | 4
37 |  8–9 |  7–8 |  7–8 |  6–8 |  6*–7 | 6 | 6 |  5–6 | 5
38 |  9–10 |  8–9 |  7–8 |  7–8 |  6–8 |  6–7 | 6 | 6 | 6
39 | 10 |  9–10 |  8–9 |  7–8 |  7–8 |  6–8 |  6–7 | 6 | 6
40 |  10–11 |  9–10 |  9–10 |  7–9 |  7*–8 |  7–8 |  6–8 |  6–7 | 6
Table 6: $d_{LCD}(n,k)$, where $33\leq n\leq 40,27\leq k\leq 34$
---
$n\backslash k$ | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34
33 | 3 |  |  |  |  |  |  |
34 | 3–4 | 3 |  |  |  |  |  |
35 | 4 | 4 | 3 |  |  |  |  |
36 | 4 | 4 | 3–4 | 3 |  |  |  |
37 | 4 | 4 | 4 | 4 | 3 |  |  |
38 | 5 | 4 | 4 | 4 | 3–4 | 3 |  |
39 |  5–6 | 5 | 4 | 4 | 4 | 4 |  2–3 |
40 |  5–6 | 6 | 5 | 4 | 4 | 4 |  3–4 |  2–3

## Acknowledgement

The authors would like to thank supervisor Professor Masaaki Harada for
introducing the problem, and his helpful advice and encouragement.

$A_{37,22,5}=\begin{pmatrix}000110111001011\\\ 100100110001100\\\
011100010000100\\\ 001110001000010\\\ 001001001100011\\\ 111101001100101\\\
000011111101010\\\ 100101001111001\\\ 101011001101000\\\ 111111011111010\\\
011111101111101\\\ 111000010101000\\\ 111000111011000\\\ 010010010101110\\\
000111000010101\\\ 111010001011110\\\ 110111111100001\\\ 101100011100110\\\
110010111111111\\\ 100000110101011\\\ 001101000001101\\\ 010101110011100\\\
\end{pmatrix}A_{38,13,10}=\begin{pmatrix}1101001110101100011010111\\\
0010011101001101111011001\\\ 1000111010010001010001001\\\
1101101001111111000100001\\\ 1111000000001000001110101\\\
1110010100110011101011111\\\ 1110111110101110011001010\\\
1010010001111011010110010\\\ 1010011011011100001010111\\\
1110100100010100010010111\\\ 1110100110111101100101110\\\
0111010011011110110010111\\\ 0111010011110100101111001\\\
\end{pmatrix}A_{40,22,6}=\begin{pmatrix}010111001011101001\\\
011100100111010110\\\ 001011001011101100\\\ 110010010010001001\\\
000110111001101111\\\ 000001100100101101\\\ 001111111001011111\\\
100101000000000011\\\ 101011011001101010\\\ 010100011111010000\\\
101100110000010100\\\ 011010010101010110\\\ 011011011101000100\\\
001010001100001010\\\ 010100000000111001\\\ 110110011100011011\\\
111000011001011111\\\ 000000110101101010\\\ 101101110011011100\\\
001001010100000011\\\ 010110110100111100\\\ 001001101110010111\\\
\end{pmatrix}$ Figure 1: Matrices $A_{37,22,5},A_{38,13,10},A_{40,22,6}$

## References

  * [1] M. Araya and M. Harada, On the minimum weights of binary linear complementary dual codes, Cryptogr. Commun. 12 (2020), 285–300.
  * [2] M. Araya and M. Harada, On the classification of quaternary optimal Hermitian LCD codes, arXiv:2011.04139.
  * [3] M. Araya, M. Harada and K. Saito, Characterization and classification of optimal LCD codes, Des. Codes Cryptogr. 89 (2021), 617–640.
  * [4] M. Araya, M. Harada and K. Saito, On the minimum weights of binary LCD codes and ternary LCD codes, Finite Fields Appl. (to appear), arXiv:1908.08661.
  * [5] V. K. Bhargava, G. Young and A. K. Bhargava, A characterization of a $(56,28)$ extremal self-dual code, IEEE Trans. Inform. Theory 27 (1981), 258–260.
  * [6] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), 235–265.
  * [7] S. Bouyuklieva, Optimal binary LCD codes, arXiv:2010.13399.
  * [8] F. C. Bussemarker and V. D. Tonchev, New extremal doubly-even codes of length $56$ derived from Hadamard matrices of order $28$, Discrete Math. 76 (1989), 45–49.
  * [9] C. Carlet and S. Guilley, Complementary dual codes for counter-measures to side-channel attacks, Adv. Math. Commun. 10 (2016), 131–150.
  * [10] S. T. Dougherty, J.-L. Kim, B. Ozkaya, L. Sok and P. Solé, The combinatorics of LCD codes: linear programming bound and orthogonal matrices, Int. J. Inf. Coding Theory 4 (2017), 116–128.
  * [11] L. Galvez, J.-L. Kim, N. Lee, Y. G. Roe and B.-S. Won, Some bounds on binary LCD codes, Cryptogr. Commun. 10 (2018), 719–728.
  * [12] M. Grassl, Code tables: Bounds on the parameters of various types of codes, Available online at http://www.codetables.de/, Accessed on 5 Aug 2021.
  * [13] K. Guenda, S. Jitman and T. A. Gulliver, Constructions of good entanglement-assisted quantum error correcting codes, Des. Codes Cryptogr. 86 (2018), 121–136.
  * [14] M. Harada, Existence of new extremal doubly-even codes and extremal singly-even codes, Des. Codes Cryptogr. 8 (1996), 273–283.
  * [15] M. Harada, Self-orthogonal $3$-$(56,12,65)$ designs and extremal doubly-even self-dual codes of length $56$, Des. Codes Cryptogr. 38 (2006), 5–16.
  * [16] M. Harada, Construction of binary LCD codes, ternary LCD codes and quaternary Hermitian LCD codes, Des. Codes Cryptogr. (to appear) https://doi.org/10.1007/s10623-021-00916-1, arXiv:2101.11821.
  * [17] M. Harada, T. A. Gulliver and H. Kaneta, Classification of extremal double-circulant self-dual codes of length up to $62$, Discrete Math. 188 (1998), 127–136.
  * [18] M. Harada and H. Kimura, New extremal doubly-even $[64,32,12]$ codes, Des. Codes Cryptogr. 6 (1995), 91–96.
  * [19] M. Harada and K. Saito, Binary linear complementary dual codes, Cryptogr. Commun. 11 (2019), 677–696.
  * [20] W. C. Huffman and V. Pless, Fundamentals of error-correcting codes, Cambridge University Press (2010).
  * [21] H. Kimura, Extremal doubly even $(56,28,12)$ codes and Hadamard matrices of order $28$, Australas. J. Combin. 10 (1994), 171–180.
  * [22] C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control. 22 (1973), 188–200.
  * [23] J. L. Massey, Linear codes with complementary duals, Discrete Math. 106/107 (1992), 337–342.
  * [24] N. Yankov and M. H. Lee, New binary self-dual codes of length $50$–$60$, Des. Codes Cryptogr. 73 (2014), 983–996.
  * [25] N. Yankov and R. Russeva, Binary self-dual codes of length $52$ to $60$ with an automorphism of order $7$ or $13$, IEEE Trans. Inform. Theory 56 (2011), 7498–7506.
  * [26] V. Yorgov, A method for constructing inequivalent self-dual codes with application to length $56$, IEEE Trans. Inform. Theory 33 (1987), 72–82.


# BiHom-Akivis algebras

Sylvain Attan Département de Mathématiques, Université d’Abomey-Calavi 01 BP
4521, Cotonou 01, Bénin. E.mail<EMAIL_ADDRESS>
###### Abstract

BiHom-Akivis algebras are introduced. The BiHom-commutator-BiHom-associator
algebra of a regular BiHom-algebra is a BiHom-Akivis algebra. It is shown that
BiHom-Akivis algebras can be obtained from Akivis algebras by twisting along
two algebra endomorphisms. It is pointed out that a BiHom-Akivis algebra
associated to a regular BiHom-alternative algebra is a BiHom-Malcev algebra.

2010 Mathematics Subject Classification: 17A30, 17D10, 17D99

Keywords: Akivis algebra, BiHom-associative algebra, BiHom-Lie algebra, BiHom-
Akivis algebra, BiHom-Malcev algebra.

##  1 Introduction

An Akivis algebra $(A,\\{-,-\\},\\{-,\textperiodcentered,-\\})$ is a vector
space $A$ together with a bilinear skew-symmetric map
$(x,y)\longrightarrow\\{x,y\\}$ and a trilinear map
$(x,y,z)\longrightarrow\\{x,y,z\\}$ satisfying the following so-called Akivis
identity for all $x,y,z\in A:$

$\displaystyle\circlearrowleft_{x,y,z}\\{x,\\{y,z\\}\\}=\circlearrowleft_{x,y,z}\\{x,y,z\\}-\circlearrowleft_{x,y,z}\\{y,x,z\\}$
(1.1)

These algebras were introduced by M.A. Akivis ([1, 2, 3]) as a tool in the
study of some aspects of web geometry and its connection with loop theory.
They were originally called ”W-algebras” [3] and later, K.H. Hofmann and K.
Strambach [11] introduced the term “Akivis algebras” for such algebraic
objects.

The theory of Hom-algebras originated from Hom-Lie algebras introduced by J.T.
Hartwig, D. Larsson, and S.D. Silvestrov in [10] in the study of quasi-
deformations of Lie algebras of vector fields, including q-deformations of
Witt algebras and Virasoro algebras. Generalizing the relation between Lie
algebras and associative algebras, the notion of a Hom-associative algebra is
introduced in [14], where it is shown that the commutator algebra (with the
twisting map) of a Hom-associative algebra is a Hom-Lie algebra. By twisting
defining identities, other Hom-type algebras such as Hom-alternative algebras,
Hom-Jordan algebras [13, 16], Hom-Novikov algebras [17], or Hom-Malcev
algebras [16] are introduced and discussed. Moving forward in the general
theory of Hom-algebras, a study of ”binary-ternary” Hom-algebras is initiated
in [12] by defining the class of Hom-Akivis algebras as a Hom-analogue of the
class of Akivis algebras ([1, 2, 11]) which are a typical example of binary-
ternary (see also [4] and [8] for other classes of binary-ternary Hom-
algebra).

