# Minimal and Optimal binary codes obtained using $C_{D}$-construction over
the non-unital ring $I$

Vidya Sagar111email<EMAIL_ADDRESS>ID and Ritumoni Sarma222email:
<EMAIL_ADDRESS>ID
Department of Mathematics,
Indian Institute of Technology Delhi,
Hauz Khas, New Delhi-110016, India

###### Abstract

In this article, we construct linear codes over the commutative non-unital
ring $I$ of size four. We obtain their Lee-weight distributions and study
their binary Gray images. Under certain mild conditions, these classes of
binary codes are minimal and self-orthogonal. All codes in this article are
few-weight codes. Besides, an infinite class of these binary codes consists of
distance optimal codes with respect to the Griesmer bound.

Keywords: few-weight code, minimal code, optimal code, self-orthogonal code,
non-unital ring, simplicial complex

2020 Mathematics Subject Classification: Primary 94 B05 $\cdot$ Secondary 16
L30, 05 E45

##  1 INTRODUCTION

Codes over rings of size four, namely, $\mathbb{F}_{2}\times\mathbb{F}_{2}$
(in [14]), $\mathbb{F}_{2}+u\mathbb{F}_{2}$ (in [15]), $\mathbb{Z}_{4}$ (in
[21]) and $\mathbb{F}_{4}$ (in [29]), have got huge attention in the area of
algebraic coding theory in the past. These are only commutative unital rings
among the 11 rings of size four classified by Fine in [17]. Out of the 11
rings, $5$ of them are commutative non-unital (denoted as $B$, $C$, $H$, $I$
and $J$) and $2$ of them are non-commutative non-unital (denoted as $E$ and
$F$).

In 2021, Alahmadi et. al. [3] studied algebraic structure of linear codes over
the ring $I$ (see [17]) by introducing the notion of QSD codes. After that the
authors in [2] studied a construction of self-orthogonal codes over $I$, and
classified Type IV codes and quasi Type IV codes over $I$ up to length $n=6$.
Motivated by these works, Kim and Roe in [26] further investigated QSD codes,
Type IV codes and quasi Type IV codes over $I$ and extended the previous work
for lengths $n=7$ and $8$. Since then codes over non-unital rings have been
studied by many researchers (for instance, [1, 4, 25, 39, 41]).

In this article, we study the linear $I$-code $C_{D}=\\{(v\cdot d)_{d\in
D}:v\in I^{m}\\}$, where $D\subseteq I^{m}$ is an ordered finite multiset,
called the defining set for $C_{D}$. In [12], the authors introduced the
construction of $C_{D}$ in order to generalize the Mattson-Solomon transform
for cyclic codes. Study of $C_{D}$ becomes convenient, if the defining set $D$
is constructed via simplicial complexes. In fact, several interesting linear
codes have been constructed in the recent past by using simplicial complexes
(for instance, [22, 23, 32, 34, 35, 36, 37, 38, 40, 42, 46, 49]). It is
expected that for properly chosen finite fields (more generally, rings) and
defining sets, we may be able to discover codes with good parameters using the
above construction.

Yansheng et. al. in [45] studied linear codes over $\mathbb{F}_{4}$ and their
binary subfield codes, and could produce two infinite families of optimal
linear codes. Motivated by this work, the authors in [31] studied octanary
linear codes using simplicial complexes, and could obtain minimal and optimal
codes. After that the authors in [28] generalized the work of [31] for finite
fields of characteristic $2$ by using LFSR sequences. Recently, the authors in
[33] studied algebraic structure of linear codes over the ring $E$ (see [17])
using simplicial complexes for the first time and obtained several classes of
minimal codes, self-orthogonal codes and distance optimal codes.

The weight distribution (see [24]) of a linear code contains crucial
information regarding error detecting as well as error correcting capacity of
the code, and it allows the computation of the error probability of error
detection and correction with respect to certain algorithms [27]. Few-weight
codes are useful because of their connection with strongly regular graphs,
difference sets and finite geometry [8, 11]. 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 of research. In [23], the authors showed how optimal
codes can be utilized for the construction of secret sharing schemes with nice
access structures following the framework discussed in [47]. Minimal codes are
useful to construct the access structure of the secret sharing schemes [13,
30]. This special class of codes is also important as these can be decoded by
using the minimum distance decoding rule [5]. Normally, it is difficult to
identify all the minimal codewords of a given code even over a finite field of
characteristic $2$. In view of this, researchers began to investigate minimal
codes. Self-orthogonal codes are also essential class of codes. For instance,
they can be utilized for the construction of quantum error-correcting codes
[6, 9]. For the construction of self-orthogonal codes we refer to [16, 18, 43,
44, 48].

