# Linear codes using simplicial complexes

Vidya Sagar  (V. Sagar)Department of Mathematics, IIT Delhi, New Delhi, India
110016<EMAIL_ADDRESS>and  Ritumoni Sarma  (R. Sarma)Department of
Mathematics, IIT Delhi, New Delhi, India 110016<EMAIL_ADDRESS>
###### Abstract.

Certain simplicial complexes are used to construct a subset $D$ of
$\mathbb{F}_{2^{n}}^{m}$ and $D$, in turn, defines the linear code $C_{D}$
over $\mathbb{F}_{2^{n}}$ that consists of $(v\cdot d)_{d\in D}$ for
$v\in\mathbb{F}_{2^{n}}^{m}$. Here we deal with the case $n=3$, that is, when
$C_{D}$ is an octanary code. We establish a relation between $C_{D}$ and its
binary subfield code $C_{D}^{(2)}$ with the help of a generator matrix. For a
given length and dimension, a code is called distance optimal if it has the
highest possible distance. With respect to the Griesmer bound, five infinite
families of distance optimal codes are obtained, and sufficient conditions for
certain linear codes to be minimal are established.

###### Key words and phrases:

octanary linear code, subfield code, Griesmer code, minimal code, simplicial
complex

###### 2010 Mathematics Subject Classification:

94B05, 94B60, 94B65, 05E45

##  1\. INTRODUCTION

Throughout the manuscript, $\mathbb{F}_{q}$ is the field with $q$ elements;
however, in this, we work over $\mathbb{F}_{2^{3}}$ and more generally
$\mathbb{F}_{2^{n}}$ for $n\in\mathbb{N}$. Given a code $C$ over
$\mathbb{F}_{q^{n}}$, the code $C^{(q)}$ over $\mathbb{F}_{q}$ is constructed
using the trace map in [5]. The code $C^{(q)}$ is referred to as subfield code
with respect to $C$. Hyun et al. [13] studied linear codes by choosing some
specific defining set obtained using simplicial complexes and produced certain
families of binary optimal linear codes. In [12], Hyun et al. used posets in
place of simplicial complexes to study linear codes. They constructed binary
linear codes that are minimal as well as optimal, but the Ashikhmin-Barg
condition [1] does not hold for them. In 2021, Zhu et al. studied codes over
$\mathbb{F}_{4}$ and produced minimal optimal codes. In [17], the authors
studied linear codes over $\mathbb{F}_{4}$ and the corresponding subfield
codes, and obtained their weight distributions. They used simplicial complexes
to construct defining set and produced two infinite families of optimal linear
codes. The weight distribution of a linear code is an important research topic
in coding theory because it can be used to calculate the error correcting
capability as well as the error probability of error detection and correction
[8, 6, 14]. The minimum Hamming distance of linear codes are well known for
their importance in determining error-correcting capacity. As a result,
finding optimal linear codes has become one of the central topics for
researchers. A lot of work has been done in the recent past. In [13], Hyun et
al. shows how optimal codes can be utilized for the construction of secret
sharing schemes having nice access structures following the framework
discussed in [19]. The secret sharing schemes are essential nowadays; in fact,
these are used heavily in electronic voting systems, cryptographic protocols,
banking systems etc. Minimal codes are helpful to construct the access
structure of the secret sharing schemes [8, 15].

The study of minimal codes is an active research topic nowadays because of its
applications in secure two-party computation and secret sharing schemes [2, 4,
19, 16]. This special class of codes is also important as these can be decoded
by using the minimum distance decoding rule [1]. Normally, it is difficult to
identify all the minimal codewords of a given code even over a finite field of
characteristic $2$. So researchers started looking into minimal codes.

Motivated by the above research, we in this manuscript, study octanary linear
codes whose defining set is obtained from certain simplicial complexes. We
also investigate binary subfield codes with respect to these codes and obtain
their weight distributions. If defining sets are properly chosen, these codes
are minimal and distance optimal. Up to some extent, we generalize these
results to codes over $\mathbb{F}_{2^{n}}$. We also propose a few conjectures
that generalizes results on octanary codes. We give a few examples that
support our results.

We arrange the remaining part of the manuscript as follows. Preliminaries are
presented in the next section. With the help of simplicial complexes, octanary
linear codes are investigated in section 3. We obtain sufficient conditions
for some of these codes to be minimal. In this section, we prove these codes
are distance optimal if the defining sets are chosen properly. Section 4
studies subfield codes. We prove these codes are Griesmer code and obtain
sufficient condition for these codes to be minimal. Weight distributions of
these codes are also obtained in this section. Section 5 concludes this
manuscript.

##  2\. Definitions and Preliminaries

Throughout the manuscript, for $m\in\mathbb{N}$, we shall write $[m]$ to
denote the set $\\{1,2,\dots,m\\}$. Suppose $C$ is an $[n,k,d]$-linear code
over $\mathbb{F}_{q}$, where $n,k,d\in\mathbb{N}$. Denote by $A_{i}$, the
cardinality of the set that contains all codewords of $C$ having (Hamming)
weight $i$, $0\leq i\leq n$. Then the string $(1,A_{1},\dots,A_{n})$ is called
the weight distribution of $C$. In addition, if the total number of $i\geq 1$
such that $A_{i}\neq 0$ is $l$, then $C$ is called an $l$-weight linear code.
The code $C$ is called distance optimal if there exist no $[n,k,d+1]$-linear
code (see[10]). Next we recall the Griesmer bound.

######  Lemma 2.1.

[9] (Griesmer Bound) If $C$ is an $[n,k,d]$-linear code over $\mathbb{F}_{q}$,
then we have

(2.1) $\sum\limits_{i=0}^{k-1}\left\lceil\frac{d}{q^{i}}\right\rceil\leq n,$

where $\lceil\cdot\rceil$ denotes the ceiling function.

A linear code is called a Griesmer code if equality holds in Equation 2.1.
Note that every Griesmer code is distance optimal.

###  2.1. Construction of linear codes

For any $n,m\in\mathbb{N}$, let $D=\\{d_{1}<d_{2}<\dots<d_{n}\\}$ be a subset
of $\mathbb{F}_{q}^{m}$ of cardinality $n$. Define [10],

(2.2) $C_{D}=\\{(v\cdot d_{1},v\cdot d_{2},\dots,v\cdot
d_{n}):v\in\mathbb{F}_{q}^{m}\\}$

where $x\cdot y=\sum\limits_{i=1}^{m}x_{i}y_{i}$ for
$x,y\in\mathbb{F}_{q}^{m}$.
The ordered set $D$ is termed as the defining set of $C_{D}$. Note that on
changing the ordering of $D$ we will get a permutation equivalent code. If $D$
is chosen appropriately, $C_{D}$ may possess good parameters. For
$w\in\mathbb{F}_{q}^{m}$, the set

$\textnormal{Supp}(w)=\\{i\in[m]:w_{i}\neq 0\\}$

is called the support of $w$. Note that the Hamming weight of
$w\in\mathbb{F}_{q}^{m}$ is $wt(w)=|\textnormal{Supp}(w)|$.

For $v,w\in\mathbb{F}_{q}^{m}$, one says that $v$ covers $w$ if
$\textnormal{Supp}(w)\subseteq\textnormal{Supp}(v)$. If $v$ covers $w$ we
write $w\preceq v$. Let $C$ be a linear code over $\mathbb{F}_{q}$. An element
$v_{0}\in C\setminus\\{0\\}$ is called minimal if $v\preceq v_{0}$ and $v\in
C\setminus\\{0\\}$ imply $v=\lambda v_{0}$ for some
$\lambda\in\mathbb{F}_{q}^{\ast}$. If every nonzero codeword of a code is a
minimal codeword then the code is termed as a minimal code. Now we recall a
result from [1] regarding a sufficient condition for a linear code to be
minimal.

######  Lemma 2.2.

[1](Ashikhmin-Barg) If the ratio of the minimum and the maximum (Hamming)
weights of nonzero codewords in a linear code $C$ over $\mathbb{F}_{q}$ is
larger than $\frac{q-1}{q}$, then $C$ is a minimal code.

Consider the map $\phi:\mathbb{F}_{2}^{m}\longrightarrow 2^{[m]}$ is defined
as $\phi(w)=\textnormal{Supp}(w)$, where $2^{[m]}$ denotes the power set of
$[m]$. Note that $\phi$ is a bijective map.
A subset $\Delta$ of $\mathbb{F}_{2}^{m}$ is called a simplicial complex if
$u\in\Delta$ then $w\in\mathbb{F}_{2}^{m}\implies w\in\Delta$ whenever
$w\preceq u$. An element $v\in\Delta$ is called maximal if $v\preceq w$ and
$w\in\Delta$ imply $v=w$. Let $L\subseteq[m]$. The simplicial complex
generated by $L$ is denoted by $\Delta_{L}$ and is defined as

(2.3) $\Delta_{L}=\\{w\in\mathbb{F}_{2}^{m}|\textnormal{ Supp}(w)\subseteq
L\\}.$

Note that $|\Delta_{L}|=|2^{L}|=2^{|L|}$.

Given a subset $P$ of $\mathbb{F}_{2}^{m}$, define the polynomial (referred to
as m-variable generating function, see [3])
$\mathcal{H}_{P}(y_{1},y_{2},\dots,y_{m})$ by

(2.4) $\mathcal{H}_{P}(y_{1},y_{2},\dots,y_{m})=\sum\limits_{v\in
P}\prod_{i=1}^{m}y_{i}^{v_{i}}\in\mathbb{Z}[y_{1},y_{2},\dots,y_{m}],$

where $v=(v_{1},\dots,v_{m})$.
We recall a lemma from [3].

######  Lemma 2.3.

[3] Suppose $\Delta\subset\mathbb{F}_{2}^{m}$ is a simplicial complex. If
$\mathcal{F}$ is the collection of all maximal elements of $\Delta$, then

(2.5)
$\mathcal{H}_{\Delta}(y_{1},y_{2},\dots,y_{m})=\sum\limits_{\emptyset\neq
S\subseteq\mathcal{F}}(-1)^{|S|+1}\prod_{i\in\cap S}(1+y_{i}),$

where $\cap S=\bigcap\limits_{F\in S}\textnormal{Supp}(F)$. In particular, we
have

$|\Delta|=\sum\limits_{\emptyset\neq
S\subseteq\mathcal{F}}(-1)^{|S|+1}2^{|\cap S|}.$

######  Example 2.4.

Consider the simplicial complex

$\Delta=\\{(0,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,0,1),(1,1,0,0),(0,1,0,1)\\}\subset\mathbb{F}_{2}^{4}.$

Then $\mathcal{F}=\\{F_{1},F_{2}\\}$, where $F_{1}=(1,1,0,0),F_{2}=(0,1,0,1)$.
So

$\begin{split}\mathcal{H}_{\Delta}(y_{1},y_{2},y_{3},y_{4})=&\prod_{i\in\textnormal{Supp}(F_{1})}(1+y_{i})+\prod_{i\in\textnormal{Supp}(F_{2})}(1+y_{i})-(1+y_{2})\\\
=&(1+y_{1})(1+y_{2})+(1+y_{2})(1+y_{4})-(1+y_{2})\\\
=&1+y_{1}+y_{2}+y_{4}+y_{1}y_{2}+y_{2}y_{4}\end{split}$

and $|\Delta|=6.$

###  2.2. Construction of subfield codes

Let $\mathbb{F}_{8}=\mathbb{F}_{2}(\omega)$, where $\omega\in\mathbb{F}_{8}$
satisfies the polynomial $y^{3}+y+1=0$. Now for each $v\in\mathbb{F}_{8}^{m}$,
where $m\in\mathbb{N}$, there exist unique $a,b,c\in\mathbb{F}_{2}^{m}$ such
that $v=a+\omega b+\omega^{2}c$. Suppose that $C$ is a $k$-dimensional code of
length $n$ over $\mathbb{F}_{q^{m}}$ generated by $G$ and $\mathcal{B}$ is an
ordered basis of $\mathbb{F}_{q^{m}}$ over $\mathbb{F}_{q}$. The code
$C^{(q)}$ over $\mathbb{F}_{q}$ generated by the matrix which is obtained by
replacing each entry of $G$ by its column representation (a column vector in
$\mathbb{F}_{q}^{m}$) with respect to $\mathcal{B}$, is termed as a subfield
code. Now we recall a result from [5].