Generalizing the approah in [6] the authors of [9] introduce BiHom-algebras,
which are algebras where the identities defining the structure are twisted by
two homomorphisms $\alpha$ and $\beta$. This class of algebras can be viewed
as an extension of the class of Hom-algebras since, when the two linear maps
of a BiHom-algebra are the same, it reduces to a Hom-algebra. These algebraic
structures include BiHom-associative algebras, BiHom-Lie algebras and BiHom-
bialgebras, BiHom-alternative algebras, BiHom-Jordan algebras, Bihom-Malev
algebras $\dots$

As for BiHom-associative, BiHom-Lie, BiHom-alternative, BiHom-Jordan, Bihom-
Malev algebras $\dots$ we consider in this paper a twisted version by two
commuting linear maps of the Akivis identity which defines the so-called
Akivis algebras. We call ”BiHom-Akivis algebra” this twisted Akivis algebra.
It is known [3] that the commutator-associator algebra of a nonassociative
algebra is an Akivis algebra. The Hom-version is this result an be found in
[12]. This led us to consider “non-BiHom-associative algebras” i.e. BiHom-
nonassociative algebras or nonassociative BiHom-algebras and we point out that
the BiHom-commutator-BiHom-associator algebra of a regular non-BiHom-
associative algebra has a BiHom-Akivis structure. Also the class of BiHom-
Akivis algebras contains the one of BiHom-Lie algebras in the same way as the
class of Akivis (resp. Hom-Akivis) algebras contains the one of Lie (resp.
Hom-Le) algebras.

The rest of the present paper is organized as follows. In Section 2 we recall
basic definitions and results about BiHom-algebras. Here, we prove that any
two of the three conditions left BiHom-alternative, right Bihom-alternative
and BiHom-flexible in a regular BiHom-algebra, imply the third (Proposition
2.4, Proposition 2.5 and Proposition 2.6 ). In Section 3, BiHom-Akivis
algebras are considered. Two methods of producing BiHom-Akivis algebras are
provided starting with either a regular BHom-algebras (Theorem 3.2) or
classical Akivis algebras along with twisting maps (Corollary 3.5). BiHom-
Akivis algebras are shown to be closed under twisting by self-morphisms
(Theorem 3.3). In Section 4, BiHom-Akivis algebras associated to a regular
BiHom-alternative algebras are shown to be BiHom-Malcev algebras (these later
algebraic objects are recently introduced [7]). This could be seen as a
generalization of the construction of Malcev (resp. Hom-Malcev) algebras from
alternative [15] (resp. Hom-alternative [16]) algebras.

Throughout this paper, all vector spaces and algebras are meant over a ground
field $\mathbb{K}$ of characteristic 0.

##  2 Preliminaries

In the sequel, a BiHom-algebra refers to a quadruple $(A,\mu,\alpha,\beta),$
where $\mu:A\otimes A\longrightarrow A,$ $\alpha:A\longrightarrow A$ and
$\beta:A\longrightarrow A$ are linear maps such that
$\alpha\beta=\beta\alpha$. The composition of maps is denoted by concatenation
for simplicity and the map $\tau:A^{\otimes 2}\longrightarrow A^{\otimes 2}$
denotes the twist isomorphism $\tau(a\otimes b)=b\otimes a.$

######  Definition 2.1.

A BiHom-algebra $(A,\mu,\alpha,\beta)$ is said to be regular if $\alpha$ and
$\beta$ are bijective and multiplicative if
$\alpha\circ\mu=\mu\circ\alpha^{\otimes 2}$ and
$\beta\circ\mu=\mu\circ\beta^{\otimes 2}.$

######  Definition 2.2.

[9] Let $(A,\mu,\alpha,\beta)$ be a BiHom-algebra.

  1. 1.

A BiHom-associator of $A$ is the trilinear map $as_{\alpha,\beta}:A^{\otimes
3}\longrightarrow A$ defined by

$as_{\alpha,\beta}=\mu\circ(\mu\otimes\beta-\alpha\otimes\mu).$ (2.2)

In terms of elements, the map $as_{\alpha,\beta}$ is given by

$as_{\alpha,\beta}(x,y,z)=\mu(\mu(x,y),\beta(z))-\mu(\alpha(x),\mu(y,z)),\
\forall x,y,z\in A.$

  2. 2.

A BiHom-associative algebra [9] is a multiplicative Bihom-algebra
$(A,\mu,\alpha,\beta)$ satisfying the following BiHom-associativity condition:

$as_{\alpha,\beta}(x,y,z)=0,\ \text{for all}\ x,y,z\in A.\ $ (2.3)

Note that if $\alpha=\beta=Id$, then the BiHom-associator coincide with the
usual associator denoted by $as(,,).$ Clearly, a Hom-associative algebra
$(A,\mu,\alpha)$ can be regarded as a BiHom-associative algebra
$(A,\mu,\alpha,\alpha)$.

######  Remark 2.1.

A non-BiHom-assoiative algebra is a BiHom-algebra $(A,\mu,\alpha,\beta)$ for
which there exists $x,y,z\in A$ such that $as_{\alpha,\beta}(x,y,z)=0.$

######  Example 2.2.

Let $(A,\mu)$ be the two-dimensional algebra with basis $(e_{1},e_{2})$ and
multiplication given by

$\mu(e_{1},e_{2})=\mu(e_{2},e_{2})=e_{1}$

and all missing products are 0. Then $(A,\mu)$ is nonassociative since, e.g.,
$\mu(\mu(e_{1},e_{2}),e_{2})=e_{1}\neq 0=\mu(e_{1},\mu(e_{2},e_{2})).$ Next,
if we define for any $\lambda\neq-1,$ linear maps
$\alpha_{\lambda},\beta_{\lambda}:A\longrightarrow A$ by
$\alpha_{\lambda}(e_{1})=(\lambda+1)e_{1},\ \alpha_{\lambda}(e_{2})=\lambda
e_{1}+e_{2}$ and $\beta_{\lambda}(e_{1})=\frac{1}{\lambda+1}e_{1},\
\beta_{\lambda}(e_{2})=\frac{-\lambda}{\lambda+1}e_{1}+e_{2},$ then
$A_{\alpha_{\lambda},\beta_{\lambda}}=(A,\mu_{\alpha_{\lambda},\beta_{\lambda}}=\mu\circ(\alpha_{\lambda}\otimes\beta_{\lambda}),\alpha_{\lambda},\beta_{\lambda})$
is a non BiHom-associative algebra where the non-zero products are
$\mu_{\alpha_{\lambda},\beta_{\lambda}}(e_{1},e_{2})=\mu_{\alpha_{\lambda},\beta_{\lambda}}(e_{2},e_{2})=(\lambda+1)e_{1}$
since e.g.
$as_{\alpha_{\lambda},\beta_{\lambda}}(e_{1},e_{2},e_{2})=(\lambda+1)e_{1}\neq
0.$ Actually, $A_{\alpha_{\lambda},\beta_{\lambda}}$ is a regular BiHom-
algebra with $\beta=\alpha^{-1}.$

Let recall the notion of BiHom-alternative and BiHom-flexible algebras.

######  Definition 2.3.

[7] Let $(A,\mu,\alpha,\beta)$ be a BiHom-algebra

  1. 1.

$(A,\mu,\alpha,\beta)$ is said to be a left BiHom-alternative $($resp. right
BiHom-alternative $)$ if its satisfies the left BiHom-alternative identity,

$as_{\alpha,\beta}(\beta(x),\alpha(y),z)+as_{\alpha,\beta}(\beta(y),\alpha(x),z)=0,$
(2.4)

respectively, the right BiHom-alternative identity,

$as_{\alpha,\beta}(x,\beta(y),\alpha(z))+as_{\alpha,\beta}(x,\beta(z),\alpha(y))=0,$
(2.5)

for all $x,y,z\in A$. A BiHom-alternative algebra [7] is the one which is both
a left and right BiHom-alternative algebra.

  2. 2.

$(A,\mu,\alpha,\beta)$ is said to be BiHom-flexible [5] if its satisfies the
BiHom-flexible identity,

$\displaystyle
as_{\alpha,\beta}(\beta^{2}(x),\alpha\beta(y),\alpha^{2}(z))+as_{\alpha,\beta}(\beta^{2}(z),\alpha\beta(y),\alpha^{2}(x))=0$
(2.6)

for all $x,y,z\in A.$

Observe that when $\alpha=\beta=Id,$ a BiHom-alternative and a Bihom-flexible
algebra reduce to an alternative and flexible algebra respectively.

######  Remark 2.4.

  1. 1.

Any BiHom-associative algebra is a BiHom-alternative algebra. and BiHom-
flexible.

  2. 2.

It is proved that equations (2.4), (2.5) and (2.6) are respectively equivalent
to

$\displaystyle as_{\alpha,\beta}(\beta(x),\alpha(x),z)=0,\
as_{\alpha,\beta}(x,\beta(y),\alpha(z))=0\mbox{ and
}as_{\alpha,\beta}(\beta^{2}(x),\alpha\beta(y),\alpha^{2}(x))=0$

for all $x,y\in A.$

######  Lemma 2.3.

[7] Let $(A,\mu,\alpha,\beta)$ be a regular BiHom-algebra. Then,
$(A,\mu,\alpha,\beta)$ is a regular BiHom-alternative algebra. if and only if
the function $as_{\alpha,\beta}(\beta^{2}\otimes\alpha\beta\otimes\alpha^{2})$
is alternating.

######  Proposition 2.4.

Any regular BiHom-alternative algebra is BiHom-flexible.

###### Proof.

Follows by a direct by Lemma 2.3. ∎

######  Proposition 2.5.

Any regular left BiHom-alternative BiHom-flexible algebra is BiHom-
alternative.

###### Proof.

Let $(A,\mu,\alpha,\beta)$ be a regular left BiHom-alternative BiHom-flexible
algebra. We need just to prove $as_{\alpha,\beta}(x,\beta(y),\alpha(y))=0$
since it is equivalent to (2.5) (see Remark 2.4 ). Now, let pick $x,y\in A$
then, we have:

$\displaystyle
as_{\alpha,\beta}(x,\beta(y),\alpha(y))=as_{\alpha,\beta}(\beta^{2}(\beta^{-2}(x)),\alpha\beta(\alpha^{-1}(y)),\alpha^{2}(\alpha^{-1}(y)))$
$\displaystyle=-as_{\alpha,\beta}(\beta^{2}(\alpha^{-1}(y)),\alpha\beta(\alpha^{-1}(y)),\alpha^{2}(\beta^{-2}(x)))\
\mbox{ ( by (\ref{BiHomflex1}) \ )}$
$\displaystyle=-as_{\alpha,\beta}(\beta(\beta\alpha^{-1}(y)),\alpha(\beta\alpha^{-1}(y)),\alpha^{2}\beta^{-2}(x))=0\
\mbox{ ( by (\ref{sa1}) \ ).}$

Hence, $(A,\mu,\alpha,\beta)$ is right Bihom-alternative and therefore, it is
BiHom-alternative. ∎

Similarly, we can prove

######  Proposition 2.6.

Any regular right BiHom-alternative BiHom-flexible algebra is BiHom-
alternative.