Inspired by the work of [45], a natural endeavour is to explore the
construction of linear codes over non-unital rings with the help of simplicial
complexes and investigate the structure of these codes. We consider the
commutative non-unital ring $I$ according to the notation of Fine [17]. We, in
this article, choose a defining set $D$ with the help of certain simplicial
complexes, and study the linear codes $C_{D}$ over $I$. We obtain the Lee-
weight distributions for these codes by using Boolean functions. By
considering a Gray map, we obtain certain binary linear codes, and their
weight distributions. Most of these binary codes turn out to be self-
orthogonal. By choosing the defining set appropriately, we produce an infinite
family of optimal codes. Moreover, most of the binary codes obtained here are
minimal. We give a few examples to illustrate our results.

The remaining sections of this article are arranged as follows. Preliminaries
are presented in the next section. By using simplicial complexes, we construct
linear $I$-codes and obtain their Lee-weight distributions in Section 3.
Section 4 studies Gray images of these linear $I$-codes. In this section, we
obtain a class of distance optimal codes and establish certain conditions
under which the Gray images of linear $I$-codes are minimal and self-
orthogonal. In Section 5, we conclude this article.

##  2 Definitions and Preliminaries

Suppose $I$ is the ring given by $I=\langle
a,b~{}|~{}2a=0,2b=0,a^{2}=b,ab=0\rangle$ (notation as in [17]). Suppose the
underlying set of $I$ is $\\{0,a,b,c\\}$. Then the addition and multiplication
tables of $I$ are given by Table 1 and 2 respectively.

$+$ | 0 | a | b | c
---|---|---|---|---
0 | 0 | a | b | c
a | a | 0 | c | b
b | b | c | 0 | a
c | c | b | a | 0
Table 1: Addition table of the ring $I$ $\times$ | 0 | a | b | c
---|---|---|---|---
0 | 0 | 0 | 0 | 0
a | 0 | b | 0 | b
b | 0 | 0 | 0 | 0
c | 0 | b | 0 | b
Table 2: Multiplication table of the ring $I$

It is easy to observe from the multiplication table that $I$ is commutative
and non-unital. Consider the following action of $\mathbb{F}_{2}$ on $I$:
$x0=0x=0$ and $x1=1x=x$ for all $x\in I$. Then every element of $I$ can be
expressed as $as+bt$ for $s,t\in\mathbb{F}_{2}$ (see [3]).

######  Lemma 2.1.

[3] For any $n\in\mathbb{N}$, $I^{n}=a\mathbb{F}_{2}^{n}+b\mathbb{F}_{2}^{n}$
and the sum is direct.

Suppose $\Phi:I\longrightarrow\mathbb{F}_{2}^{2}$ is the Gray map given by

$\Phi(as+bt)=(t,s+t).$ (2.1)

This extends to a map from $I^{m}$ to $(\mathbb{F}_{2}^{m})^{2}$ component-
wise for any $m\in\mathbb{N}$.

######  Definition 2.2.

An $I$-submodule $C$ of $I^{m}$ is called a linear $I$-code of length $m$.

Let $v,w\in\mathbb{F}_{2}^{m}$. Then the Hamming weight of $v$ denoted by
$wt_{H}(v)$ is the number of non-zero entries in $v$. The Hamming distance
between $v$ and $w$ is $d_{H}(v,w)=wt_{H}(v-w)$.
Let $x=a\alpha+b\beta$, $y=a\alpha^{\prime}+b\beta^{\prime}\in I^{m}$, where
$\alpha,\beta,\alpha^{\prime},\beta^{\prime}\in\mathbb{F}_{2}^{m}$. Then the
Lee weight of $x$ is
$wt_{Lee}(x)=wt_{H}(\Phi(x))=wt_{H}(\beta)+wt_{H}(\alpha+\beta)$. The Lee
distance between $x$ and $y$ is $d_{Lee}(x,y)=wt_{Lee}(x-y)$ so that $\Phi$ is
an isometry. Note that the image of a linear $I$-code under the above Gray map
is a binary linear code. Suppose $C$ is a linear $I$-code of length $m$. Let
$A_{i}$ be the cardinality of the set that contains all codewords of $C$
having Lee weight $i$, $0\leq i\leq 2m$. Then the homogeneous polynomial in
two variables

$Lee_{C}(X,Y)=\sum_{c\in C}X^{2m-wt_{Lee}(c)}Y^{wt_{Lee}(c)}$

is called the Lee weight enumerator of $C$ and the string
$(1,A_{1},\dots,A_{2m})$ is called the Lee weight distribution of $C$. In a
similar way, we can define Hamming weight enumerator and Hamming weight
distribution of a linear code over a finite field. 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. Every $1$-weight linear code is an equidistant linear
code. Bonisoli characterized all equidistant linear codes over finite fields
in [7].