######  Lemma 2.5.

[5] Suppose $\mathcal{B}=\\{v_{1}<v_{2}<\dots<v_{m}\\}$ is an ordered basis of
$\mathbb{F}_{q^{m}}$ over $\mathbb{F}_{q}$. Suppose that $C$ is a
$k$-dimensional linear code of length $n$ over $\mathbb{F}_{q^{m}}$ generated
by

$G=\begin{pmatrix}g_{11}&g_{12}&\dots&g_{1n}\\\ g_{21}&g_{22}&\dots&g_{2n}\\\
\vdots&\vdots&\ddots&\vdots\\\ g_{k1}&g_{k2}&\dots&g_{kn}\end{pmatrix}.$

Then the subfield code $C^{(q)}$ is a binary linear code generated by

$G^{(q)}=\begin{pmatrix}G_{1}^{(q)}\\\ G_{2}^{(q)}\\\ \vdots\\\
G_{k}^{(q)}\end{pmatrix},$

where $Tr_{q^{m}/q}:\mathbb{F}_{q^{m}}\longrightarrow\mathbb{F}_{q}$ is the
trace map and, for $1\leq i\leq k$

$G_{i}^{(q)}=\begin{pmatrix}Tr_{q^{m}/q}(g_{i1}v_{1})&Tr_{q^{m}/q}(g_{i2}v_{1})&\dots&Tr_{q^{m}/q}(g_{in}v_{1})\\\
Tr_{q^{m}/q}(g_{i1}v_{2})&Tr_{q^{m}/q}(g_{i2}v_{2})&\dots&Tr_{q^{m}/q}(g_{in}v_{2})\\\
\vdots&\vdots&\ddots&\vdots\\\
Tr_{q^{m}/q}(g_{i1}v_{m})&Tr_{q^{m}/q}(g_{i2}v_{m})&\dots&Tr_{q^{m}/q}(g_{in}v_{m})\end{pmatrix}.$

Set $m=3,q=2$ and fix the basis $\\{v_{1}=1,v_{2}=\omega,v_{3}=\omega^{2}\\}$
for $\mathbb{F}_{8}$ over $\mathbb{F}_{2}$ in Lemma 2.5. Let
$g_{ij}=g_{ij}^{(1)}+\omega
g_{ij}^{(2)}+\omega^{2}g_{ij}^{(3)}\in\mathbb{F}_{8}$, where
$g_{ij}^{(1)},g_{ij}^{(2)},g_{ij}^{(3)}\in\mathbb{F}_{2}$. Then we have,
$Tr_{8/2}(g_{ij}v_{1})=g_{ij}^{(1)}$, $Tr_{8/2}(g_{ij}v_{2})=g_{ij}^{(3)}$ and
$Tr_{8/2}(g_{ij}v_{3})=g_{ij}^{(2)}$. Now, we have the following result.

######  Theorem 2.6.

Let $\mathcal{B}=\\{1<\omega<\omega^{2}\\}$ be the ordered basis of
$\mathbb{F}_{8}$ over $\mathbb{F}_{2}$. Suppose $G:=G_{1}+\omega
G_{2}+\omega^{2}G_{3}\in M_{k\times n}(\mathbb{F}_{8})$ generates the linear
code $C$ over $\mathbb{F}_{8}$, where $G_{i}\in M_{k\times
n}(\mathbb{F}_{2})$, $1\leq i\leq 3$. Then the code $C^{(2)}$, with respect to
$\mathcal{B}$, is a linear code over $\mathbb{F}_{2}$ generated by

$G^{(2)}=\begin{pmatrix}G_{1}\\\ G_{3}\\\ G_{2}\end{pmatrix}.$

In addition, $C_{D}^{(2)}=C_{D^{(2)}}$ that is, if the defining set of the
code $C_{D}$ is $D=D_{1}+\omega
D_{2}+\omega^{2}D_{3}\subset\mathbb{F}_{8}^{m}$, where
$D_{i}\subset\mathbb{F}_{2}^{m},1\leq i\leq 3$, then the defining set of
$C_{D}^{(2)}$ is

$D^{(2)}=\\{(d_{1},d_{3},d_{2})|d_{i}\in D_{i},1\leq i\leq 3\\}.$

###### Proof.

Proof follows from the above discussion. ∎

The Theorem 2.6 is illustrated by the following example.

######  Example 2.7.

Consider the ordered basis $\\{1<\omega<\omega^{2}\\}$ of $\mathbb{F}_{8}$
over $\mathbb{F}_{2}$. Let $D_{1}=\\{(1,0)\\},D_{2}=\\{(0,1),(1,1)\\}$ and
$D_{3}=\\{(1,0),(0,1),(1,1)\\}$ so that $D=D_{1}+\omega
D_{2}+\omega^{2}D_{3}\subset\mathbb{F}_{8}^{2}$ is the defining set of the
code $C_{D}$. Consider the following generator matrix of $C_{D}$:

$\begin{split}G=&\begin{pmatrix}1&1&\omega^{2}&\omega^{2}&1+\omega+\omega^{2}&1+\omega+\omega^{2}\\\
0&1&1&\omega^{2}&1&1+\omega\end{pmatrix}\\\ =&\begin{pmatrix}1&1&0&0&1&1\\\
0&1&1&0&1&1\end{pmatrix}+\omega\begin{pmatrix}0&0&0&0&1&1\\\
0&0&0&0&0&1\end{pmatrix}+\omega^{2}\begin{pmatrix}0&0&1&1&1&1\\\
0&0&0&1&0&0\end{pmatrix}.\end{split}$

Then $C_{D}$ is a $[6,2,4]$-linear code over $\mathbb{F}_{8}$ and according to
Theorem 2.6, a generator matrix and a defining set of $C^{(2)}_{D}$ are given
by

$G^{(2)}=\begin{pmatrix}1&1&0&0&1&1\\\ 0&1&1&0&1&1\\\ 0&0&1&1&1&1\\\
0&0&0&1&0&0\\\ 0&0&0&0&1&1\\\ 0&0&0&0&0&1\end{pmatrix}$

and
$D^{(2)}=\\{(1,0,0,1,1,0),(1,0,0,1,0,1),(1,0,0,1,1,1),(1,0,1,1,1,0),(1,0,1,1,0,1),(1,0,1,1,1,1)\\}$,
respectively.

##  3\. Linear codes using simplicial complexes

This section studies linear codes over $\mathbb{F}_{2^{3}}$ and its
generalization to the field $\mathbb{F}_{2^{n}}$ with the help of simplicial
complexes.

Suppose that $\mathbb{F}_{8}=\mathbb{F}_{2}(\omega)$, where
$\omega\in\mathbb{F}_{8}$ satisfying $y^{3}+y+1=0$. Let $m\in\mathbb{N}$ and
let $D_{i}\subseteq\mathbb{F}_{2}^{m},1\leq i\leq 3$. Assume that
$D=D_{1}+\omega D_{2}+\omega^{2}D_{3}\subseteq\mathbb{F}_{8}^{m}$. Define an
octanary linear code as follows:

(3.1) $C_{D^{\ast}}=\\{(v\cdot d)_{d\in D^{\ast}}|v\in\mathbb{F}_{8}^{m}\\}.$

Observe that the map $c_{D^{\ast}}:\mathbb{F}^{m}_{2^{3}}\longrightarrow
C_{D^{\ast}}$ defined by $c_{D^{\ast}}(v)=(v\cdot d)_{d\in D^{\ast}}$ is a
surjective linear transformation. By definition, $|D^{\ast}|$ is the length of
$C_{D^{\ast}}$. Assume that
$v=\alpha+\omega\beta+\omega^{2}\gamma\in\mathbb{F}_{8}^{m}$ and
$d=d_{1}+\omega d_{2}+\omega^{2}d_{3}$, where
$\alpha,\beta,\gamma\in\mathbb{F}_{2}^{m}$ and $d_{i}\in D_{i},1\leq i\leq 3$.
Then we have

$\begin{split}wt(c_{D^{\ast}}(v))=&wt\big{(}((\alpha+\omega\beta+\omega^{2}\gamma)\cdot(d_{1}+\omega
d_{2}+\omega^{2}d_{3}))_{d_{1}\in D_{1},d_{2}\in D_{2},d_{3}\in
D_{3}}\big{)}\\\ =&wt\big{(}(\alpha\cdot d_{1}+\gamma\cdot d_{2}+\beta\cdot
d_{3})+\omega\big{(}\beta\cdot d_{1}+(\alpha+\gamma)\cdot
d_{2}+(\beta+\gamma)\cdot d_{3}\big{)}+\\\ &~{}\omega^{2}\big{(}\gamma\cdot
d_{1}+\beta\cdot d_{2}+(\alpha+\gamma)\cdot d_{3}\big{)}_{d_{1}\in
D_{1},d_{2}\in D_{2},d_{3}\in D_{3}}\big{)}.\end{split}$

Now if $u=u_{1}+\omega u_{2}+\omega^{2}u_{3}\in\mathbb{F}_{8}^{m}$ with
$u_{1},u_{2},u_{3}\in\mathbb{F}_{2}^{m}$, then $wt(u)=0\iff
u_{1}=u_{2}=u_{3}=0$. Hence,

$\begin{split}wt(c_{D^{\ast}}(v))=wt(c_{D}(v))=&|D|-\frac{1}{8}\sum\limits_{d_{1}\in
D_{1}}\sum\limits_{d_{2}\in D_{2}}\sum\limits_{d_{3}\in
D_{3}}\big{(}1+(-1)^{\alpha\cdot d_{1}+\gamma\cdot d_{2}+\beta\cdot
d_{3}}\big{)}\times\\\ &~{}\big{(}1+(-1)^{\beta\cdot
d_{1}+(\alpha+\gamma)\cdot d_{2}+(\beta+\gamma)\cdot
d_{3}}\big{)}\times\big{(}1+(-1)^{\gamma\cdot d_{1}+\beta\cdot
d_{2}+(\alpha+\gamma)\cdot d_{3}}\big{)}.\end{split}$