######  Definition 2.7.

Let $(A,[-,-],\alpha,\beta)$ be a BiHom-algebra.

  1. 1.

The BiHom-Jacobiator of $A$ is the trilinear map $J_{\alpha,\beta}:A^{\times
3}\rightarrow A$ defined as

$\displaystyle
J_{\alpha,\beta}(x,y,z)=\circlearrowright_{x,y,z}[\beta^{2}(x),[\beta(y),\alpha(z)]]$
(2.7)

  2. 2.

$(A,[-,-],\alpha,\beta)$ is said to be a BiHom-Lie algebra if

    1. (a)

$[\beta(x),\alpha(y)]=-[\beta(y),\alpha(x)]$ (BiHom-skew-symmetry)

    2. (b)

$A$ satisfies a BiHom-Jacobi identity i.e.

$\displaystyle J_{\alpha,\beta}(x,y,z)=0$ (2.8)

for all $x,y,z\in A.$

  3. 3.

$(A,[-,-],\alpha,\beta)$ is said to be a BiHom-Malcev algebra if

    1. (a)

$[\beta(x),\alpha(y)]=-[\beta(y),\alpha(x)]$ (BiHom-skew-symmetry)

    2. (b)

$A$ satisfies a BiHom-Malcev identity i.e.

$\displaystyle
J_{\alpha,\beta}(\alpha\beta(x),\alpha\beta(y),[\beta(x),\alpha(z)])=[J_{\alpha,\beta}(\beta(x),\beta(y),\beta(z)),\alpha^{2}\beta^{2}(x)]$
(2.9)

for all $x,y,z\in A.$

######  Remark 2.8.

  1. 1.

If $\alpha=\beta=Id,$ a Bihom-Lie (resp. BiHom-Malcev )algebra reduces to a
Lie (resp. Malcev) algebra.

  2. 2.

Any BiHom-Lie algebra is a BiHom-Malcev algebra.

##  3 BiHom-Akivis algebras: Constructions and exemples

In this section we give the notion and an example of a BiHom-Akivis algebra.
We provide some construction methods of BiHom-Akivis algebras (the
construction from non-BiHom-associative algebras and the one from Akivis
algebras).

######  Definition 3.1.

A BiHom-Akivis algebra is a quintuuple ($V,[-,-],[-,-,-],\alpha,\beta$), where
$V$ is a vector space, $[-,-]:V\times V\rightarrow V$ a BiHom-skew-symmetric
bilinear map, $[-,-,-]:V\times V\times V\rightarrow V$ a trilinear map and
$\alpha:V\rightarrow V$ a linear map such that

$\displaystyle
J_{\alpha,\beta}(x,y,z)=\circlearrowright_{x,y,z}[x,y,z]-\circlearrowright_{x,y,z}[y,x,z]$
(3.10)

for all $x,y,z$ in $V$ where
$J_{\alpha,\beta}(x,y,z)=\circlearrowright_{x,y,z}[\beta^{2}(x),[\beta(y),\alpha(z)]]$
is the BiHom-Jacobiator of $(V,[-,-],\alpha,\beta).$

A BiHom-Akivis algebra ($V,[-,-],[-,-,-],\alpha,\beta$) is said multiplicative
if $\alpha$ and $\beta$ are endomorphisms with respect to $[-,-]$ and
$[-,-,-]$.

In analogy Akivis and Hom-Akivis cases, let call (3.10) the BiHom-Akivis
identity.

######  Remark 3.1.

  1. 1.

If $\alpha=\beta={Id}_{V}$, the BiHom-Akivis identity (3.10) is the usual
Akivis identity (1.1).

  2. 2.

The BiHom-Akivis identity (3.10) reduces to the BiHom-Jacobi identity (2.8),
when $[x,y,z]=0$, for all $x,y,z$ in $V$.

The following result shows how one can get BiHom-Akivis algebras from non-
BiHom-associative algebras.

######  Theorem 3.2.

Let $(A,\mu,\alpha,\beta)$ be a mutipliative regular BiHom-algebra. Then the
BiHom-commutator-BiHom-associator algebra of $(A,\mu,\alpha,\beta)$ is a
multiplicative BiHom-Akivis algebra.

###### Proof.

Let ($A,\mu,\alpha$) be a multiplicative regular BiHom-algebra. For any
$x,y,z$ in $A$, define the operations
$[x,y]:=\mu(x,y)-\mu(\alpha^{-1}\beta(y),\alpha\beta^{-1}(x))$ and
$[x,y,z]:=as_{\alpha,\beta}(\alpha^{-1}\beta^{2}(x),\beta(y),\alpha(z))$.
Then, by [7] (Lemma 2.1), we have

$J_{\alpha,\beta}(x,y,z)=\circlearrowright_{x,y,z}as_{\alpha,\beta}(\alpha^{-1}\beta^{2}(x),\beta(y),\alpha(z))-\circlearrowright_{x,y,z}as_{\alpha,\beta}(\alpha^{-1}\beta^{2}(y),\beta(x),\alpha(z))$

i.e.

$J_{\alpha,\beta}(x,y,z)=\circlearrowright_{x,y,z}[x,y,z]-\circlearrowright_{x,y,z}[y,x,z].$

Hence, $(A,[-,-],[-,-,-],\alpha,\beta)$ is a multiplicative BiHom-Akivis
algebra. ∎

The BiHom-Akivis algebra constructed by Theorem 3.2 is said associated (with a
given regular BiHom-algebra).

######  Example 3.2.

Consider the family of regular non-BiHom-associative algebras
$A_{\alpha_{\lambda},\beta_{\lambda}}$ of Example 2.2. By Theorem 3.2 if
define
$[x,y]:=\mu_{\alpha_{\lambda},\beta_{\lambda}}(x,y)-\mu_{\alpha_{\lambda},\beta_{\lambda}}(\alpha_{\lambda}^{-1}\beta_{\lambda}(y),\alpha_{\lambda}\beta_{\lambda}^{-1}(x))$
and
$[x,y,z]:=as_{\alpha_{\lambda},\beta_{\lambda}}(\alpha_{\lambda}^{-1}\beta_{\lambda}^{2}(x),\beta_{\lambda}(y),\alpha_{\lambda}(z))$
then, $(A,[-,-],[-,-,-],)$ are multiplicative BiHom-Akivis algebras where the
non-zero products are $[e_{1},e_{2}]=(\lambda+1)e_{1},\
[e_{2},e_{1}]=-\frac{1}{\lambda+1}e_{1},\
[e_{2},e_{2}]=\frac{\lambda^{2}+2\lambda}{\lambda+1}e_{1}$ and
$[e_{1},e_{2},e_{2}]=[e_{2},e_{2},e_{2}]=\frac{1}{\lambda+1}e_{1}.$

######  Definition 3.3.

Let $(A,[-,-],[-,-,-],\alpha,\beta)$ and
$(\tilde{A},\\{-,-\\},\\{-,-,-\\},\tilde{\alpha},\tilde{\beta})$ be biHom-
Akivis algebras. A morphism $\phi:A\rightarrow\tilde{A}$ of BiHom-Akivis
algebras is a linear map of $\mathbb{K}$-modules $A$ and $\tilde{A}$ such that
$f\circ\alpha=\tilde{\alpha}\circ f,f\circ\beta=\tilde{\beta}\circ f$ and

$\displaystyle f\circ[-,-]=\\{-,-\\}\circ f^{\otimes 2}\mbox{,
}f\circ[-,-,-]=\\{-,-,-\\}\circ f^{\otimes 3}$

For example, if take $(A,[-,-],[-,-,-],\alpha,\beta)$ as a multiplicative
BiHom-Akivis algebra, then the twisting self-maps $\alpha$ and $\beta$ are
themselfs endomorphisms of $(A,[-,-],[-,-,-])$.

The following result holds.

######  Theorem 3.3.

Let $(A,[-,-],[-,-,-],\alpha,\beta)$ be a BiHom-Akivis algebra and
$\varphi,\psi:A\rightarrow A$ self-morphisms of
$(A,[-,-],[-,-,-],\alpha,\beta)$ such that $\varphi\psi=\psi\varphi$. Define
on $A$ a bilinear operation $[-,-]_{\varphi,\psi}$ and a trilinear operation
$[-,-,-]_{\varphi,\psi}$ by

$[x,y]_{\varphi,\psi}:=[\varphi(x),\psi(y)]$,

$[x,y,z]_{\varphi,\psi}:=\varphi\psi^{2}([x,y,z])$, for all $x,y,z\in A$.
Then
$A_{\varphi,\psi}:=(A,[-,-]_{\varphi,\psi},[-,-,-]_{\varphi,\psi},\varphi\alpha,\psi\beta)$
is a BiHom-Akivis algebra.

Moreover, if $(A,[-,-],[-,-,-],\alpha,\beta)$ is multiplicative, then
$A_{\varphi,\psi}$ is also multiplicative.

###### Proof.

Clearly $[-,-]_{\varphi,\psi}$ (resp. $[-,-,-]_{\varphi,\psi}$) is a bilinear
(resp. trilinear) map and the BiHom-skew-symmetry of $[-,-]$ in
$(A,[-,-],[-,-,-],\alpha,\beta)$ implies the BiHom-skew-symmetry of
$[-,-]_{\varphi,\psi}$ in $A_{\varphi,\psi}$.