######  Theorem 2.3.

[7](Bonisoli) Suppose $C$ is a equidistant linear code over $\mathbb{F}_{q}$.
Then $C$ is equivalent to the $r$-fold replication of a simplex code, possibly
with added $0$-coordinates.

An $[n,k,d]$-linear code $C$ is called distance optimal if there exist no
$[n,k,d+1]$-linear codes (see [24]). Next we recall the Griesmer bound.

######  Lemma 2.4.

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

$\sum\limits_{i=0}^{k-1}\left\lceil\frac{d}{q^{i}}\right\rceil\leq n,$ (2.2)

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

A linear code is called a Griesmer code if equality holds in Equation (2.2).
Note that every Griesmer code is distance optimal, but not conversely.
For $m\in\mathbb{N}$, we shall write $[m]$ to denote the set
$\\{1,2,\dots,m\\}$ and $w\in\mathbb{F}_{2}^{m}$. Then the set
$\textnormal{Supp}(w)=\\{i\in[m]:w_{i}=1\\}$ is called the support of $w$.
Note that the Hamming weight of $w\in\mathbb{F}_{2}^{m}$ is
$wt_{H}(w)=|\textnormal{Supp}(w)|$. For $v,w\in\mathbb{F}_{2}^{m}$, one says
that $v$ covers $w$ if $\textnormal{Supp}(w)\subseteq\textnormal{Supp}(v)$. If
$v$ covers $w$, we write $w\preceq v$.
Consider the map $\psi:\mathbb{F}_{2}^{m}\longrightarrow 2^{[m]}$ is defined
as $\psi(w)=\textnormal{Supp}(w)$, where $2^{[m]}$ denotes the power set of
$[m]$. Note that $\psi$ is a bijective map. Now onwards, we will write $w$
instead of Supp($w$) whenever we require.

######  Definition 2.5.

Let $C$ be a linear code over $\mathbb{F}_{2}$. An element $v\in
C\setminus\\{0\\}$ is called minimal if $w\preceq v$ and $w\in
C\setminus\\{0\\}$ $\implies$ $w=v$. If each nonzero codeword of $C$ is
minimal then $C$ is called a minimal code.

Now we recall a result from [5] which is a sufficient condition for a linear
code over $\mathbb{F}_{q}$ to be minimal.

######  Lemma 2.6.

[5](Ashikhmin-Barg) Let $C$ be a linear code over $\mathbb{F}_{q}$ with
$wt_{o}$ and $wt_{\infty}$ as minimum and maximum weights of its non-zero
codewords. If $\frac{wt_{0}}{wt_{\infty}}>\frac{q-1}{q}$, then $C$ is minimal.

Here we recall the definition of simplicial complex.

######  Definition 2.7.

A subset $\Delta$ of $\mathbb{F}_{2}^{m}$ is called a simplicial complex if
$v\in\Delta,w\in\mathbb{F}_{2}^{m}$ and $w\preceq v$ $\implies$ $w\in\Delta$.
An element $v\in\Delta$ is called a maximal element of $\Delta$ if for
$w\in\Delta$, $v\preceq w$ $\implies$ $v=w$.

A simplicial complex can have more than one maximal elements. Let
$M\subseteq[m]$. The simplicial complex generated by $M$ is denoted by
$\Delta_{M}$ and is defined as

$\Delta_{M}=\\{w\in\mathbb{F}_{2}^{m}|\textnormal{ Supp}(w)\subseteq
M\\}=\psi^{-1}(2^{M}).$ (2.3)

Note that $\psi^{-1}(M)$ is the only maximal element of $\Delta_{M}$, and
$|\Delta_{M}|=2^{|M|}$. Here $\Delta_{M}$ is a vector space over
$\mathbb{F}_{2}$ of dimension $|M|$.

Given a subset $P$ of $\mathbb{F}_{2}^{m}$, define the polynomial (referred to
as an $m$-variable generating function, see [10])
$\mathcal{H}_{P}(y_{1},y_{2},\dots,y_{m})$ by

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

where $v=(v_{1},\dots,v_{m})$ and $\mathbb{Z}$ denotes the ring of integers.
We recall a lemma from [10].