Define for $x\in\mathbb{F}_{2}^{m}$,

(3.2) $\chi_{x}:2^{\mathbb{F}^{m}_{2}}\longrightarrow\mathbb{Z}\textnormal{ by
}\chi_{x}(P)=\sum\limits_{y\in P}(-1)^{x\cdot y}.$

Then,

(3.3)
$\begin{split}wt(c_{D^{\ast}}(v))=&\frac{7}{8}|D|-\frac{1}{8}\big{\\{}\chi_{\alpha}(D_{1})\chi_{\gamma}(D_{2})\chi_{\beta}(D_{3})+\chi_{\beta}(D_{1})\chi_{\alpha+\gamma}(D_{2})\chi_{\beta+\gamma}(D_{3})+\\\
&~{}\chi_{\gamma}(D_{1})\chi_{\beta}(D_{2})\chi_{\alpha+\gamma}(D_{3})+\chi_{\alpha+\beta}(D_{1})\chi_{\alpha}(D_{2})\chi_{\gamma}(D_{3})+\\\
&~{}\chi_{\alpha+\gamma}(D_{1})\chi_{\beta+\gamma}(D_{2})\chi_{\alpha+\beta+\gamma}(D_{3})+\chi_{\beta+\gamma}(D_{1})\chi_{\alpha+\beta+\gamma}(D_{2})\chi_{\alpha+\beta}(D_{3})+\\\
&~{}\chi_{\alpha+\beta+\gamma}(D_{1})\chi_{\alpha+\beta}(D_{2})\chi_{\alpha}(D_{3})\big{\\}}.\end{split}$

Denoted by $\delta$ the Kronecker delta function and set
$D^{c}=\mathbb{F}_{8}^{m}\setminus D$. Then

(3.4)
$D^{c}=\big{(}D_{1}^{c}+\omega\mathbb{F}_{2}^{m}+\omega^{2}\mathbb{F}_{2}^{m}\big{)}\bigsqcup\big{(}D_{1}+\omega
D_{2}^{c}+\omega^{2}\mathbb{F}_{2}^{m}\big{)}\bigsqcup\big{(}D_{1}+\omega
D_{2}+\omega^{2}D_{3}^{c}\big{)},$

where $\bigsqcup$ indicates disjoint union.
By using the identity $\chi_{x}(\mathbb{F}_{2}^{m})=2^{m}\delta_{0,x}$ and
$\chi_{x}(P)+\chi_{x}(P^{c})=\chi_{x}(\mathbb{F}_{2}^{m})$ for
$x\in\mathbb{F}_{2}^{m},P\subset\mathbb{F}_{2}^{m}$, we get

(3.5)
$\begin{split}wt(c_{D^{c}}(v))=&\frac{7}{8}\big{(}|D^{c}|-2^{3m}\delta_{0,v}\big{)}+\frac{1}{8}\big{\\{}\chi_{\alpha}(D_{1})\chi_{\gamma}(D_{2})\chi_{\beta}(D_{3})+\chi_{\beta}(D_{1})\chi_{\alpha+\gamma}(D_{2})\chi_{\beta+\gamma}(D_{3})+\\\
&~{}\chi_{\gamma}(D_{1})\chi_{\beta}(D_{2})\chi_{\alpha+\gamma}(D_{3})+\chi_{\alpha+\beta}(D_{1})\chi_{\alpha}(D_{2})\chi_{\gamma}(D_{3})+\\\
&~{}\chi_{\alpha+\gamma}(D_{1})\chi_{\beta+\gamma}(D_{2})\chi_{\alpha+\beta+\gamma}(D_{3})+\chi_{\beta+\gamma}(D_{1})\chi_{\alpha+\beta+\gamma}(D_{2})\chi_{\alpha+\beta}(D_{3})+\\\
&~{}\chi_{\alpha+\beta+\gamma}(D_{1})\chi_{\alpha+\beta}(D_{2})\chi_{\alpha}(D_{3})\big{\\}}.\end{split}$

By Equation 3.3 and 3.5, we have the following lemma.

######  Lemma 3.1.

Let the symbols be same as above. For $v\in\mathbb{F}_{8}^{m}$, we have

(3.6) $\begin{split}wt(c_{D^{c}}(v))+wt(c_{D^{\ast}}(v))=7\times
2^{3(m-1)}\big{(}1-\delta_{0,v}\big{)}.\end{split}$

By a similar fashion, we can generalize the above relation for codes over
$\mathbb{F}_{2^{n}}$.
Let $\mathbb{F}_{2^{n}}=\mathbb{F}_{2}(\eta)$ be the finite field of order
$2^{n}$, where $\eta\in\mathbb{F}_{2^{n}}$. Consider the defining set
$D=D_{1}+\eta D_{2}+\cdots+\eta^{n-1}D_{n}\subset\mathbb{F}_{2^{n}}^{m}$,
where $m\in\mathbb{N}$ and $D_{i}\subset\mathbb{F}_{2}^{m}$ for $1\leq i\leq
n$. Then

(3.7)
$\begin{split}D^{c}=&\big{(}D_{1}^{c}+\eta\mathbb{F}_{2}^{m}+\cdots+\eta^{n-1}\mathbb{F}_{2}^{m}\big{)}\bigsqcup\\\
&\big{(}D_{1}+\eta
D_{2}^{c}+\eta^{2}\mathbb{F}_{2}^{m}\cdots+\eta^{n-1}\mathbb{F}_{2}^{m}\big{)}\bigsqcup\\\
&~{}\vdots\\\ &\big{(}D_{1}+\eta
D_{2}+\cdots+\eta^{n-2}D_{n-1}^{c}+\eta^{n-1}\mathbb{F}_{2}^{m}\big{)}\bigsqcup\\\
&\big{(}D_{1}+\eta
D_{2}+\cdots+\eta^{n-2}D_{n-1}+\eta^{n-1}D_{n}^{c}\big{)}.\end{split}$

Now we conjecture that the relation in Lemma 3.1 get extended for codes over
any finite field of characteristic $2$.

######  Conjecture 3.2.

Let the symbols be same as above. For $v\in\mathbb{F}_{2^{n}}^{m}$, we have

(3.8) $\begin{split}wt(c_{D^{c}}(v))+wt(c_{D^{\ast}}(v))=&(2^{n}-1)\times
2^{n(m-1)}\big{(}1-\delta_{0,v}\big{)}.\end{split}$

In the next result we find the parameters of a linear code over
$\mathbb{F}_{8}$.

######  Proposition 3.3.

Consider the defining set
$D=\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N}\subset\mathbb{F}_{8}^{m}$,
where $m\in\mathbb{N}$ and $L,M,N$ are nonempty subsets of $[m]$. Assume that
at least two of the sets $L\setminus(M\cup N),M\setminus(N\cup L),$ and
$N\setminus(L\cup M)$ are nonempty. Then the code $C_{D^{\ast}}$ is a
$3$-weight linear code over $\mathbb{F}_{8}$ of length $2^{|L|+|M|+|N|}-1$,
dimension $|L\cup M\cup N|$ and distance $2^{|L|+|M|+|N|-1}$. In particular,
$C_{D^{\ast}}$ have codewords of weights $0,2^{|L|+|M|+|N|-1},6\times
2^{|L|+|M|+|N|-3}$ and $7\times 2^{|L|+|M|+|N|-3}$. If
$(1,A_{1},A_{2},\dots,A_{|D^{\ast}|})$ is the weight distribution of
$C_{D^{\ast}}$ and
$Z_{i}=|\\{v\in\mathbb{F}_{2^{3}}^{m}:wt(c_{D^{\ast}}(v))=i\\}|$ for $0\leq
i\leq|D^{\ast}|$, then

$Z_{0}=2^{3(m-|L\cup M\cup N|)}\textnormal{ and }Z_{i}=Z_{0}A_{i}.$

###### Proof.

Observe the length of the code $C_{D^{\ast}}$ is $2^{|L|+|M|+|N|}-1$. Consider
the map $c_{D^{\ast}}:\mathbb{F}_{2^{3}}^{m}\longrightarrow C_{D^{\ast}}$
defined by $c_{D^{\ast}}(x)=(x\cdot d)_{d\in D^{\ast}}$ which is a surjective
linear transformation and

(3.9)
$\begin{split}\ker(c_{D^{\ast}})=&\\{v=(v_{i})\in\mathbb{F}_{2^{3}}^{m}:v\cdot
d=0~{}\forall~{}d\in D^{\ast}\\}\\\
=&\\{v=(v_{i})\in\mathbb{F}_{2^{3}}^{m}:v_{i}=0\textnormal{ for }i\in L\cup
M\cup N\\}.\end{split}$

Therefore, $|\ker(c_{D^{\ast}})|=2^{3({m-|L\cup M\cup N|})}=Z_{0}$. By the
first isomorphism theorem of groups, we have
$|C_{D^{\ast}}|=\frac{|\mathbb{F}_{2^{3}}^{m}|}{|\ker(c_{D^{\ast}})|}=2^{3|L\cup
M\cup N|}$. Hence $\dim(C_{D^{\ast}})=|L\cup M\cup N|$.
Note that $\textnormal{Supp}(d)\subseteq L\cup M\cup N$ for $d\in D$. Let
$Y_{i}=\\{c\in C_{D^{\ast}}:wt(c)=i\\}$ so that $|Y_{i}|=A_{i}$ and let
$X_{i}=\\{v\in\mathbb{F}_{2^{3}}^{m}:wt(c_{D^{\ast}}(v))=i\\}$ so that
$|X_{i}|=Z_{i}$ for $0\leq i\leq|D^{\ast}|$. Since
$Y_{i}=c_{D^{\ast}}(X_{i})$, we have
$X_{i}=c_{D^{\ast}}^{-1}(Y_{i})=\displaystyle{\bigsqcup_{c\in
Y_{i}}c^{-1}_{D^{\ast}}(c)}$. Let $c\in C_{D^{\ast}}$ then
$|c^{-1}_{D^{\ast}}(c)|=|c^{-1}_{D^{\ast}}(0)|=|\ker(c_{D^{\ast}})|=Z_{0}$.
Therefore $Z_{i}=|X_{i}|=\sum\limits_{c\in
Y_{i}}|c^{-1}_{D^{\ast}}(c)|=Z_{0}A_{i}$.
For $Y\subseteq[m]$, define the Boolean function
$\varphi(\cdot|Y):\mathbb{F}_{2}^{m}\longrightarrow\mathbb{F}_{2}$ as

$\varphi(x|Y)=\prod_{i\in Y}(1-x_{i})=\begin{cases}1,\text{ if }\textnormal{
Supp}(x)\cap Y=\emptyset;\\\ 0,\text{ if }\textnormal{ Supp}(x)\cap
Y\neq\emptyset.\end{cases}$

For $x=(x_{1},\dots,x_{m})\in\mathbb{F}_{2}^{m}$ and $\emptyset\neq
L\subset[m]$, we have