Next, we have (by the Hom-Akivis identity (3.10)),

$\displaystyle\circlearrowright_{x,y,z}[(\psi\beta)^{2}(x),[\psi\beta(y),\varphi\alpha(z)]_{\varphi,\psi}]_{\varphi,\psi}=\circlearrowright_{x,y,z}[\varphi\psi^{2}\beta^{2}(x),\psi([\varphi\psi\beta(y),\psi\varphi\alpha(z)])]$
$\displaystyle=\circlearrowright_{x,y,z}(\varphi\psi^{2}([\beta^{2}(x),[\beta(y),\alpha(z)]]))=\circlearrowright_{x,y,z}(\varphi\psi^{2}([x,y,z])-\varphi\psi^{2}([y,x,z]))$
$\displaystyle=\circlearrowright_{x,y,z}([x,y,z]_{\varphi,\psi}-[y,x,z]_{\varphi,\psi})$

The second assertion is proved as follows:

$\displaystyle\varphi\alpha([x,y]_{\varphi,\psi})=\varphi\alpha([\varphi(x),\psi(y)])=[\varphi\alpha\varphi(x),\varphi\alpha\psi(y)]$
$\displaystyle=[\varphi(\varphi\alpha(x)),\psi(\varphi\alpha(y))]=[\varphi\alpha(x),\varphi\alpha(y)]_{\varphi,\psi},$
$\displaystyle\psi\beta([x,y]_{\varphi,\psi})=\psi\beta([\varphi(x),\psi(y)])=[\psi\beta\varphi(x),\psi\beta\psi(y)]$
$\displaystyle=[\varphi(\psi\beta(x)),\psi(\psi\beta(y))]=[\psi\beta(x),\psi\beta(y)]_{\varphi,\psi},$

and

$\displaystyle\varphi\alpha([x,y,z]_{\varphi,\psi})=\varphi\alpha\varphi\psi^{2}([x,y,z])=\varphi\psi^{2}([\varphi\alpha(x),\varphi\alpha(y),\varphi\alpha(z)])=[\varphi\alpha(x),\varphi\alpha(y),\varphi\alpha(z)]_{\varphi,\psi}$
$\displaystyle\psi\beta([x,y,z]_{\varphi,\psi})=\psi\beta\varphi\psi^{2}([x,y,z])=\varphi\psi^{2}([\psi\beta(x),\psi\beta(y),\psi\beta(z)])=[\psi\beta(x),\psi\beta(y),\psi\beta(z)]_{\varphi,\psi}$

This completes the proof. ∎

######  Corollary 3.4.

If $(A,[-,-],[-,-,-],\alpha,\beta)$ is a multiplicative BiHom-Akivis algebra,
then so is $A_{\alpha^{n},\beta^{m}}$ for all $n,m\in\mathbb{N}$.

###### Proof.

This follows from Theorem 3.3 if take $\varphi=\alpha^{n}$ and
$\psi=\beta^{m}$. ∎

######  Corollary 3.5.

Let $(A,[-,-],[-,-,-])$ be an Akivis algebra and $\alpha,\beta$ endomorphisms
of $(A,[-,-],[-,-,-])$. Define on $A$ a bilinear operation
$[-,-]_{\alpha,\beta}$ and a trilinear operation $[-,-,-]_{\alpha,\beta}$ by

$[x,y]_{\alpha,\beta}:=[\alpha(x),\beta(y)]$,

$[x,y,z]_{\alpha,\beta}:=\alpha\beta^{2}([x,y,z]))$,
for all $x,y,z\in A$ Then
$A_{\alpha,\beta}=(A,[-,-]_{,\alpha,\beta},[-,-,-]_{\alpha,\beta},\alpha,\beta)$
is a multiplicative BiHom-Akivis algebra.

Moreover, suppose that $(B,\\{-,-\\},\\{-,-,-\\}$ is another Akivis algebra
and that $\varphi,\psi$ are endomorphisms of $B$. If $f:A\rightarrow B$ is an
Akivis algebra morphism satisfying $f\circ\alpha=\varphi\circ f$ and
$f\circ\beta=\psi\circ f$, then
$f:(A,[-,-]_{\alpha,\beta},[-,-,-]_{\alpha,\beta},\alpha,\beta)\rightarrow(B,\\{-,-\\}_{\varphi,\psi},\\{-,-,-\\}_{\varphi,\psi},\varphi,\psi)$
is a morphism of multiplicative BiHom-Akivis algebras.

###### Proof.

The first of this theorem is a special case of Theorem3.3 above when
$\alpha=\beta=id$. The second part is proved in the same way as in Theorem
3.3. For completeness, we repeat it as follows:

$\displaystyle
f([x,y]_{\alpha,\beta})=f([\alpha(x),\beta(y)])=\\{f\alpha(x),f\beta(y)\\}=\\{\varphi
f(x),\psi f(y)\\}=\\{f(x),f(y)\\}_{\varphi,\psi}$ $\displaystyle
f([x,y,z]_{\alpha,\beta})=f\alpha\beta^{2}([x,y,z])=\varphi\psi^{2}(\\{f(x),f(y),f(z)\\})=\\{f(x),f(y),f(z)\\}_{\varphi,\psi}$

This completes the proof. ∎

######  Example 3.4.

Consider the two-dimensional Akivis algebra $(A,[-,-],[-,-,-])$ with basis
$(e_{1},e_{2})$ given by

$[e_{1},e_{2}]=[e_{1},e_{2},e_{2}]=[e_{2},e_{2},e_{2}]=e_{1}$

and all missing products are 0 (see Example 4.7 in [12]). For any
$r,s\in\mathbb{R},$ the maps $\alpha_{r}$ and $\beta_{s}$ defined by
$\alpha_{r}(e_{1})=(r+1)e_{1},\alpha_{r}(e_{2})=re_{1}+e_{2}$ and
$\beta_{s}(e_{1})=(s+1)e_{1},\beta_{s}(e_{2})=se_{1}+e_{2}$ are commuting
morphisms of $A.$ Note that $\alpha_{r}\neq\beta_{s}$ if and only if $r\neq
s.$ Next, if we define the operations $[−,−]_{\alpha_{r},\beta_{s}}$ and
$[−,−,−]_{\alpha_{r},\beta_{s}}$ with non-zero products by

$[e_{1},e_{2}]_{\alpha_{r},\beta_{s}}=(r+1)e_{1}\mbox{ and
}[e_{1},e_{2},e_{2}]_{\alpha_{r},\beta_{s}}=[e_{2},e_{2},e_{2}]_{\alpha_{r},\beta_{s}}=(r+1)(s+1)^{2}e_{1},$

we get, by Corollary 3.5, that
$A_{\alpha_{r},\beta_{s}}=(A,[−,−]_{\alpha_{r},\beta_{s}},[−,−,−]_{\alpha_{r},\beta_{s}}\alpha_{r},\beta_{s})$
are BiHom-Akivis algebras.

##  4 BiHom-Malcev algebras from BiHom-Akivis algebras

In this section we define BiHom-alternative and BiHom-flexible BiHom-Akivis
algebras, and we give a characterization of BiHom-alternative algebras through
associate BiHom-Akivis algebras. The main result here is that the BiHom-Akivis
algebra associated with a BiHom-alternative algebra has a BiHom-Malcev
structure (this could be seen as another version of Theorem in [7]).

######  Definition 4.1.

A BiHom-Akivis algebra ${\cal A}:=(A,[-,-],[-,-,-],\alpha,\beta)$ is said:

(i) BiHom-flexible, if

$\displaystyle[\alpha(x),\alpha(y),\alpha(z)]+[\alpha(z),\alpha(y),\alpha(x)]=0\mbox{
for all $x,y\in A$ }$ (4.11)

(ii) BiHom-alternative, if

$\displaystyle[\alpha^{2}(x),\alpha^{2}(y),\beta(z)]+[\alpha^{2}(y),\alpha^{2}(x),\beta(z)]=0\mbox{
for all $x,y\in A$ }$ (4.12)
$\displaystyle[\alpha(x),\beta^{2}(y),\beta^{2}(z)]+[\alpha(x),\beta^{2}(z),\beta^{2}(y)]=0\mbox{
for all $x,y\in A$ }$ (4.13)

######  Remark 4.1.

  1. 1.

The BiHom-flexible law (4.11) in ${\cal A}$ is equivalent to
$[\alpha(x),\alpha(y),\alpha(x)]=0$ for all $x,y\in A.$

  2. 2.

The identities (4.12) and (4.13) are respectively called the left BiHom-
alternativity and the right alternativity. They are respectively equivalent to
$[\alpha^{2}(x),\alpha^{2}(x),\beta(y)]=0\ and\
[\alpha(x),\beta^{2}(y),\beta^{2}(y)]=0\ \ for\ all\ x,y\in{\cal A}$.

The following result is an immediate consequence of Theorem 3.2 and Definition
4.1.

######  Proposition 4.2.

Let ${\cal A}=(A,\mu,\alpha,\beta)$ be a multiplicative regular BiHom-algebra
and
${\cal
A_{K}}=(A,[-,-]=\mu-\mu\circ(\alpha^{-1}\beta\otimes\alpha\beta^{-1})\circ\tau,[-,-,-]=as_{\alpha,\beta}\circ(\alpha^{-1}\beta^{2}\otimes\beta\otimes\alpha),\alpha,\beta)$
its associate BiHom-Akivis algebra.

  1. 1.

If $(A,\mu,\alpha,\beta)$ is BiHom-flexible, then ${\cal A_{K}}$ is BiHom-
flexible.

  2. 2.

If $(A,\mu,\alpha,\beta)$ is BiHom-alternative, then so is ${\cal A_{K}}$.

We have the following characterization of BiHom-Lie algebras in terms of
BiHom-Akivis algebras.

######  Proposition 4.3.

Let ${\cal A}:=(A,[-,-],[-,-,-],\alpha,\beta)$ be a BiHom-flexible BiHom-
Akivis algebra such that $\alpha$ is surjective. Then ${\cal
A}_{L}=(A,[-,-],\alpha,\beta)$ is a BiHom-Lie algebra if, and only if
$\circlearrowright_{x,y,z}[x,y,z]=0$, for all $x,y,z\in A.$

###### Proof.

Pick $x,y,z$ in $A.$ Then there exists $a,b,c\in A$ such that
$x=\alpha(a),y=\alpha(b),z=\alpha(c)$ since $\alpha$ is surjective. Therefore,
the BiHom-Akivis identity (3.10) and the BiHom-flexibility (4.11) in ${\cal
A}$ imply

$\displaystyle\circlearrowright_{x,y,z}[\beta^{2}(x),[\beta(y),\alpha(z)]]=\circlearrowright_{x,y,z}[x,y,z]-\circlearrowright_{x,y,z}[y,x,z]=\circlearrowright_{a,b,c}[\alpha(a),\alpha(b),\alpha(c)]$
$\displaystyle-\circlearrowright_{a,b,c}[\alpha(b),\alpha(a),\alpha(c)]=2\circlearrowright_{a,b,c}[\alpha(a),\alpha(b),\alpha(c)]=2\circlearrowright_{x,y,z}[x,y,z].$

Hence, $\circlearrowright_{x,y,z}[\beta^{2}(x),[\beta(y),\alpha(z)]]=0$ if and
only if $\circlearrowright_{x,y,z}[x,y,z]=0$ (recall that the ground field
$\mathbb{K}$ is of characteristic 0). ∎

The following result is a slight generalization of Proposition 2.1 in [7],
which in turn generalizes a similar well-known result in alternative rings.
The reader can also see Proposition 3.17 in [16] for the Hom-version of this
result.