(3.10)
$\begin{split}\chi_{x}(\Delta_{L})=&\sum\limits_{y\in\Delta_{L}}(-1)^{x\cdot
y}=\mathcal{H}_{\Delta_{L}}((-1)^{x_{1}},\dots,(-1)^{x_{m}})\\\ =&\prod_{i\in
L}\big{(}1+(-1)^{x_{i}}\big{)}=\prod_{i\in L}(2-2x_{i})\\\
=&2^{|L|}\prod_{i\in L}(1-x_{i})=2^{|L|}\varphi(x|L).\end{split}$

Let $v=\alpha+\omega\beta+\omega^{2}\gamma\in\mathbb{F}_{8}^{m}$. By Equation
3.3 and 3.10, we have

$\begin{split}wt(c_{D^{\ast}}(v))=&wt(c_{D}(v))\\\
=&\frac{7}{8}|D|-2^{|L|+|M|+|N|-3}\big{\\{}\varphi(\alpha|L)\varphi(\gamma|M)\varphi(\beta|N)+\varphi(\beta|L)\varphi(\alpha+\gamma|M)\varphi(\beta+\gamma|N)+\\\
&\varphi(\gamma|L)\varphi(\beta|M)\varphi(\alpha+\gamma|N)+\varphi(\alpha+\beta|L)\varphi(\alpha|M)\varphi(\gamma|N)+\\\
&\varphi(\alpha+\gamma|L)\varphi(\beta+\gamma|M)\varphi(\alpha+\beta+\gamma|N)+\varphi(\beta+\gamma|L)\varphi(\alpha+\beta+\gamma|M)\varphi(\alpha+\beta|N)+\\\
&\varphi(\alpha+\beta+\gamma|L)\varphi(\alpha+\beta|M)\varphi(\alpha|N)\big{\\}}.\end{split}$

Let

(3.11)
$\begin{split}\theta=&\varphi(\alpha|L)\varphi(\gamma|M)\varphi(\beta|N)+\varphi(\beta|L)\varphi(\alpha+\gamma|M)\varphi(\beta+\gamma|N)+\varphi(\gamma|L)\varphi(\beta|M)\varphi(\alpha+\gamma|N)+\\\
&\varphi(\alpha+\beta|L)\varphi(\alpha|M)\varphi(\gamma|N)+\varphi(\alpha+\gamma|L)\varphi(\beta+\gamma|M)\varphi(\alpha+\beta+\gamma|N)+\\\
&\varphi(\beta+\gamma|L)\varphi(\alpha+\beta+\gamma|M)\varphi(\alpha+\beta|N)+\varphi(\alpha+\beta+\gamma|L)\varphi(\alpha+\beta|M)\varphi(\alpha|N).\end{split}$

Case: $\theta=7$
If $\theta=7$ then $wt(c_{D^{\ast}}(v))=0$ and
$\theta=7\iff\varphi(\alpha|L)=\varphi(\beta|L)=\varphi(\gamma|L)=\varphi(\alpha|M)=\varphi(\beta|M)=\varphi(\gamma|M)=\varphi(\alpha|N)=\varphi(\beta|N)=\varphi(\gamma|N)=1$.
Therefore, $\textnormal{Supp}(\alpha)\cap(L\cup M\cup
N)=\textnormal{Supp}(\beta)\cap(L\cup M\cup
N)=\textnormal{Supp}(\gamma)\cap(L\cup M\cup N)=\emptyset$.
Hence, the number of such $v=\alpha+\omega\beta+\omega^{2}\gamma$ is
$8^{m-|L\cup M\cup N|}$ and an example of such $v$ for which $\theta=7$ is
$v=(0,0,\dots,0)$.
Case: $\theta=6$
Let $\theta=6$. Then the Equation 3.11 holds if and only if exactly one of the
term in it is $0$ and rest are all equal to $1$. For example

$\begin{split}\varphi(\alpha|L)\varphi(\gamma|M)\varphi(\beta|N)&=0,\\\
\varphi(\beta|L)\varphi(\alpha+\gamma|M)\varphi(\beta+\gamma|N)&=1,\\\
\varphi(\gamma|L)\varphi(\beta|M)\varphi(\alpha+\gamma|N)&=1,\\\
\varphi(\alpha+\beta|L)\varphi(\alpha|M)\varphi(\gamma|N)&=1,\\\
\varphi(\alpha+\gamma|L)\varphi(\beta+\gamma|M)\varphi(\alpha+\beta+\gamma|N)&=1,\\\
\varphi(\beta+\gamma|L)\varphi(\alpha+\beta+\gamma|M)\varphi(\alpha+\beta|N)&=1,\\\
\varphi(\alpha+\beta+\gamma|L)\varphi(\alpha+\beta|M)\varphi(\alpha|N)&=1.\end{split}$

One can verify that these equations can not hold simultaneously. Similarly
other six possibilities are also not possible. So that the case $\theta=6$ is
rejected. The same argument can be used to reject the possibilities of
$\theta=5,4$ and $2$.
Case: $\theta=3$
If $\theta=3$ then we have

$\begin{split}wt(c_{D^{\ast}}(v))=&\frac{7}{8}|D|-\frac{3}{8}2^{|L|+|M|+|N|}\\\
=&2^{|L|+|M|+|N|-1}\end{split}$

and an example of such $v$ for which $\theta=3$ is $v=\alpha$ with
$\textnormal{Supp}(\alpha)=\\{i\\}$ where $i\in L\setminus(M\cup N)$.
Case: $\theta=1$
If $\theta=1$ then we have

$\begin{split}wt(c_{D^{\ast}}(v))=&\frac{7}{8}|D|-\frac{1}{8}2^{|L|+|M|+|N|}\\\
=&6\times 2^{|L|+|M|+|N|-3}\end{split}$

and an example of such $v$ for which $\theta=1$ is $v=\alpha$ with
$\textnormal{Supp}(\alpha)=\\{i,j\\}$ where $i\in L\setminus(M\cup N),j\in
M\setminus(N\cup L)$.
Case: $\theta=0$
If $\theta=0$ then we have $wt(c_{D^{\ast}}(v))=7\times 2^{|L|+|M|+|N|-3}$ and
an example of such $v$ for which $\theta=0$ is $v=\alpha=(1,1,\dots,1)$ where
$\textnormal{Supp}(\alpha)\cap L,\textnormal{Supp}(\alpha)\cap
M,\textnormal{Supp}(\alpha)\cap N$ are each nonempty. ∎

If all the three subsets $L,M$ and $N$ are nonempty and equal then we have the
following.

######  Theorem 3.4.

Let $\emptyset\neq L=M=N\subsetneq[m]$ and let the defining set
$D=\Delta_{L}+\omega\Delta_{L}+\omega^{2}\Delta_{L}\subset\mathbb{F}_{8}^{m}$,
where $m\in\mathbb{N}$. Then the code $C_{D^{\ast}}$ is a $1$-weight linear
code over $\mathbb{F}_{8}$ of length $2^{3|L|}-1$, dimension $|L|$ and
distance $7\times 2^{3(|L|-1)}$. In particular, it is a minimal code.
Moreover, $C_{D^{\ast}}$ is a Griesmer code and hence it is a distance
optimal. If $(1,A_{1},A_{2},\dots,A_{|D^{\ast}|})$ is the weight distribution
of $C_{D^{\ast}}$ and
$Z_{i}=|\\{v\in\mathbb{F}_{2^{3}}^{m}:wt(c_{D^{\ast}}(v))=i\\}|$ for $0\leq
i\leq|D^{\ast}|$, then

$Z_{0}=2^{3(m-|L|)}\textnormal{ and }Z_{i}=Z_{0}A_{i}.$

Proposition 3.3 and Theorem 3.4 are illustrated in the following examples.

######  Example 3.5.

Set $m=4$ and $L=\\{1,2\\},M=\\{2,3\\},N=\\{2\\}\subseteq[4]=\\{1,2,3,4\\}$
and consider the defining set
$D=\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N}\subset\mathbb{F}_{8}^{3}$,
where

$\begin{split}\Delta_{L}=&\\{(0,0,0,0),(1,0,0,0),(0,1,0,0),(1,1,0,0)\\},\\\
\Delta_{M}=&\\{(0,0,0,0),(0,1,0,0),(0,0,1,0),(0,1,1,0)\\},\\\
\Delta_{N}=&\\{(0,0,0,0),(0,1,0,0)\\}.\end{split}$

Note that $C_{D^{\ast}}$ is a $[31,3,16]$ linear code over $\mathbb{F}_{8}$
which is not optimal; in fact an optimal octanary linear code of length $31$
and dimension $3$ has distance $26$ (see [7]).

######  Example 3.6.

Set $m=3$ and $L=\\{1,2\\}=M=N$. Let
$D=\Delta_{L}+\omega\Delta_{L}+\omega^{2}\Delta_{L}\subset\mathbb{F}_{8}^{2}$,
where

$\begin{split}\Delta_{L}=&\\{(0,0,0),(1,0,0),(0,1,0),(1,1,0)\\}.\end{split}$

Note that $C_{D^{\ast}}$ is a $[63,2,56]$ linear code over $\mathbb{F}_{8}$
which is distance optimal from [7].

The following result gives parameters of codes over the field
$\mathbb{F}_{2^{n}}$.

######  Proposition 3.7.

Let $\mathbb{F}_{2^{n}}=\mathbb{F}_{2}(\eta)$, where
$\eta\in\mathbb{F}_{2^{n}}$. Consider the defining set
$D=\Delta_{M_{1}}+\eta\Delta_{M_{2}}+\cdots+\eta^{n-1}\Delta_{M_{n}}\subset\mathbb{F}_{2^{n}}^{m}$,
where $m\in\mathbb{N}$ and $\emptyset\neq M_{i}\subsetneq[m],1\leq i\leq n$.
Then the code $C_{D^{\ast}}$ is a
$[2^{\sum_{j=1}^{n}|M_{j}|}-1,|\cup_{j=1}^{n}M_{j}|]$ linear code over
$\mathbb{F}_{2^{n}}$. If $(1,A_{1},A_{2},\dots,A_{|D^{\ast}|})$ is the weight
distribution of $C_{D^{\ast}}$ and
$Z_{i}=|\\{v\in\mathbb{F}_{2^{n}}^{m}:wt(c_{D^{\ast}}(v))=i\\}|$, where $0\leq
i\leq|D^{\ast}|$, then

$Z_{0}=2^{n(m-|\cup_{j=1}^{n}M_{j}|)}\textnormal{ and
}Z_{i}=Z_{0}A_{i}\textnormal{ where }1\leq i\leq|D^{\ast}|.$

###### Proof.

Proof follows from the argument used in the proof of Proposition 3.3. ∎

Based on the computations done for codes over the fields
$\mathbb{F}_{2^{r}},r=1,2,3$ and $4$, we conjecture the following.

######  Conjecture 3.8.

Let $\emptyset\neq M_{i}=M\subsetneq[m],1\leq i\leq n$ and let the defining
set
$D=\Delta_{M}+\eta\Delta_{M}+\cdots+\eta^{n-1}\Delta_{M}\subset\mathbb{F}_{2^{n}}^{m}$,
where $m\in\mathbb{N}$. Then the code $C_{D^{\ast}}$ is a $1$-weight linear
code over $\mathbb{F}_{2^{n}}$ of length $2^{n|M|}-1$, dimension $|M|$ and
distance $(2^{n}-1)\times 2^{n(|M|-1)}$. In particular, it is a minimal code.
Moreover, $C_{D^{\ast}}$ is a Griesmer code. If
$(1,A_{1},A_{2},\dots,A_{|D^{\ast}|})$ is the weight distribution of
$C_{D^{\ast}}$ and
$Z_{i}=|\\{v\in\mathbb{F}_{2^{n}}^{m}:wt(c_{D^{\ast}}(v))=i\\}|$, where $0\leq
i\leq|D^{\ast}|$, then

$Z_{0}=2^{n(m-|M|)}\textnormal{ and }Z_{i}=Z_{0}A_{i}.$

In the next result, we find the parameters of a code whose defining set is the
complement of the defining set in Proposition 3.3.

######  Theorem 3.9.

Let $m\in\mathbb{N}$. Suppose that $L,M,N$ are nonempty subsets of $[m]$ such
that $L\cup M\cup N\subsetneq[m]$. Further, assume that at least two of the
sets $L\setminus(M\cup N),M\setminus(N\cup L),$ and $N\setminus(L\cup M)$ are
nonempty. Let
$D=\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N}\subset\mathbb{F}_{8}^{m}$
so that
$D^{c}=\big{(}\Delta_{L}^{c}+\omega\mathbb{F}_{2}^{m}+\omega^{2}\mathbb{F}_{2}^{m}\big{)}\bigsqcup\big{(}\Delta_{L}+\omega\Delta_{M}^{c}+\omega^{2}\mathbb{F}_{2}^{m}\big{)}\bigsqcup\big{(}\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N}^{c}\big{)}$.
Then $C_{D^{c}}$ is a $[2^{3m}-2^{|L|+|M|+|N|},m,7\times 2^{3(m-1)}-7\times
2^{|L|+|M|+|N|-3}]$ octanary $4$-weight linear code. In particular,
$C_{D^{c}}$ have codewords of weights $0,7\times 2^{3(m-1)}-7\times
2^{|L|+|M|+|N|-3},7\times 2^{3(m-1)}-6\times 2^{|L|+|M|+|N|-3},7\times
2^{3(m-1)}-4\times 2^{|L|+|M|+|N|-3}$ and $7\times 2^{3(m-1)}$. Moreover,
$C_{D^{c}}$ is a Griesmer code and hence it is a distance optimal. In fact, it
is a minimal code if $|L|+|M|+|N|\leq 3m-4$.

###### Proof.

The code $C_{D^{c}}$ is a $4$-weight octanary linear code with the required
parameters by using Lemma 3.1.
Now we have

$\begin{split}\sum\limits_{i=0}^{m-1}\left\lceil\frac{d}{q^{i}}\right\rceil=&\sum\limits_{i=0}^{m-1}\left\lceil\frac{7\times
2^{3m-3}-7\times 2^{|L|+|M|+|N|-3}}{8^{i}}\right\rceil\\\
=&\sum\limits_{i=0}^{m-1}\frac{7\times
2^{3m-3}}{8^{i}}-\sum\limits_{i=0}^{m-1}\left\lfloor\frac{7\times
2^{|L|+|M|+|N|-3}}{8^{i}}\right\rfloor\\\
=&7\times\big{(}2^{3m-3}+2^{3m-6}+\cdots+1\big{)}-\\\ &\big{(}7\times
2^{|L|+|M|+|N|-3}+7\times 2^{|L|+|M|+|N|-6}+\cdots+X+Y\big{)}\end{split}$

where,
if $|L|+|M|+|N|-3\equiv 0\textnormal{ mod }3$ then $X=7,Y=0$;
if $|L|+|M|+|N|-3\equiv 1\textnormal{ mod }3$ then $X=7\times 2,Y=1$;
if $|L|+|M|+|N|-3\equiv 2\textnormal{ mod }3$ then $X=7\times 2^{2},Y=3$.
Then

$\begin{split}\sum\limits_{i=0}^{m-1}\left\lceil\frac{d}{q^{i}}\right\rceil=&\big{(}2^{3m}-1\big{)}-\big{(}2^{|L|+|M|+|N|}-1\big{)}\\\
=&2^{3m}-2^{|L|+|M|+|N|}.\end{split}$

Hence $C_{D^{c}}$ is a Griesmer code. By Lemma 2.2, we have

$\begin{split}\frac{wt_{min}}{wt_{max}}=&\frac{7\times 2^{3m-3}-7\times
2^{|L|+|M|+|N|-3}}{7\times 2^{3m-3}}\\\ =&1-2^{|L|+|M|+|N|-3m}\end{split}$

and $1-2^{|L|+|M|+|N|-3m}>\frac{7}{8}\iff|L|+|M|+|N|\leq 3m-4$.
Hence, $C_{D^{c}}$ is minimal if $|L|+|M|+|N|\leq 3m-4$. ∎

If all the three subsets $L,M$ and $N$ of $[m]$ are nonempty and equal then we
have the following.

######  Theorem 3.10.

Let $\emptyset\neq L=M=N\subsetneq[m]$ and let the defining set
$D=\Delta_{L}+\omega\Delta_{L}+\omega^{2}\Delta_{L}\subset\mathbb{F}_{8}^{m}$,
where $m\in\mathbb{N}$. The code $C_{D^{c}}$ is a $2$-weight linear code over
$\mathbb{F}_{8}$ of length $2^{3m}-2^{3|L|}$, dimension $m$ and distance
$7\times 2^{3(m-1)}-7\times 2^{3(|L|-1)}$. In particular, $C_{D^{c}}$ have
codewords of weights $0,7\times 2^{3(m-1)}-7\times 2^{3(|L|-1)}$ and $7\times
2^{3(m-1)}$. Moreover, $C_{D^{c}}$ is a Griesmer code and hence it is a
distance optimal. Further, it is a minimal code if $3(m-|L|)-4\geq 0$.

Here are examples illustrating Theorem 3.9 and Theorem 3.10.

######  Example 3.11.

Set $m=4$ and $L=\\{1,2\\},M=\\{2,3\\},N=\\{2\\}\subseteq[4]=\\{1,2,3,4\\}$.
Let the defining set
$D=\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N}\subset\mathbb{F}_{8}^{4}$
so that
$D^{c}=\big{(}\Delta_{L}^{c}+\omega\mathbb{F}_{2}^{4}+\omega^{2}\mathbb{F}_{2}^{4}\big{)}\bigsqcup\big{(}\Delta_{L}+\omega\Delta_{M}^{c}+\omega^{2}\mathbb{F}_{2}^{4}\big{)}\bigsqcup\big{(}\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N}^{c}\big{)}$,
where

$\begin{split}\Delta^{c}_{L}=&\\{(0,0,1,0),(1,0,1,0),(0,1,1,0),(1,1,1,0),(0,0,0,1),(1,0,0,1),(0,1,0,1),(1,1,0,1),\\\
&~{}(0,0,1,1),(1,0,1,1),(0,1,1,1),(1,1,1,1)\\},\\\
\Delta^{c}_{M}=&\\{(1,0,0,0),(1,1,0,0),(1,0,1,0),(1,1,1,0),(0,0,0,1),(0,1,0,1),(0,0,1,1),(0,1,1,1),\\\
&~{}(1,0,0,1),(1,1,0,1),(1,0,1,1),(1,1,1,1)\\},\\\
\Delta^{c}_{N}=&\\{(1,0,0,0),(1,1,0,0),(0,0,1,0),(0,1,1,0),(0,0,0,1),(0,1,0,1),(1,0,1,0),(1,1,1,0),\\\
&~{}(1,0,0,1),(1,1,0,1),(0,0,1,1),(0,1,1,1),(1,0,1,1),(1,1,1,1)\\}.\end{split}$

Note that $C_{D^{c}}$ is a $[4064,4,3556]$ linear code over $\mathbb{F}_{8}$
and it is a $4$-weight linear code. In particular, $C_{D^{c}}$ have codewords
of weights $0,3556,3560,3568,3584$. Observe $C_{D^{c}}$ is a Griesmer code and
hence it is a distance optimal code. Since $\frac{3556}{3584}>\frac{7}{8}$, it
is a minimal code by using Lemma 2.2.

######  Example 3.12.

Set $m=4$ and $L=\\{1,2\\}=M=N$. Let
$D=\Delta_{L}+\omega\Delta_{L}+\omega^{2}\Delta_{L}\subset\mathbb{F}_{8}^{4}$
so that
$D^{c}=\big{(}\Delta_{L}^{c}+\omega\mathbb{F}_{2}^{4}+\omega^{2}\mathbb{F}_{2}^{4}\big{)}\bigsqcup\big{(}\Delta_{L}+\omega\Delta_{L}^{c}+\omega^{2}\mathbb{F}_{2}^{4}\big{)}\bigsqcup\big{(}\Delta_{L}+\omega\Delta_{L}+\omega^{2}\Delta_{L}^{c}\big{)}$,
where

$\begin{split}\Delta^{c}_{L}=&\\{(0,0,1,0),(1,0,1,0),(0,1,1,0),(1,1,1,0),(0,0,0,1),(1,0,0,1),(0,1,0,1),(1,1,0,1),\\\
&~{}(0,0,1,1),(1,0,1,1),(0,1,1,1),(1,1,1,1)\\}.\end{split}$

The code $C_{D^{c}}$ is a $[4032,4,3528]$ octanary $2$-weight linear code. In
particular, $C_{D^{c}}$ have codewords of weights $0,3528,3584$. Observe that
$C_{D^{c}}$ is a Griesmer code and hence it is a distance optimal code. Since
$\frac{3528}{3584}>\frac{7}{8}$, it is a minimal code by using Lemma 2.2.

######  Remark 3.13.

Let $\mathbb{F}_{2^{n}}=\mathbb{F}_{2}(\eta)$, where
$\eta\in\mathbb{F}_{2^{n}}$. Consider the defining set
$D=\Delta_{M_{1}}+\eta\Delta_{M_{2}}+\cdots+\eta^{n-1}\Delta_{M_{n}}\subset\mathbb{F}_{2^{n}}^{m}$,
where $m\in\mathbb{N}$ and $\emptyset\neq M_{i}\subsetneq[m],1\leq i\leq n$
such that $\cup_{i=1}^{n}M_{i}\subsetneq[m]$. The code $C_{D^{c}}$ is a linear
code of length $2^{nm}-2^{\sum_{j=1}^{n}|M_{j}|}$ over $\mathbb{F}_{2^{n}}$.

Based on the computations done for codes over the fields
$\mathbb{F}_{2^{r}},r=1,2,3$ ans $4$, we conjecture the following.