######  Proposition 4.4.

Let ${\cal A}:=(A,[-,-],[-,-,-],\alpha,\beta)$ be a BiHom-alternative BiHom-
Akivis algebra such that $\alpha$ and $\beta$ are surjective. Then

$\displaystyle\circlearrowright_{x,y,z}[\beta^{2}(x),[\beta(y),\alpha(z)]]=6[x,y,z]$
(4.14)

for all $x,y,z\in A.$

###### Proof.

Pick $x,y,z\in A.$ Then, there exists $a,b,c\in A$ such that
$x=\alpha^{2}(a),y=\alpha^{2}(b),z=\alpha(c).$ Hence, the application to
(3.10) of the BiHom-alternativity in ${\cal A}$ gives :

$\displaystyle\circlearrowright_{x,y,z}[\beta^{2}(x),[\beta(y),\alpha(z)]]=\circlearrowright_{x,y,z}[x,y,z]-\sigma[y,x,z]=\circlearrowright_{x,y,z}[x,y,z]-\circlearrowright_{x,y,z}[\alpha^{2}(b),\alpha^{2}(a),\beta(c)]$
$\displaystyle\circlearrowright_{x,y,z}[x,y,z]-\circlearrowright_{x,y,z}[\alpha^{2}(a),\alpha^{2}(b),\beta(c)]=2\circlearrowright_{x,y,z}[x,y,z]$
(4.15)

Next, again by the BiHom-alternativity in ${\cal A}$ and surjectivity of
$\alpha$ and $\beta$, we prove that $\sigma[x,y,z]=3[x,y,z].$ Therefore

$\displaystyle\circlearrowright_{x,y,z}[\beta^{2}(x),[\beta(y),\alpha(z)]]=6[x,y,z]$

∎

First, let recall the following

######  Definition 4.2.

[7] Let $(A,\mu,\alpha,\beta)$ be a regular BiHom-algebra. Define the BiHom-
Bruck-Kleinfeld function $f:A^{\otimes 4}\longrightarrow A$ as the multilinear
map

$\displaystyle
f(w,x,y,z)=as_{\alpha,\beta}(\beta^{2}(w)\alpha\beta(x),\alpha^{2}\beta(y),\alpha^{3}(z))-as_{\alpha,\beta}(\beta^{2}(x),\alpha\beta(y),\alpha^{2}(z))\alpha^{3}\beta(w)$
$\displaystyle-\alpha^{2}\beta^{2}(x)as_{\alpha,\beta}(\alpha\beta(w),\alpha^{2}(y),\alpha^{3}\beta^{-1}(z)).$
(4.16)

The following result is very useful.

######  Lemma 4.5.

[7] Let $(A,\mu,\alpha,\beta)$ be a regular BiHom-alternative algebra. Then
the BiHom-Bruck-Kleinfeld function $f$ is alternating.

######  Proposition 4.6.

Let $(A,\mu,\alpha,\beta)$ be a regular BiHom-alternative algebra. Then

$\displaystyle
as_{\alpha,\beta}(\beta^{3}(x),\alpha\beta^{2}(y),\alpha\beta(x)\alpha^{2}(z))=as_{\alpha,\beta}(\alpha^{-1}\beta^{3}(x),\beta^{2}(y),\alpha\beta(z))\alpha^{2}\beta^{2}(x)$
(4.17) $\displaystyle
as_{\alpha,\beta}(\beta^{3}(x),\alpha\beta^{2}(y),\alpha\beta(z)\alpha^{2}(x))=\alpha\beta^{3}(x)as_{\alpha,\beta}(\beta^{2}(x),\alpha\beta(y),\alpha^{2}(z))$
(4.18)

for all $x,y,z\in A.$

###### Proof.

For (4.17), we compute as follows

$\displaystyle
as_{\alpha,\beta}(\beta^{3}(x),\alpha\beta^{2}(y),\alpha\beta(x)\alpha^{2}(z))=as_{\alpha,\beta}(\beta^{2}(\beta(x)),\alpha\beta(\beta(y)),\alpha^{2}(\alpha^{-1}\beta(x)z))$
$\displaystyle=as_{\alpha,\beta}(\beta^{2}(\alpha^{-1}\beta(x)z),\alpha\beta(\beta(x)),\alpha^{2}(\beta(y)))\
\ \mbox{( by alternativity of
$as_{\alpha,\beta}(\beta^{2}\otimes\alpha\beta\otimes\alpha^{2})$ \ \ )}$
$\displaystyle=as_{\alpha,\beta}(\beta^{2}(\alpha^{-1}\beta(x))\alpha\beta(\alpha^{-1}\beta(z)),\alpha^{2}\beta(\alpha^{-1}\beta(x)),\alpha^{3}(\alpha^{-1}\beta(y)))$
$\displaystyle=f(\alpha^{-1}\beta(x),\alpha^{-1}\beta(z),\alpha^{-1}\beta(x),\alpha^{-1}\beta(y))$
$\displaystyle+as_{\alpha,\beta}(\beta^{2}(\alpha^{-1}\beta(z)),\alpha\beta(\alpha^{-1}\beta(x)),\alpha^{2}(\alpha^{-1}\beta(y)))\alpha^{3}\beta(\alpha^{-1}\beta(x))$
$\displaystyle+\alpha^{2}\beta^{2}(\alpha^{-1}\beta(z))as_{\alpha,\beta}(\alpha\beta(\alpha^{-1}\beta(x)),\alpha^{2}(\alpha^{-1}\beta(x)),\alpha^{3}\beta^{-1}(\alpha^{-1}\beta(y)))$
$\displaystyle=as_{\alpha,\beta}(\alpha^{-1}\beta^{3}(x),\beta^{2}(y),\alpha\beta(z))\alpha^{2}\beta^{2}(x)\
\ \mbox{( by alternativity of $f$ and
$as_{\alpha,\beta}(\beta^{2}\otimes\alpha\beta\otimes\alpha^{2}).$ \ \ )}$

This finishes the proof of (4.17).

For (4.18), we compute as follows

$\displaystyle
as_{\alpha,\beta}(\beta^{3}(x),\alpha\beta^{2}(y),\alpha\beta(z)\alpha^{2}(x))=as_{\alpha,\beta}(\beta^{2}(\beta(x)),\alpha\beta(\beta(y)),\alpha^{2}(\alpha^{-1}\beta(z)x))$
$\displaystyle=as_{\alpha,\beta}(\beta^{2}(\alpha^{-1}\beta(z)x),\alpha\beta(\beta(x)),\alpha^{2}(\beta(y)))\
\ \mbox{( by alternativity of
$as_{\alpha,\beta}(\beta^{2}\otimes\alpha\beta\otimes\alpha^{2})$ \ \ )}$
$\displaystyle=as_{\alpha,\beta}(\beta^{2}(\alpha^{-1}\beta(z))\alpha\beta(\alpha^{-1}\beta(x)),\alpha^{2}\beta(\alpha^{-1}\beta(x)),\alpha^{3}(\alpha^{-1}\beta(y)))$
$\displaystyle=f(\alpha^{-1}\beta(z),\alpha^{-1}\beta(x),\alpha^{-1}\beta(x),\alpha^{-1}\beta(y))$
$\displaystyle+as_{\alpha,\beta}(\beta^{2}(\alpha^{-1}\beta(x)),\alpha\beta(\alpha^{-1}\beta(x)),\alpha^{2}(\alpha^{-1}\beta(y)))\alpha^{3}\beta(\alpha^{-1}\beta(z))$
$\displaystyle+\alpha^{2}\beta^{2}(\alpha^{-1}\beta(x))as_{\alpha,\beta}(\alpha\beta(\alpha^{-1}\beta(z)),\alpha^{2}(\alpha^{-1}\beta(x)),\alpha^{3}\beta^{-1}(\alpha^{-1}\beta(y)))$
$\displaystyle=\alpha\beta^{3}(x)as_{\alpha,\beta}(\beta^{2}(x),\alpha\beta(y),\alpha^{2}(z))\
\ \mbox{( by alternativity of $f$ and
$as_{\alpha,\beta}(\beta^{2}\otimes\alpha\beta\otimes\alpha^{2}).$ \ \ )}$

This finishes the proof of (4.18). ∎

######  Corollary 4.3.

Let $(A,\mu,\alpha,\beta)$ be a regular BiHom-alternative algebra. Then

$\displaystyle
as_{\alpha,\beta}(\beta^{3}(x),\alpha\beta^{2}(y),[\alpha\beta(x),\alpha^{2}(z)])=[as_{\alpha,\beta}(\alpha^{-1}\beta^{3}(x),\beta^{2}(y),\alpha\beta(z)),\alpha^{2}\beta^{2}(x)]$
(4.19)

for all $x,y,z\in A$ where
$[-,-]=\mu-\mu\circ(\alpha^{-1}\beta\otimes\alpha\beta^{-1})\circ\tau$ is the
BiHom-commutator bracket

###### Proof.

Indeed, we have

$\displaystyle
as_{\alpha,\beta}(\beta^{3}(x),\alpha\beta^{2}(y),[\alpha\beta(x),\alpha^{2}(z)])$
$\displaystyle=as_{\alpha,\beta}(\beta^{3}(x),\alpha\beta^{2}(y),\alpha\beta(x)\alpha^{2}(z))-as_{\alpha,\beta}(\beta^{3}(x),\alpha\beta^{2}(y),\alpha\beta(z)\alpha^{2}(x))$
$\displaystyle=as_{\alpha,\beta}(\alpha^{-1}\beta^{3}(x),\beta^{2}(y),\alpha\beta(z))\alpha^{2}\beta^{2}(x)-\alpha\beta^{3}(x)as_{\alpha,\beta}(\beta^{2}(x),\alpha\beta(y),\alpha^{2}(z))$
( by (4.17) and (4.18) )
$\displaystyle=[as_{\alpha,\beta}(\alpha^{-1}\beta^{3}(x),\beta^{2}(y),\alpha\beta(z)),\alpha^{2}\beta^{2}(x)]$

as desired. ∎

We now come to the main result of this section, which is Theorem 2.2 in [7]
but from a point of view of BiHom-Akivis algebras.

######  Theorem 4.7.

Let $(A,\mu,\alpha,\beta)$ be a regular BiHom-alternative BiHom-algebra and
${\cal
A_{K}}=(A,[-,-]=\mu-\mu\circ(\alpha^{-1}\beta\otimes\alpha\beta^{-1})\circ\tau,[-,-,-]=as_{\alpha,\beta}\circ(\alpha^{-1}\beta^{2}\otimes\beta\otimes\alpha),\alpha,\beta)$
its associate BiHom-Akivis algebra. Then $(A,[-,-],\alpha,\beta)$ is a BiHom-
Malcev algebra.

###### Proof.

From Proposition 4.2 we get that ${\cal A_{K}}$ is Hom-alternative so that
(4.14) implies

$\displaystyle
J_{\alpha,\beta}(\alpha\beta(x),\alpha\beta(y),[\beta(x),\alpha(y)])=6[\alpha\beta(x),\alpha\beta(y),[\beta(x),\alpha(z)]]$
$\displaystyle=6as_{\alpha,\beta}(\beta^{3}(x),\alpha\beta^{2}(y),[\alpha\beta(x),\alpha^{2}(z)])$
$\displaystyle=[6as_{\alpha,\beta}(\alpha^{-1}\beta^{3}(x),\beta^{2}(y),\alpha\beta(z)),\alpha^{2}\beta^{2}(x)]\mbox{\
( by (\ref{f3}) \ )}$
$\displaystyle=[[\beta(x),\beta(y),\beta(z)],\alpha^{2}\beta^{2}(x)]=[J_{\alpha,\beta}(\beta(x),\beta(y),\beta(z)),\alpha^{2}\beta^{2}(x)]$

and one recognizes the BiHom-Malcev identity (2.9). Therefore, we get that
$(A,[-,-],\alpha,\beta)$ is a BiHom-Malcev algebra. ∎

## References

  * [1] M.A. Akivis , Three-webs of multidimensional surfaces, Trudy Geom. Sem. 2 (1969), 7-31 (Russian).
  * [2] M.A. Akivis , Local differentiable quasigroups and three-webs of multidimensional surfaces, in “Studies in the Theory of Quasigroups and Loops”, 3-12, Izdat. “Shtiintsa”, Kishinev, 1973 (Russian).
  * [3] M.A. Akivis , Local algebras of a multidimensional three-web, Siberian Math. J. 17 (1976), no. 1, 3-8.
  * [4] S. Attan and A. N. Issa, Hom-Bol algebras, Quasigroups Rel. Syst. 21 (2013), 131-146.
  * [5] S. Attan and I. Laraiedh, Structures of BiHom-Poisson algebras arXiv:2008.04763v2 [math.RA].
  * [6] S. Caenepeel and I. Goyvaerts, Monoidal Hom-Hopf algebras, Comm. Algebra. 39 (2011), 2216-2240.
  * [7] T. Chtioui, S. Mabrouk and A. Makhlouf, BiHom-alternative, BiHom-Malcev and BiHom-Jordan algebras. arXiv:1811.10394v1 [math.RA]
  * [8] D. Gaparayi and A.N. Issa, A twisted generalization of Lie-Yamaguti algebras, Int. J. Algebra 6(7) (2012), 339-352.
  * [9] G. Graziani, A. Makhlouf, C. Menini and F. Panaite, BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras. SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper 086, 34 pp.
  * [10] J. T. Hartwig, D. Larsson and S. D. Silvestrov, Deformations of Lie algebras using $\sigma$-derivations, J. Algebra, 292 (2006), 314-361.
  * [11] Hofmann K.H., Strambach K., Lie’s fundamental theorems for local analytical loops, Pacific J. Math. 123 (1986), no. 2, 301-327.
  * [12] A.N. Issa, Hom-Akivis algebras, Comment. Math. Univ. Carolinae, 52 (2011), no. 4, 485-500.
  * [13] A. Makhlouf , Hom-Alternative algebras and Hom-Jordan algebras, Int. Elect. J. Alg., 8 (2010), 177-190.
  * [14] A. Makhlouf, S. D. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2 (2008), 51-64.
  * [15] A. I. Maltsev, Analytic loops, Mat. Sb. 36 (1955), 569-576.
  * [16] D. Yau, Hom-Maltsev, Hom-alternative and Hom-Jordan algebras, Int. Elect. J. Alg., 11 (2012), 177-217.
  * [17] D. Yau , Hom-Novikov algebras, arXiv:0909.0726v1 [math.RA] (2009).

.


In the last few years, researchers have investigated the relationships among
LPMLN and several logic formalisms such as ASP, MLN, LPOD, P-log [BGR09],
ProbLog [DKT07] etc. On the one hand, these results provide a translation
based approach to implementing LPMLN solvers, which is used in several
implementations such as LPMLN2ASP, LPMLN2MLN [LTW17], and LPMLN-Models
[WXZ+18]. On the other hand, these relationships help us to investigate new
properties of a logic formalism by using existing results in other logic
formalisms, such as the investigations of splitting set theorems [WZXS18] and
weight learning of LPMLN [LW18].

Under the circumstance, one may ask whether the strong equivalence of other
formalisms can be investigated in LPMLN. From the discussion of ASPwc and
LPOD, we can observe that the key of the question is the translation from a
logic program to an LPMLN program. There are some common properties of the
translations $\tau^{c}$ and $\tau^{\times}$ used for ASPwc and LPOD. Firstly,
a translation should be semantics-preserving, i.e. original program and the
translated program have the same inference results. Obviously, this is a
necessary property for a translation, and both $\tau^{c}$ and $\tau^{\times}$
are semantics-preserving. Secondly, a translation should be modular. A
translation $\tau^{*}$ is modular, if for logic programs $P$ and $Q$, we have
$\tau^{*}(P\cup Q)=\tau^{*}(P)\cup\tau^{*}(Q)$. By investigating the strong
equivalences for ASPwc and LPOD, we have shown that for a logic program $P$,
if there is a modular LPMLN translation $\tau^{\prime}$ such that the
semantics of $P$ can be characterized by $\tau^{\prime}(P)$, then we can at
least find a sufficient condition for characterizing the strong equivalence of
the program $P$. In other words, if we cannot find a modular translation for
the program, then its strong equivalence may not be investigated via using the
results obtained in this paper.

In addition, the LPMLN translations $\tau^{c}$ and $\tau^{\times}$ for ASPwc
and LPOD have other good properties. For example, for a logic program $P$ and
a translation $\tau^{*}(P)$, $\tau^{*}$ is called fixed, if $\tau^{*}(P)$ does
not introduce new atoms, i.e. $lit(P)=lit(\tau^{*}(Q))$. It is easy to check
that both the translations $\tau^{c}$ and $\tau^{\times}$ are fixed. But it is
still unknown whether the property affects the translation based investigation
of strong equivalences. For P-log programs, there exists a translation
$\tau^{p}$ from P-log to LPMLN [LY17], and there also exists a translation
$\tau^{m}$ from LPMLN to P-log [BG16]. Both of $\tau^{p}$ and $\tau^{m}$ are
semantics-preserving and modular, but they are not fixed. Intuitively, since
there is a one-to-one map between P-log programs and LPMLN programs, it is
very likely the strong equivalence for P-log can be characterized by the
strong equivalence for LPMLN.

In a word, our results provide a method to investigate the strong equivalence
for some logic formalisms by translating them into LPMLN programs. And for the
method, some properties of translations are important, such as the semantics-
preserving and modular properties. But it is unclear whether newly introduced
literals in the translation affect the translation based investigation of
strong equivalences.

##  7\. Simplifying LPMLN Programs

Program simplification is an important technology for improving the
implementations of logic programs. In this section, we investigate the
simplification of LPMLN programs via introducing the notions of semi-valid and
valid rules, which are two kinds of redundant LPMLN rules based on the semi-
strong and p-strong equivalences. Firstly, we present an algorithm to simplify
and solve LPMLN programs by eliminating these redundant LPMLN rules. Then, to
decide the redundant rules efficiently, we present some syntactic conditions
that characterize the semi-valid and valid LPMLN rules.

According to whether an LPMLN rule is semi-strongly or p-strongly equivalent
to the empty program $\emptyset$, the notions of semi-valid and valid LPMLN
rules are defined as follows. An LPMLN rule $w:r$ is called semi-valid, if
$w:r$ is semi-strongly equivalent to $\emptyset$; the rule is called valid, if
$w:r$ is p-strongly equivalent to $\emptyset$.

Obviously, a valid LPMLN rule can be eliminated from any LPMLN programs, while
a semi-valid LPMLN rule cannot. By the definition, eliminating a semi-valid
LPMLN rule does not change the stable models of original programs, but changes
the probability distributions of the stable models. Furthermore, it may change
the probabilistic stable models of original programs, which can be observed
from Example 7.

Consider three LPMLN programs $P=\\{\alpha:a\leftarrow a.\\}$,
$Q=\\{\alpha:\leftarrow a.\\}$, and $R=\\{1:a.\\}$. It is easy to check that
rules in $P$ and $Q$ are valid and semi-valid, respectively. Table 8 shows the
stable models and their probability degrees of LPMLN programs $P\cup R$,
$Q\cup R$, and $R$. It is easy to check that eliminating the rule of $P$ from
program $P\cup R$ does not affect the inference results of $P\cup R$. By
contrast, eliminating the rule of $Q$ from $Q\cup R$ changes both of the MAP
and MPD inference results of $Q\cup R$.

Table 8. Computing Results in Example 7 Stable Model $S$ | $Pr(P\cup R,S)$ | $Pr(Q\cup R,S)$ | $Pr(R,S)$
---|---|---|---
$\emptyset$ | $0.27$ | $1$ | $0.27$
$\\{a\\}$ | $0.73$ | $0$ | $0.73$

Algorithm 1 provides a framework to simplify and solve LPMLN programs based on
the notions of semi-valid and valid LPMLN rules. Firstly, simplify an LPMLN
program $P$ by removing all semi-valid and valid rules (line 2 - 8). Then,
compute the stable models of the simplified LPMLN program via using some
existing LPMLN solvers, such as LPMLN2ASP, LPMLN2MLN [LTW17], and LPMLN-Models
[WXZ+18] ect. Finally, compute the probability degrees of the stable models
w.r.t. the simplified program and all semi-valid rules (line 9 - 12). The
correctness of the algorithm can be proven by corresponding definitions.

Input: an LPMLN program $P$

Output: stable models of $P$ and their probability degrees

1 $Q=\emptyset$, $P^{\prime}=P$ ;

2 foreach  _$w:r\in P$_ do

3 if  _$w:r$ is valid_ then

4 $P^{\prime}=P^{\prime}-\\{w:r\\}$ ;

5

6 else if  _$w:r$ is semi-valid_ then

7 $Q=Q\cup\\{w:r\\}$ ;

8 $P^{\prime}=P^{\prime}-\\{w:r\\}$ ;

9

10

11

12$SM^{L}(P)=call\text{-}lpmln\text{-}solver(P^{\prime})$;

13

14foreach  _$X\in SM^{L}(P)$_ do

15 $W^{\prime}(P,X)=exp\left(\sum_{w:r\in P^{\prime}\cup Q\text{ and }X\models
w:r}~{}w\right)$ ;

16

17

18Compute probability degrees for each stable model $X$ by Equation (12) and
$W^{\prime}(P,X)$;

return _$SM^{L}(P)$ and corresponding probability degrees_

Algorithm 1 Simplify and Solve LPMLN Programs Table 9. Syntactic Conditions for Valid and Semi-valid LPMLN Rules Name | Definition | Strong Equivalence
---|---|---
TAUT | $h(r)\cap b^{+}(r)\neq\emptyset$ | p, semi
CONTRA | $b^{+}(r)\cap b^{-}(r)\neq\emptyset$ | p, semi
CONSTR1 | $h(r)=\emptyset$ | semi
CONSTR2 | $h(r)\subseteq b^{-}(r)$ | semi
CONSTR3 |  $~{}~{}~{}~{}h(r)=\emptyset$, $b^{+}(r)=\emptyset$, and $b^{-}(r)=\emptyset~{}~{}~{}~{}$ | p, semi

In Algorithm 1, a crucial problem is to decide whether an LPMLN rule is valid
or semi-valid. Theoretically, it can be done by checking the SE-models of each
rule. However, the model-theoretical approach is highly complex in
computation. Therefore, we investigate the syntactic conditions for the
problem. Table 9 shows five syntactic conditions for a rule $r$, where TAUT
and CONTRA have been introduced to investigate the program simplification of
ASP [ONA01, EFTW04], CONSTR1 means the rule $r$ is a constraint, and CONSTR3
is a special case of CONSTR1. Rules satisfying CONSTR2 is usually used to
eliminate constraints in ASP. For example, rule “$\leftarrow a.$” is
equivalent to rule “$p\leftarrow a,~{}not~{}p.$”, if the atom $p$ does not
occur in other rules. Based on these conditions, the following theorems
provide the characterizations for semi-valid and valid LPMLN rules.