######  Conjecture 3.14.

Let $\mathbb{F}_{2^{n}}=\mathbb{F}_{2}(\eta)$, where
$\eta\in\mathbb{F}_{2^{n}}$. Consider the defining set
$D=\Delta_{M_{1}}+\eta\Delta_{M_{2}}+\cdots+\eta^{n-1}\Delta_{M_{n}}\subset\mathbb{F}_{2^{n}}^{m}$,
where $m\in\mathbb{N}$ and $\emptyset\neq M_{i}\subsetneq[m],1\leq i\leq n$
such that $\cup_{i=1}^{n}M_{i}\subsetneq[m]$. The code $C_{D^{c}}$ in remark
3.13 is an $m$-dimensional code with distance
$(2^{n}-1)\times\big{(}2^{n(m-1)}-2^{\sum_{j=1}^{n}|M_{j}|-n}\big{)}$ and it
is a $(t+1)$-weight linear code over $\mathbb{F}_{2^{n}}$ provided
$C_{D^{\ast}}$ is a $t$-weight linear code over $\mathbb{F}_{2^{n}}$.
Moreover, it is a Griesmer code. In fact, $C_{D^{c}}$ is a minimal code if
$\sum_{j=1}^{n}|M_{j}|\leq nm-(n+1)$.

##  4\. Subfield code with respect to a linear code

In this section, we study subfield codes over $\mathbb{F}_{2}$ with respect to
codes discussed earlier.

######  Theorem 4.1.

Let $L,M,N$ be nonempty subsets of $[m]$ and let
$D=\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N}\subset\mathbb{F}_{8}^{m}$
so that
$D^{(2)}=\\{(d_{1},d_{3},d_{2}):d_{1}\in\Delta_{L},d_{2}\in\Delta_{M},d_{3}\in\Delta_{N}\\}\subset(\mathbb{F}_{2}^{m})^{3}$.
Then the code $C^{(2)}_{D^{\ast}}$ is a
$[2^{|L|+|M|+|N|}-1,|L|+|M|+|N|,2^{|L|+|M|+|N|-1}]$ linear $1$-weight code
over $\mathbb{F}_{2}$ and its weight distribution is recorded in Table 1. In
particular, $C^{(2)}_{D^{\ast}}$ is a minimal code. Further, it is a Griesmer
code and hence it is a distance optimal.

Hamming weight | Number of codewords
---|---
$0$ | $1$
$2^{|L|+|M|+|N|-1}$ | $2^{|L|+|M|+|N|}-1$
Table 1. Weight distribution in Theorem 4.1

If
$Z_{i}=|\\{(\alpha,\beta,\gamma)\in(\mathbb{F}_{2}^{m})^{3}:wt(c^{(2)}_{D^{\ast}}(\alpha,\beta,\gamma))=i\\}|$,
where $i=0,2^{|L|+|M|+|N|-1}$, then

$Z_{0}=2^{3m-|L|-|M|-|N|}\textnormal{ and
}Z_{j}=2^{3m-|L|-|M|-|N|}\times\big{(}2^{|L|+|M|+|N|}-1\big{)}.$

###### Proof.

Note that the length of $C^{(2)}_{D^{\ast}}$ is
$|D^{\ast}|=|{D^{(2)}}^{\ast}|=2^{|L|+|M|+|N|}-1$. Observe that the map
$c^{(2)}_{D^{\ast}}:(\mathbb{F}_{2}^{m})^{3}\longrightarrow
C^{(2)}_{D^{\ast}}$ defined by $c^{(2)}_{D^{\ast}}(x)=\big{(}x\cdot
d\big{)}_{d\in D^{(2)^{\ast}}}$ is a surjective linear transformation. By
Theorem 2.6,

$C^{(2)}_{D^{\ast}}=\\{c^{(2)}_{D^{\ast}}(\alpha,\beta,\gamma)=\big{(}(\alpha,\beta,\gamma)\cdot(d_{1},d_{3},d_{2})\big{)}_{d_{1}\in\Delta_{L},d_{2}\in\Delta_{M},d_{3}\in\Delta_{N}}:\alpha,\beta,\gamma\in\mathbb{F}_{2}^{m}\\}.$

Now, we have

(4.1)
$\begin{split}wt\big{(}c^{(2)}_{D^{\ast}}(\alpha,\beta,\gamma)\big{)}=&wt\big{(}c^{(2)}_{D}(\alpha,\beta,\gamma)\big{)}\\\
=&|D|-\frac{1}{2}\sum\limits_{d_{1}\in\Delta_{L}}\sum\limits_{d_{2}\in\Delta_{M}}\sum\limits_{d_{3}\in\Delta_{N}}\big{(}1+(-1)^{\alpha
d_{1}+\gamma d_{2}+\beta d_{3}}\big{)}\\\
=&\frac{1}{2}|D|-\frac{1}{2}\sum\limits_{d_{1}\in\Delta_{L}}(-1)^{\alpha
d_{1}}\sum\limits_{d_{2}\in\Delta_{M}}(-1)^{\gamma
d_{2}}\sum\limits_{d_{3}\in\Delta_{N}}(-1)^{\beta d_{3}}\\\
=&2^{|L|+|M|+|N|-1}\big{(}1-\varphi(\alpha|L)\varphi(\gamma|M)\varphi(\beta|N)\big{)}.\end{split}$

Case: (1)
$wt(c^{(2)}_{D^{\ast}}(\alpha,\beta,\gamma))=0\iff\varphi(\alpha|L)\varphi(\gamma|M)\varphi(\beta|N)=1\iff\textnormal{
Supp}(\alpha)\cap L=\textnormal{Supp}(\gamma)\cap
M=\textnormal{Supp}(\beta)\cap N=\emptyset$. Hence, $Z_{0}=2^{m-|L|}\times
2^{m-|M|}\times 2^{m-|N|}=2^{3m-|L|-|M|-|N|}=|\ker(c^{(2)}_{D^{\ast}})|$. By
the first isomorphism theorem of groups, we have
$|C^{(2)}_{D^{\ast}}|=\frac{|(\mathbb{F}_{2}^{m})^{3}|}{|\ker(c^{(2)}_{D^{\ast}})|}=2^{|L|+|M|+|N|}$.
Hence $\dim(C^{(2)}_{D^{\ast}})=|L|+|M|+|N|$.
Case: (2)
$wt(c^{(2)}_{D^{\ast}}(\alpha,\beta,\gamma))=2^{|L|+|M|+|N|-1}\iff\varphi(\alpha|L)\varphi(\gamma|M)\varphi(\beta|N)=0$.
Therefore, $Z_{j}=2^{3m-|L|-|M|-|N|}\times\big{(}2^{|L|+|M|+|N|}-1\big{)}$,
where $j=2^{|L|+|M|+|N|-1}$.
Since $C_{D^{\ast}}^{(2)}$ is a $1$-weight code, it is a minimal code. Now we
have

$\begin{split}\sum\limits_{i=0}^{|L|+|M|+|N|-1}\left\lceil\frac{2^{|L|+|M|+|N|-1}}{2^{i}}\right\rceil=&\big{(}2^{|L|+|M|+|N|}-1\big{)}.\end{split}$

Therefore, $C^{(2)}_{D^{c}}$ is a Griesmer code. ∎

Here are examples illustrating Theorem 4.1.

######  Example 4.2.

Set $m=3$ and $L=\\{1,2\\},M=\\{2,3\\},N=\\{2\\}\subseteq[3]=\\{1,2,3\\}$. Let
the defining set
$D=\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N}\subset\mathbb{F}_{8}^{3}$
so that
$D^{(2)}=\\{(d_{1},d_{3},d_{2}):d_{1}\in\Delta_{L},d_{2}\in\Delta_{M},d_{3}\in\Delta_{N}\\}$,
where

$\begin{split}\Delta_{L}=&\\{(0,0,0),(1,0,0),(0,1,0),(1,1,0)\\},\\\
\Delta_{M}=&\\{(0,0,0),(0,1,0),(0,0,1),(0,1,1)\\},\\\
\Delta_{N}=&\\{(0,0,0),(0,1,0)\\}.\end{split}$

Note that $C^{(2)}_{D^{\ast}}$ is a $[31,5,16]$ linear $1$-weight code over
$\mathbb{F}_{2}$ which is distance optimal from [7]. Since
$C^{(2)}_{D^{\ast}}$ is a $1$-weight code, it is a minimal code.

######  Example 4.3.

Set $m=2$ and $L=\\{1\\},M=\\{2\\},N=\\{2\\}\subseteq[2]=\\{1,2\\}$. Let the
defining set
$D=\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N}\subset\mathbb{F}_{8}^{2}$
so that
$D^{(2)}=\\{(d_{1},d_{3},d_{2}):d_{1}\in\Delta_{L},d_{2}\in\Delta_{M},d_{3}\in\Delta_{N}\\}$,
where

$\begin{split}\Delta_{L}=&\\{(0,0),(1,0)\\},\\\
\Delta_{M}=&\\{(0,0),(0,1)\\},\\\ \Delta_{N}=&\\{(0,0),(0,1)\\}.\end{split}$

Note that $C^{(2)}_{D^{\ast}}$ is a $[7,3,4]$ linear $1$-weight code over
$\mathbb{F}_{2}$ which is distance optimal from [7]. Since
$C^{(2)}_{D^{\ast}}$ is a $1$-weight code, it is a minimal code.

The following result gives parameters of subfield codes with respect to linear
codes over $\mathbb{F}_{2^{n}}$.

######  Proposition 4.4.

Let $M_{i}$ be nonempty subsets of $[m]$ for $1\leq i\leq n$. Suppose
$D=\Delta_{M_{1}}+\eta\Delta_{M_{2}}+\cdots+\eta^{n-1}\Delta_{M_{n}}\subset\mathbb{F}_{2^{n}}^{m}$,
where $n,m\in\mathbb{N}$. Then the code $C^{(2)}_{D^{\ast}}$ is a $1$-weight
linear code over $\mathbb{F}_{2}$ of length $2^{\sum_{j=1}^{n}|M_{j}|}-1$ and
distance $2^{\sum_{j=1}^{n}|M_{j}|-1}$. In particular, $C^{(2)}_{D^{\ast}}$ is
a minimal code.

###### Proof.

Observe $|D^{\ast}|=|D^{(2)^{\ast}}|=2^{\sum_{j=1}^{n}|M_{j}|}-1$. Therefore
the length of the code $C^{(2)}_{D^{\ast}}$ with respect to the code in
Proposition 3.7 is $2^{\sum_{j=1}^{n}|M_{j}|}-1$. Consider the map
$c^{(2)}_{D^{\ast}}:(\mathbb{F}_{2}^{m})^{n}\longrightarrow
C^{(2)}_{D^{\ast}}$ defined by $c^{(2)}_{D^{\ast}}(x)=\big{(}x\cdot
d\big{)}_{d\in D^{(2)^{\ast}}}$, which is a surjective linear transformation.
Now,

$\begin{split}wt\big{(}c^{(2)}_{D^{\ast}}(x)\big{)}=&wt\big{(}c^{(2)}_{D}(x)\big{)}\\\
=&|D^{(2)}|-\frac{1}{2}\sum\limits_{d\in D^{(2)}}\big{(}1+(-1)^{x\cdot
d}\big{)}\\\ =&\frac{1}{2}|D^{(2)}|-\frac{1}{2}\sum\limits_{d\in
D^{(2)}}(-1)^{x\cdot d}\\\
=&2^{\sum_{j=1}^{n}|M_{j}|-1}\big{(}1-\theta\big{)}\textnormal{, where
}\theta\textnormal{ is a product of some Boolean functions}.\end{split}$

Case: $\theta=0$
$wt(c^{(2)}_{D^{\ast}}(x))=0$ if and only if $\theta=1$ and an example of such
$x$ for which $\theta=1$ is $x=(0,\dots,0)$. Case: $\theta=1$
$wt(c^{(2)}_{D^{\ast}}(x))=2^{\sum_{j=1}^{n}|M_{j}|-1}$ if and only if
$\theta=0$. Since $C^{(2)}_{D^{\ast}}\neq 0$, there exists $x$ for which
$\theta=0$. Thus $C_{D^{\ast}}^{(2)}$ is a $1$-weight code. ∎

Based on the computations done for subfield codes with respect to codes over
$\mathbb{F}_{2^{r}},r=1,2,3$ and $4$, we conjecture the following.

######  Conjecture 4.5.

Let $M_{i}$ be nonempty subsets of $[m]$ for $1\leq i\leq n$. Suppose
$D=\Delta_{M_{1}}+\eta\Delta_{M_{2}}+\cdots+\eta^{n-1}\Delta_{M_{n}}\subset\mathbb{F}_{2^{n}}^{m}$,
where $n,m\in\mathbb{N}$. Then the dimension of the code $C^{(2)}_{D^{\ast}}$
in Proposition 4.4 is $\sum_{j=1}^{n}|M_{j}|$. Moreover, $C^{(2)}_{D^{\ast}}$
is a Griesmer code. If
$Z_{i}=|\\{x\in(\mathbb{F}_{2}^{m})^{n}:wt(c^{(2)}_{D^{\ast}}(x))=i\\}|$ for
$i=0,2^{\sum_{j=1}^{n}|M_{j}|-1}$, then

$Z_{0}=2^{nm-\sum_{j=1}^{n}|M_{j}|}\textnormal{ and }Z_{j}=Z_{0}A_{j}.$

Suppose
$D=\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N}\subset\mathbb{F}_{8}^{m}$
then by using Equation 3.4, we have

$D^{c}=\big{(}\Delta_{L}^{c}+\omega\mathbb{F}_{2}^{m}+\omega^{2}\mathbb{F}_{2}^{m}\big{)}\bigsqcup\big{(}\Delta_{L}+\omega\Delta_{M}^{c}+\omega^{2}\mathbb{F}_{2}^{m}\big{)}\bigsqcup\big{(}\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N}^{c}\big{)},$

where $\sqcup$ denotes disjoint union.
If $(\alpha,\beta,\gamma)\in(\mathbb{F}_{2}^{m})^{3}$ then we have

$\begin{split}wt\big{(}c^{(2)}_{D^{c}}(\alpha,\beta,\gamma)\big{)}=&|D^{c}|-\frac{1}{2}\sum\limits_{d_{1}\in\Delta_{L}^{c}}\sum\limits_{d_{2}\in\mathbb{F}_{2}^{m}}\sum\limits_{d_{3}\in\mathbb{F}_{2}^{m}}\big{(}1+(-1)^{\alpha
d_{1}+\gamma d_{2}+\beta d_{3}}\big{)}\\\
&-\frac{1}{2}\sum\limits_{d_{1}\in\Delta_{L}}\sum\limits_{d_{2}\in\Delta_{M}^{c}}\sum\limits_{d_{3}\in\mathbb{F}_{2}^{m}}\big{(}1+(-1)^{\alpha
d_{1}+\gamma d_{2}+\beta d_{3}}\big{)}\\\
&-\frac{1}{2}\sum\limits_{d_{1}\in\Delta_{L}}\sum\limits_{d_{2}\in\Delta_{M}}\sum\limits_{d_{3}\in\Delta_{N}^{c}}\big{(}1+(-1)^{\alpha
d_{1}+\gamma d_{2}+\beta d_{3}}\big{)}\\\
=&\frac{1}{2}\big{(}|D^{c}|-2^{3m}\delta_{0,\alpha}\delta_{0,\beta}\delta_{0,\gamma}\big{)}+\frac{1}{2}\sum\limits_{d_{1}\in\Delta_{L}}(-1)^{\alpha
d_{1}}\sum\limits_{d_{2}\in\Delta_{M}}(-1)^{\gamma
d_{2}}\sum\limits_{d_{3}\in\Delta_{N}}(-1)^{\beta d_{3}}\\\
=&\frac{1}{2}\big{(}|D|+|D^{c}|-2^{3m}\delta_{0,\alpha}\delta_{0,\beta}\delta_{0,\gamma}\big{)}-wt\big{(}c^{(2)}_{D^{\ast}}(\alpha,\beta,\gamma)\big{)}\textnormal{,
By using Equation }\ref{weightCd2F8}\\\
=&2^{3m-1}\times\big{(}1-\delta_{0,\alpha}\delta_{0,\beta}\delta_{0,\gamma}\big{)}-wt\big{(}c^{(2)}_{D^{\ast}}(\alpha,\beta,\gamma)\big{)}.\end{split}$

######  Lemma 4.6.

Let the symbols be the same as above. For
$(\alpha,\beta,\gamma)\in(\mathbb{F}_{2}^{m})^{3}$, we have

(4.2)
$wt\big{(}c^{(2)}_{D^{c}}(\alpha,\beta,\gamma)\big{)}+wt\big{(}c^{(2)}_{D^{\ast}}(\alpha,\beta,\gamma)\big{)}=2^{3m-1}\times\big{(}1-\delta_{0,\alpha}\delta_{0,\beta}\delta_{0,\gamma}\big{)},$

where $\delta$ denotes the Kronecker delta function.

######  Theorem 4.7.

Suppose $L,M,N$ are nonempty subsets of $[m]$ such that at least one subset is
proper. Let
$D=\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N}\subset\mathbb{F}_{8}^{m}$.
Then the code $C_{D^{c}}^{(2)}$ is a
$[2^{3m}-2^{|L|+|M|+|N|},3m,2^{3m-1}-2^{|L|+|M|+|N|-1}]$ binary $2$-weight
linear code and its weight distribution is recorded in Table 2. Moreover, it
is a Griesmer code and hence it is distance optimal. Further, if
$|L|+|M|+|N|\leq 3m-2$ then $C_{D^{c}}^{(2)}$ is a minimal code.

Hamming weight | Number of codewords
---|---
$0$ | $1$
$2^{3m-1}-2^{|L|+|M|+|N|-1}$ | $2^{3m-|L|-|M|-|N|}\times\big{(}2^{|L|+|M|+|N|}-1\big{)}$
$2^{3m-1}$ | $2^{3m-|L|-|M|-|N|}-1$
Table 2. Weight distribution in Theorem 4.7

###### Proof.

By Theorem 2.6,
$C^{(2)}_{D^{c}}=\\{c^{(2)}_{D^{c}}(\alpha,\beta,\gamma)=((\alpha,\beta,\gamma)\cdot
d)_{d\in({D^{c}})^{(2)}}:\alpha,\beta,\gamma\in\mathbb{F}_{2}^{m}\\}$.
Clearly, the length of the code $C_{D^{c}}^{(2)}$ is
$|D^{c}|=2^{3m}-2^{|L|+|M|+|N|}$. By Lemma 4.6 and the weight distribution of
$C^{(2)}_{D^{\ast}}$, we have the weight distribution of $C^{(2)}_{D^{c}}$
which is recorded in Table 2.
Now we have

$\begin{split}\sum\limits_{i=0}^{3m-1}\left\lceil\frac{2^{3m-1}-2^{|L|+|M|+|N|-1}}{2^{i}}\right\rceil=&\sum\limits_{i=0}^{3m-1}\frac{2^{3m-1}}{2^{i}}-\sum\limits_{i=0}^{3m-1}\left\lfloor\frac{2^{|L|+|M|+|N|-1}}{2^{i}}\right\rfloor\\\
=&\big{(}2^{3m}-1\big{)}-\big{(}2^{|L|+|M|+|N|}-1\big{)}\\\
=&2^{3m}-2^{|L|+|M|+|N|}.\end{split}$

Therefore, $C^{(2)}_{D^{c}}$ is a Griesmer code. By Lemma 2.2, we have

$\begin{split}\frac{wt_{min}}{wt_{max}}=&\frac{2^{3m-1}-2^{|L|+|M|+|N|-1}}{2^{3m-1}}\\\
=&1-2^{|L|+|M|+|N|-3m}\end{split}$

and $1-2^{|L|+|M|+|N|-3m}>\frac{1}{2}\iff|L|+|M|+|N|\leq 3m-2$.
Hence, $C^{(2)}_{D^{c}}$ is minimal if $|L|+|M|+|N|\leq 3m-2$. ∎