######  Theorem 34.

An LPMLN rule $w:r$ is semi-valid, iff the rule satisfies one of TAUT, CONTRA,
CONSTR1, and CONSTR2.

######  Proof 7.1.

The if direction of Theorem 34 is straightforward. For each of TAUT, CONTRA,
CONSTR1, and CONSTR2, it can be verified by Lemma 5 and the definition of SE-
models. Here, we only present the proof of TAUT, the proofs of other
conditions are similar.

For the condition TAUT, let $P=\\{w:r\\}$, to prove the condition, we need to
show that $SE^{L}(P)=SE^{L}(\emptyset)$. It is easy to observe that for any
SE-interpretation $(X,Y)$, $(X,Y)\in SE^{L}(\emptyset)$, therefore, we only
need to show that for any SE-interpretation $(X,Y)$, $(X,Y)\in SE^{L}(P)$. We
use proof by contradiction. Assume that there is an SE-interpretation $(X,Y)$
such that $(X,Y)\not\in SE^{L}(P)$, by the definition, we have
$X\not\models\left(\overline{{P}_{Y}}\right)^{Y}$. For the program
$P=\\{w:r\\}$, since $h(r)\cap b^{+}(r)\neq\emptyset$, we have $w:r$ can be
satisfied by $Y$, which means $\overline{P_{Y}}=\overline{P}$. For the GL-
reduct $\left(\overline{{P}_{Y}}\right)^{Y}$, there are two cases: (1)
$\left(\overline{{P}_{Y}}\right)^{Y}=\emptyset$ and (2)
$\left(\overline{{P}_{Y}}\right)^{Y}\neq\emptyset$.

Case 1. If $\left(\overline{{P}_{Y}}\right)^{Y}=\emptyset$, we have $Y\cap
b^{-}(r)\neq\emptyset$. It is obvious that
$X\models\left(\overline{{P}_{Y}}\right)^{Y}$, which contradicts with the
assumption.

Case 2. If $\left(\overline{{P}_{Y}}\right)^{Y}\neq\emptyset$, we have $Y\cap
b^{-}(r)=\emptyset$. Since $X\not\models\left(\overline{{P}_{Y}}\right)^{Y}$,
we have $b^{+}(r)\subseteq X$ and $h(r)\cap X=\emptyset$, which means
$h(r)\cap b^{+}(r)=\emptyset$. It contradicts with the condition TAUT.

Combining above results, the condition TAUT is proven.

To prove the only-if direction, we need to show that if a rule $w:r$ satisfies
none of TAUT, CONTRA, CONSTR1, and CONSTR2, then an LPMLN program
$P=\\{w:r\\}$ is not semi-strongly equivalent to $\emptyset$, which means
$h(r)\cap b^{+}(r)=\emptyset$, $b^{+}(r)\cap b^{-}(r)=\emptyset$,
$h(r)\neq\emptyset$, and $h(r)\not\subseteq b^{-}(r)$. There are four cases
according to whether the body of $r$ is empty, we need to show that for each
case, there exists an SE-interpretation that is not an SE-model of $P$.

Case 1. If $b^{+}(r)=\emptyset$ and $b^{-}(r)=\emptyset$, $r$ is a fact.
Obviously, $(\emptyset,h(r))$ is not an SE-model of $P$, therefore, $P$ is not
semi-strongly equivalent to $\emptyset$.

Case 2. If $b^{+}(r)=\emptyset$ and $b^{-}(r)\neq\emptyset$, it is easy to
check that $(\emptyset,h(r)-b^{-}(r))$ is not an SE-model of $P$, therefore,
$P$ is not semi-strongly equivalent to $\emptyset$.

Case 3. If $b^{+}(r)\neq\emptyset$ and $b^{-}(r)=\emptyset$, it is easy to
check that $(b^{+}(r),h(r)\cup b^{+}(r))$ is not an SE-model of $P$,
therefore, $P$ is not semi-strongly equivalent to $\emptyset$.

Case 4. If $b^{+}(r)\neq\emptyset$ and $b^{-}(r)\neq\emptyset$, it is easy to
check that $(b^{+}(r),(h(r)-b^{-}(r))\cup b^{+}(r))$ is not an SE-model of
$P$, therefore, $P$ is not semi-strongly equivalent to $\emptyset$.

Combining above results, the only-if direction is proven.

######  Theorem 35.

An LPMLN rule $w:r$ is valid, iff one of the following conditions is satisfied

  * -

rule $w:r$ satisfies one of CONSTR1 and CONSTR2, and $w=0$; or

  * -

rule $w:r$ satisfies one of TAUT, CONTRA, and CONSTR3.

######  Proof 7.2.

For the if direction, the proof is divided into three cases, in each case, we
prove some conditions in Theorem 35.

Case 1. If $w:r$ is a rule of the form $CONSTR1$ or $CONSTR2$ and $w=0$. By
Theorem 34, $w:r$ is semi-valid. Since $w=0$, we have for any SE-models
$(X,Y)$ of $\\{w:r\\}$, $W(\\{w:r\\},(X,Y))=e^{0}=1$, therefore, $w:r$ is
valid.

Case 2. If $w:r$ is a rule of the form $TAUT$ or $CONSTRA$, it is easy to
check that $w:r$ is semi-valid, and for any SE-model $(X,Y)$, $Y\models w:r$.
Therefore, we have for any SE-model $(X,Y)$, $W(\\{w:r\\},(X,Y))=e^{w}$, which
means $w:r$ is valid.

Case 3. If $w:r$ is a rule of the form $CONSTR3$, it is easy to check that
$w:r$ is semi-valid, and for any SE-model $(X,Y)$, $Y\not\models w:r$.
Therefore, we have for any SE-model $(X,Y)$, $W(\\{w:r\\},(X,Y))=e^{0}=1$,
which means $w:r$ is valid.

For the only-if direction, we use proof by contradiction. Assume $w:r$ is a
valid LPMLN rule satisfying none of conditions in Theorem 35, i.e. (1) $w:r$
does not satisfy all of $TAUT$, $CONSTRA$, $CONSTR1$, $CONSTR2$, and
$CONSTR3$; (2) $w:r$ satisfies $CONSTR1$ or $CONSTR2$, and $w\neq 0$.

Case 1. If $w:r$ does not satisfy all of $TAUT$, $CONSTRA$, $CONSTR1$,
$CONSTR2$, and $CONSTR3$, by Theorem 34, $w:r$ is not semi-valid, therefore,
$w:r$ is not valid, which contradicts with the assumption.

Case 2. If $w:r$ satisfies $CONSTR1$ or $CONSTR2$, and $w\neq 0$, by the
definition of SE-models, there exist SE-models $(X_{1},Y_{1})$ and
$(X_{2},Y_{2})$ such that $Y_{1}\models w:r$ and $Y_{2}\not\models w:r$, such
as $Y_{1}=h(r)$ and $Y_{2}=b^{+}(r)-b^{-}(r)$. Therefore, we have
$W(\\{w:r\\},(X_{1},Y_{1}))=e^{w}$ and $W(\\{w:r\\},(X_{2},Y_{2}))=e^{0}$, but
$W(\\{w:r\\},(X_{1},Y_{1}))=W(\\{w:r\\},(X_{2},Y_{2}))=e^{0}$, which means
$w:r$ is not valid and contradicts with the assumption.

Combining above results, Theorem 35 is proven.

Theorem 34 and Theorem 35 can be used to check the validity of an LPMLN rule
efficiently, which makes Algorithm 1 an alternative approach to enhance LPMLN
solvers. In addition, Theorem 34 and Theorem 35 also contribute to the field
of knowledge acquiring. On the one hand, although it is impossible that rules
of the form TAUT, CONTRA1, and CONSTR3 are constructed by a skillful knowledge
engineer, these rules may be obtained by rule learning. Therefore, we can use
TAUT, CONTRA, and CONSTR3 as heuristic information to improve the results of
rule learning. On the other hand, it is worth noting that conditions CONSTR1
and CONSTR2 mean the only effect of constraints in LPMLN is to change the
probability distribution of inference results, which can be observed in
Example 3.3. Therefore, for the problem modeling in LPMLN, we can encode
objects and relations by LPMLN facts and rules, and adjust the certainty
degrees of inference results by LPMLN constraints.

##  8\. Conclusion and Future Work

In this paper, we investigate the notions of strong equivalences for LPMLN
programs and study several properties of the notions from four aspects. First
of all, we present the notion of p-ordinary equivalence for LPMLN, that is,
two p-ordinarily equivalent LPMLN programs have the same stable models and the
same probability distribution of their stable models, which means the programs
have the same MAP and MPD inference results. Based on the p-ordinary
equivalence, we present the notion of p-strong equivalence, that is, two
p-strongly equivalent LPMLN programs are p-ordinary equivalent under any
extensions. Then, we present a sufficient and necessary condition for
characterizing the p-strong equivalence, i.e. the PSE-condition, which can be
regarded as a generalization of the SE-model approach in ASP. Due to hard
rules can be violated in the original LPMLN semantics, the necessity of the
PSE-condition is quite difficult to prove. To this end, we introduce the
notions of necessary extensions and flattening extensions, and show that the
PSE-condition is necessary, since we can construct a necessary extension by
selecting proper flattening extensions. Since the conditions of p-strong
equivalence are somewhat strict, we study the notion of q-strong equivalence.
Unfortunately, our results show that the q-strong equivalence is identical to
the p-strong equivalence.

After the characterization, we further study the properties of the p-strong
equivalence from four aspects. Firstly, we present two relaxed notions of the
p-strong equivalence, i.e. the sp-strong and p-uniform equivalence, and
discuss their characterizations, which are useful in many real-world scenarios
such as decision making and knowledge graph based applications etc. Secondly,
we analyze the computational complexities of deciding strong equivalences. Our
results show that deciding all of the semi-strong, the p-strong, and the sp-
strong equivalence is co-NP-complete, and deciding the semi-uniform
equivalence is in $\Pi_{2}^{p}$. As a by-product, Lin and Chen’s work [LC07]
implies that our method of proving the complexity results could be used to
discover syntactic conditions deciding semi-strong equivalence. Thirdly, we
investigate the relationships among the strong equivalences for LPMLN and two
important extensions of ASP: ASPwc and LPOD. Our results show that the strong
equivalences for ASPwc and LPOD can be studied by translating them into LPMLN,
which provides a viable way to study the strong equivalences for other logic
formalisms such as ProbLog and P-log etc. Finally, we investigate the program
simplification based on the p-strong equivalence. Specifically, we present a
characterization for two kinds of redundant LPMLN rules, which can be used to
improve LPMLN solvers.

For the future, we plan to continue the unsolved problems in this paper, i.e.
the characterization of p-uniform equivalence and its computational
complexity. And we will further investigate the strong equivalences for other
logic formalisms by translating them into LPMLN programs. In addition, the
investigations of approximate strong equivalences and the syntactic conditions
deciding strong equivalences are also valuable topics of the field.

## Acknowledgments

We are grateful to the anonymous referees for their useful comments on the
earlier version of this paper. The work was supported by the National Key
Research and Development Plan of China (Grant No.2017YFB1002801).

## References

  * [BET11] Gerhard Brewka, Thomas Eiter, and Mirosław Truszczyński.  Answer Set Programming at a Glance.  Communications of the ACM, 54(12):92–103, 2011.
  * [BG16] Evgenii Balai and Michael Gelfond.  On the Relationship between P-log and LPMLN.  In Subbarao Kambhampati, editor, Proceedings of the 25th International Joint Conference on Artificial Intelligence, pages 915–921, 2016\.
  * [BGR09] Chitta Baral, Michael Gelfond, and Nelson Rushton.  Probabilistic reasoning with answer sets.  Theory and Practice of Logic Programming, 9(1):57–144, 2009.
  * [BLR00] Francesco Buccafurri, Nicola Leone, and Pasquale Rullo.  Enhancing disjunctive datalog by constraints.  IEEE Transactions on Knowledge and Data Engineering, 12(5):845–860, 2000.
  * [Bre02] Gerhard Brewka.  Logic Programming with Ordered Disjunction.  In Proceedings of the 9th International Workshop on Non-Monotonic Reasoning, pages 67–76, Toulouse, France, 2002.
  * [Bre05] Gerhard Brewka.  Preferences in Answer Set Programming.  In Proceedings of the 11th Conference of the Spanish Association for Artificial Intelligence on Current Topics in Artificial Intelligence, pages 1–10, 2005.
  * [CFG+12] Francesco Calimeri, Wolfgang Faber, Martin Gebser, Giovambattista Ianni, Roland Kaminski, Thomas Krennwallner, Nicola Leone, Francesco Ricca, and Torsten Schaub.  ASP-Core-2 Input language format.  ASP Standardization Working Group, 2012.
  * [DKT07] Luc De Raedt, Angelika Kimmig, and Hannu Toivonen.  ProbLog: A probabilistic prolog and its application in link discovery.  In Manuela M. Veloso, editor, Proceedings of the 20th International Joint Conference on Artificial Intelligence, pages 2468–2473, 2007\.
  * [EF03] Thomas Eiter and Michael Fink.  Uniform Equivalence of Logic Programs under the Stable Model Semantics.  In Proceedings of the 19th International Conference on Logic Programming, pages 224–238, Mumbai, India, 2003.
  * [EFFW07] Thomas Eiter, Wolfgang Faber, Michael Fink, and Stefan Woltran.  Complexity results for answer set programming with bounded predicate arities and implications.  Annals of Mathematics and Artificial Intelligence, 51(2-4):123–165, dec 2007.
  * [EFTW04] Thomas Eiter, Michael Fink, Hans Tompits, and Stefan Woltran.  Simplifying Logic Programs Under Uniform and Strong Equivalence.  In Proceedings of the 7th International Conference on Logic Programming and Nonmonotonic Reasoning, pages 87–99, 2004.
  * [EFW07] Thomas Eiter, Michael Fink, and Stefan Woltran.  Semantical characterizations and complexity of equivalences in answer set programming.  ACM Transactions on Computational Logic, 8(3):1–53, 2007.
  * [EK16] Thomas Eiter and Tobias Kaminski.  Exploiting Contextual Knowledge for Hybrid Classification of Visual Objects.  In Jürgen Dix, Luís Fariñas del Cerro, and Ulrich Furbach, editors, Proceedings of the 15th European Conference on Logics in Artificial Intelligence, volume 10021 of Lecture Notes in Computer Science, pages 223–239, Berlin, Heidelberg, 2016. Springer Berlin Heidelberg.
  * [EW16] Lisa Ehrlinger and Wolfram Wöß.  Towards a definition of knowledge graphs.  In Joint Proceedings of the Posters and Demos Track of the 12th International Conference on Semantic Systems and the 1st International Workshop on Semantic Change & Evolving Semantics, pages 1–4, Leipzig, Germany, 2016.
  * [FTW08] Wolfgang Faber, Hans Tompits, and Stefan Woltran.  Notions of Strong Equivalence for Logic Programs with Ordered Disjunction.  In Proceedings of the Eleventh International Conference on, pages 433–443, 2008.
  * [GL88] Michael Gelfond and Vladimir Lifschitz.  The Stable Model Semantics for Logic Programming.  In Robert A. Kowalski and Kenneth A. Bowen, editors, Proceedings of the Fifth International Conference and Symposium on Logic Programming, pages 1070–1080. MIT Press, 1988.
  * [IS04] Katsumi Inoue and Chiaki Sakama.  Equivalence of Logic Programs Under Updates.  In Proceedings of the 9th European Workshop on Logics in Artificial Intelligence, volume 3229, pages 174–186. 2004.
  * [LC07] Fangzhen Lin and Yin Chen.  Discovering Classes of Strongly Equivalent Logic Programs.  Journal of Artificial Intelligence Research, 28:431–451, apr 2007\.
  * [Lin02] Fangzhen Lin.  Reducing Strong Equivalence of Logic Programs to Entailment in Classical Propositional Logic.  In Proceedings of the 8th International Conference on Principles of Knowledge Representation and Reasoning (KR-02), pages 170–176, 2002.
  * [LL19] Joohyung Lee and Man Luo.  Strong Equivalence for LPMLN Programs.  In Proceedings of the 35th International Conference on Logic Programming, pages 196–209, 2019.
  * [LPV01] Valdimir Lifschitz, David Pearce, and Agustín Valverde.  Strongly equivalent logic programs.  ACM Transactions on Computational Logic, 2(4):526–541, oct 2001\.
  * [LTW17] Joohyung Lee, Samidh Talsania, and Yi Wang.  Computing LPMLN using ASP and MLN solvers.  Theory and Practice of Logic Programming, 17(5-6):942–960, sep 2017\.
  * [LW16] Joohyung Lee and Yi Wang.  Weighted Rules under the Stable Model Semantics.  In Chitta Baral, James P. Delgrande, and Frank Wolter, editors, Proceedings of the Fifteenth International Conference on Principles of Knowledge Representation and Reasoning:, pages 145–154. AAAI Press, 2016.
  * [LW18] Joohyung Lee and Yi Wang.  Weight Learning in a Probabilistic Extension of Answer Set Programs.  In Proceedings of the 16th International Conference on the Principles of Knowledge Representation and Reasoning, pages 22–31, aug 2018\.
  * [LY17] Joohyung Lee and Zhun Yang.  LPMLN, Weak Constraints, and P-log.  In Satinder P. Singh and Shaul Markovitch, editors, Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, pages 1170–1177. AAAI Press, 2017.
  * [ONA01] Mauricio Osorio, Juan Antonio Navarro, and José Arrazola.  Equivalence in Answer Set Programming.  In Proceedings of the 11th International Workshop on Logic Based Program Synthesis and Transformation,, pages 57–75, 2001.
  * [PT09] Jörg Pührer and Hans Tompits.  Casting Away Disjunction and Negation under a Generalisation of Strong Equivalence with Projection.  In Proceedings of the 10th International Conference on Logic Programming and Nonmonotonic Reasoning, volume 5753, pages 264–276, 2009.
  * [RD06] Matthew Richardson and Pedro Domingos.  Markov logic networks.  Machine Learning, 62(1-2):107–136, feb 2006.
  * [Tur01] Hudson Turner.  Strong Equivalence for Logic Programs and Default Theories (Made Easy).  In Proceedings of the 6th International Conference on Logic Programming and Nonmonotonic Reasoning, pages 81–92, 2001.
  * [Wol10] S. Woltran.  Equivalence between extended datalog programs - A brief survey.  In Datalog Reloaded - First International Workshop, pages 106–119, 2010.
  * [WSZZ19] Bin Wang, Jun Shen, Shutao Zhang, and Zhizheng Zhang.  On the Strong Equivalences of LPMLN Programs.  In Proceedings of the 35th International Conference on Logic Programming, pages 114–125, 2019.
  * [WXZ+18] Wei Wu, Hongxiang Xu, Shutao Zhang, Jiaqi Duan, Bin Wang, Zhizheng Zhang, Chenglong He, and Shiqiang Zong.  LPMLNModels: A Parallel Solver for LPMLN.  In Proceedings of IEEE 30th International Conference on Tools with Artificial Intelligence, pages 794–799. IEEE, nov 2018.
  * [WZ17] Bin Wang and Zhizheng Zhang.  A Parallel LPMLN Solver: Primary Report.  In Bart Bogaerts and Amelia Harrison, editors, Proceedings of the 10th Workshop on Answer Set Programming and Other Computing Paradigms, pages 1–14, Espoo, Finland, 2017. CEUR-WS.
  * [WZX+18] Bin Wang, Shutao Zhang, Hongxiang Xu, Zhizheng Zhang, Wei Wu, and Xiaodong Li.  Handling Preferences in LPMLN : A Preliminary Report.  In Proceedings of the 8th International Workshop on Combinations of Intelligent Methods and Applications, volume 2252, pages 81–92, 2018.
  * [WZXS18] Bin Wang, Zhizheng Zhang, Hongxiang Xu, and Jun Shen.  Splitting an LPMLN Program.  In Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, pages 1997–2004, 2018.