Reference $q$-ary Result Defining set $[n,k,d]$-code #Weight Bound [17] 4
Proposition $4.2$ $\Delta_{A}+\omega\Delta_{B}$ $[2^{|A|+|B|}-1,|A\cup
B|,2^{|A|+|B|-1}]$ 2 Proposition $4.7$ $[(2^{|A|}+2^{|B|}-2^{|A\cap
B|})^{2}-1,|A\cup B|,d]$ $\leq 10$ Theorem $4.4$
$\mathbb{F}_{4}^{m}\setminus(\Delta_{A}+\omega\Delta_{B})$
$[4^{m}-2^{|A|+|B|},m,3\times 2^{2m-2}-3\times 2^{|A|+|B|-2}]$ 3 Griesmer
Theorem $4.10$ $[4^{m}-(2^{|A|}+2^{|B|}-2^{|A\cap B|})^{2},m]$ $\leq 11$ 2
Proposition $5.1$ $\Delta_{A}+\omega\Delta_{B}$
$[2^{|A|+|B|}-1,|A|+|B|,2^{|A|+|B|-1}]$ 1 Theorem $5.2$
$[2^{2m}-2^{|A|+|B|},2m,2^{2m-1}-2^{|A|+|B|-1}]$ $2$ Griesmer Proposition
$5.3$ $\Delta+\omega\Delta$ $[(2^{|A|}+2^{|B|}-2^{|A\cap B|})^{2}-1,2|A\cup
B|]$ $\leq 10$ Theorem $5.5$ $[(2^{|A|}+2^{|B|}-2^{|A\cap
B|})^{2}-1,(2^{|A|}+2^{|B|}-2^{|A\cap B|})^{2}-1-2|A\cup B|,3]$ Sphere packing
Proposition $5.6$ $[4^{m}-(2^{|A|}+2^{|B|}-2^{|A\cap B|})^{2},2m]$ $\leq 11$
[20] 4 Theorem $3.1$
$\mathbb{F}_{4}^{m}\setminus(\Delta_{A}+\omega\Delta_{B})$
$[(2^{m}-2^{|A|})2^{|B|},m,3(2^{m+|B|-2}-2^{|A|+|B|-2})]$ 5 Corollary $3.2$
$\mathbb{F}_{2}^{m}+\omega\Delta_{B}$ $[2^{m+|B|},m,2^{m+|B|-1}]$ 2 Corollary
$3.3$ $\mathbb{F}_{2}^{m}\setminus\Delta_{A}+\omega\mathbb{F}_{2}^{m}$
$[(2^{m}-2^{|A|})2^{m},m,3(2^{2m-2}-2^{|A|+m-2})]$ 2 Griesmer [13] 2 Lemma $7$
$\Delta^{\ast}$ $[2^{|A|}-1,|A|,2^{|A|-1}]$ 1 Griesmer
$\mathbb{F}_{2}^{m}\setminus\Delta$ $[2^{m}-2^{|A|},m,2^{m-1}-2^{|A|-1}]$ 2
Griesmer Ex. $10$ $[2^{m-1},m,4]$ Sphere Packing Corollary $21$
$[2^{m}-2\sum_{i=1}^{s}2^{|A_{i}|-1}+s-1,m,2^{m-1}-\sum_{i=1}^{s}2^{|A_{i}|-1}]$
Griesmer Lemma $26$ $\Delta^{\ast}$ $[2^{|A_{1}|}+2^{|A_{2}|}-2,|A_{1}\cup
A_{2}|,2^{|A_{1}|-1}]$ 3 $[2^{|A_{1}|}+2^{|A_{2}|}-2^{|A_{1}\cap
A_{2}|}-1,|A_{1}\cup A_{2}|,2^{|A_{1}|-1}]$ 4 Theorem $27$
$\mathbb{F}_{2}^{m}\setminus\Delta$
$[2^{m}-2^{|A_{1}|}-2^{|A_{2}|}+1,m,2^{m-1}-2^{|A_{1}|-1}2^{|A_{2}|-1}]$ 3 or
4 Griesmer $[2^{m}-2^{|A_{1}|}-2^{|A_{2}|}+2^{|A_{1}\cap
A_{2}|}-1,m,2^{m-1}-2^{|A_{1}|-1}-2^{|A_{2}|-1}]$ 4 or 5 [18] 2 Theorem $5$
$\Delta_{A}\setminus\Delta_{B}$ $[2^{|A|}-2^{|B|},|A|,2^{|A|-1}-2^{|B|-1}]$ 2
Griesmer Theorem $6$ $[2^{|A|}-2^{|B|},2^{|A|}-2^{|B|}-|A|,3\text{ or }4]$
[11] $p$ Theorem $4.1$ $\mathbb{F}_{p}^{m}\setminus\Delta$
$[p^{m}-r-1,m,(p-1)p^{m-1}-r]$ 2 Griesmer Theorem $4.4$
$[p^{m}-2r-2,m,(p-1)p^{m-1}-2r-1]$ 4 Griesmer Theorem $4.7$
$[p^{m}-3(r+1),m,(p-1)p^{m-1}-3r-2]$ 5 Griesmer Theorem $4.11$
$[p^{m}-(r+1)(p-1),m,(p-1)p^{m-1}-(r+1)p+2r+1]$ 4 Griesmer Theorem $4.14$
$[p^{m}-(r+1)(p-2),m,(p-1)p^{m-1}-(r+1)p+3r+1]$ 5 Griesmer

Table 3. Linear codes from Simplicial complexes before this article

$q$-ary Result Defining set $[n,k,d]$-code #Weight Distance optimal Minimal 8
Proposition 3.3
$(\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N})\setminus\\{0\\}$
$[2^{|L|+|M|+|N|}-1,|L\cup M\cup N|,2^{|L|+|M|+|N|-1}]$ 3 Theorem 3.4
$(\Delta_{L}+\omega\Delta_{L}+\omega^{2}\Delta_{L})\setminus\\{0\\}$
$[2^{3|L|}-1,|L|,7\times 2^{3(|L|-1)}]$ 1 Yes Yes Theorem 3.9
$\mathbb{F}_{8}^{m}\setminus(\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N})$
$[2^{3m}-2^{|L|+|M|+|N|},m,7\times 2^{3(m-1)}-7\times 2^{|L|+|M|+|N|-3}]$ 4
Yes Yes, if $|L|+|M|+|N|\leq 3m-4$ Theorem 3.10
$\mathbb{F}_{8}^{m}\setminus(\Delta_{L}+\omega\Delta_{L}+\omega^{2}\Delta_{L})$
$[2^{3m}-2^{3|L|},m,7\times 2^{3(m-1)}-7\times 2^{3(|L|-1)}]$ 2 Yes Yes, if
$3(m-|L|)-4\geq 0$ 2 Theorem 4.1
$(\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N})\setminus\\{0\\}$
$[2^{|L|+|M|+|N|}-1,|L|+|M|+|N|,2^{|L|+|M|+|N|-1}]$ 1 Yes Yes Theorem 4.7
${\mathbb{F}_{8}^{m}\setminus(\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N})}$
$[2^{3m}-2^{|L|+|M|+|N|},3m,2^{3m-1}-2^{|L|+|M|+|N|-1}]$ 2 Yes Yes, if
$|L|+|M|+|N|\leq 3m-2$

Table 4. Linear codes from Simplicial complexes in this article

Here are examples illustrating Theorem 4.7.

######  Example 4.8.

Set $m=3$ and $L=\\{1,2\\},M=\\{2,3\\},N=\\{2\\}\subseteq[3]=\\{1,2,3\\}$. Let
the defining set
$D=\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N}\subset\mathbb{F}_{8}^{3}$
so that
$D^{c}=\big{(}\Delta_{L}^{c}+\omega\mathbb{F}_{2}^{3}+\omega^{2}\mathbb{F}_{2}^{3}\big{)}\bigsqcup\big{(}\Delta_{L}+\omega\Delta_{M}^{c}+\omega^{2}\mathbb{F}_{2}^{3}\big{)}\bigsqcup\big{(}\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N}^{c}\big{)}$,
where

$\begin{split}\Delta_{L}^{c}=&\\{(0,0,1),(1,0,1),(0,1,1),(1,1,1)\\},\\\
\Delta_{M}^{c}=&\\{(1,0,0),(1,1,0),(1,0,1),(1,1,1)\\},\\\
\Delta_{N}^{c}=&\\{(1,0,0),(1,1,0),(0,0,1),(0,1,1),(1,0,1),(1,1,1)\\}.\end{split}$

Note that $C^{(2)}_{D^{c}}$ is a $[480,9,240]$ linear code over
$\mathbb{F}_{2}$ and it is a $2$-weight linear code. In particular,
$C^{(2)}_{D^{c}}$ have codewords of weights $0,240,256$. Observe
$C^{(2)}_{D^{c}}$ is a Griesmer code and hence it is a distance optimal code.
Since $\frac{240}{256}>\frac{1}{2}$, it is a minimal code by using Lemma 2.2.

######  Example 4.9.

Set $m=2$ and $L=\\{1\\},M=\\{2\\},N=\\{2\\}\subseteq[2]$. Let the defining
set
$D=\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N}\subset\mathbb{F}_{8}^{2}$
so that
$D^{c}=\big{(}\Delta_{L}^{c}+\omega\mathbb{F}_{2}^{2}+\omega^{2}\mathbb{F}_{2}^{2}\big{)}\bigsqcup\big{(}\Delta_{L}+\omega\Delta_{M}^{c}+\omega^{2}\mathbb{F}_{2}^{2}\big{)}\bigsqcup\big{(}\Delta_{L}+\omega\Delta_{M}+\omega^{2}\Delta_{N}^{c}\big{)}$,
where

$\begin{split}\Delta_{L}^{c}=&\\{(0,1),(1,1)\\},\\\
\Delta_{M}^{c}=&\\{(1,0),(1,1)\\},\\\
\Delta_{N}^{c}=&\\{(1,0),(1,1)\\}.\end{split}$

Note that $C^{(2)}_{D^{c}}$ is a $[56,6,28]$ linear code over $\mathbb{F}_{2}$
and it is a $2$-weight linear code. In particular, $C^{(2)}_{D^{c}}$ have
codewords of weights $0,28,32$. The code $C^{(2)}_{D^{c}}$ is optimal from
[7]. Since $\frac{28}{32}>\frac{1}{2}$, it is a minimal code by using Lemma
2.2.

Now we conjecture that the relation in Lemma 4.6 get extended for subfield
codes with respect to codes over any finite field of characteristic $2$.

######  Conjecture 4.10.

With the notations as above. For
$x=(x_{1},\dots,x_{n})\in(\mathbb{F}_{2}^{m})^{n}$, we have

(4.3)
$\begin{split}wt\big{(}c^{(2)}_{D^{c}}(x)\big{)}+wt\big{(}c^{(2)}_{D^{\ast}}(x)\big{)}=2^{nm-1}\times\big{(}1-\delta_{0,x_{1}}\delta_{0,x_{2}}\cdots\delta_{0,x_{n}}\big{)}.\end{split}$

######  Remark 4.11.

Let $\mathbb{F}_{2^{n}}=\mathbb{F}_{2}(\eta)$, where
$\eta\in\mathbb{F}_{2^{n}}$. Consider the defining set
$D=\Delta_{M_{1}}+\eta\Delta_{M_{2}}+\cdots+\eta^{n-1}\Delta_{M_{n}}\subset\mathbb{F}_{2^{n}}^{m}$,
where $m\in\mathbb{N}$ and $\emptyset\neq M_{i}\subset[m],1\leq i\leq n$. Let
$M_{i}\subsetneq$ for some $i,1\leq i\leq n$. The code $C^{(2)}_{D^{c}}$ is a
linear code of length $2^{nm}-2^{\sum_{j=1}^{n}|M_{j}|}$ over
$\mathbb{F}_{2}$.

Based on the computations done for subfield codes with respect to codes over
$\mathbb{F}_{2^{r}},r=1,2,3$ and $4$, we conjecture the following.

######  Conjecture 4.12.

With the notations as above, the code $C^{(2)}_{D^{c}}$ in remark 4.11 is an
$nm$-dimensional code with distance $2^{nm-1}-2^{\sum_{j=1}^{n}|M_{j}|-1}$
over $\mathbb{F}_{2}$ and it is $2$-weight linear code. Moreover,
$C^{(2)}_{D^{c}}$ is a Griesmer code. Further, it is a minimal code if
$\sum_{j=1}^{n}|M_{j}|\leq nm-2$.

##  5\. Conclusion

In this manuscript, we used simplicial complexes to construct linear codes
over $\mathbb{F}_{2^{3}}$. We study the algebraic structure of these codes and
the corresponding subfield codes over $\mathbb{F}_{2}$. We produce five
infinite families of linear codes that are distance optimal. The weight
distributions are obtained for the binary codes considered in this manuscript,
and boolean functions are used in these computations. Moreover, we obtain
sufficient conditions for some of these codes to be minimal. We give a few
examples to illustrate our results. Further, we partially extend these results
to codes over any finite field of characteristic $2$, and propose a few
conjectures. For the reader’s convenience, we include lists of recent works on
linear codes with the help of simplicial complexes and our work, respectively,
in Table 3 and 4.

In future, apart from proving the conjectures, one can find weight
distributions of octanary codes considered in this manuscript by devising an
elegant method.

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