######  Lemma 2.8.

[10] Suppose $\Delta\subseteq\mathbb{F}_{2}^{m}$ is a simplicial complex and
$\mathcal{F}$ consists of its maximal elements. Then

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

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 2.9.

Consider the simplicial complex

$\Delta=\\{(0,0,0,0),(0,0,1,0),(0,0,0,1),(0,0,1,1),(0,1,0,0),(0,1,0,1)\\}.$

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

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

and $|\Delta|=6.$

For $M\subseteq[m]$, define a Boolean function
$\Psi(\cdot|M):\mathbb{F}_{2}^{m}\longrightarrow\mathbb{F}_{2}$ by

$\Psi(\alpha|M)=\prod_{i\in M}(1-\alpha_{i})=\begin{cases}1,\text{ if
}\textnormal{ Supp}(\alpha)\cap M=\emptyset;\\\ 0,\text{ if }\textnormal{
Supp}(\alpha)\cap M\neq\emptyset.\end{cases}$ (2.6)

We recall a lemma from [32].

######  Lemma 2.10.

[32] Suppose $M$ is a subset of $[m]$. Then

  1. 1.

$|\\{v\in\mathbb{F}_{2}^{m}:\Psi(v|M)=1\\}|=2^{m-|M|}$.

  2. 2.

$|\\{v\in\mathbb{F}_{2}^{m}:\Psi(v|M)=0\\}|=(2^{|M|}-1)\times 2^{m-|M|}$.

Now we recall a result from [35].

######  Lemma 2.11.

[35] Let $\alpha\in\mathbb{F}_{2}^{m}$ and let $\Delta_{M}$ be the simplicial
complex generated by $M\subseteq[m]$. Then

$\begin{split}\sum_{t\in\Delta_{M}^{\textnormal{c}}}(-1)^{\alpha
t}=2^{m}\delta_{0,\alpha}-\sum_{t\in\Delta_{M}}(-1)^{\alpha t},\end{split}$

where $\delta$ is the Kronecker delta function, and
$\Delta_{M}^{\textnormal{c}}=\mathbb{F}_{2}^{m}\setminus\Delta_{M}$, the
complement of $\Delta_{M}$.

In the forthcoming sections, we study the algebraic structures of linear
$I$-codes and their binary Gray images.

##  3 Construction of linear $I$-codes via simplicial complexes

For any $n,m\in\mathbb{N}$, let $D=\\{d_{1}<d_{2}<\dots<d_{n}\\}\subseteq
I^{m}$ be an ordered multiset. Let $D_{i}\subseteq\mathbb{F}_{2}^{m},i=1,2$.
Assume that $D=aD_{1}+bD_{2}\subseteq I^{m}$. Define

$C_{D}=\\{c_{D}(v)=\big{(}v\cdot d\big{)}_{d\in D}~{}|~{}v\in I^{m}\\}.$ (3.1)

where $x\cdot y=\sum\limits_{i=1}^{m}x_{i}y_{i}$ for $x,y\in I^{m}$.
Then $C_{D}$ is a linear $I$-code of length $|D|$. The ordered set $D$ is
called the defining set of $C_{D}$. Throughout this article we consider a
defining set to be an ordered multiset. Note that on changing the ordering of
$D$ we will get a code which is permutation equivalent (see [24]) to $C_{D}$.
Observe that $c_{D}:I^{m}\longrightarrow C_{D}\subseteq I^{|D|}$ defined by

$c_{D}(v)=\big{(}v\cdot d\big{)}_{d\in D}$ (3.2)

is a surjective homomorphism of $I$-modules.
Assume that $x=a\alpha+b\beta\in I^{m}$ and $d=at_{1}+bt_{2}\in
D=aD_{1}+bD_{2}$, where $\alpha,\beta\in\mathbb{F}_{2}^{m}$ and $t_{i}\in
D_{i},i=1,2$. Then the Lee weight of $c_{D}(x)$ is

$\begin{split}wt_{Lee}(c_{D}(x))=&wt_{Lee}\big{(}\big{(}\big{(}a\alpha+b\beta\big{)}\cdot\big{(}at_{1}+bt_{2}\big{)}\big{)}_{t_{i}\in
D_{i}}\big{)}\\\ =&wt_{Lee}\big{(}\big{(}b(\alpha t_{1})\big{)}_{t_{i}\in
D_{i}}\big{)}\\\ =&wt_{H}\big{(}\Phi\big{(}(b(\alpha t_{1}))_{t_{i}\in
D_{i}}\big{)}\big{)}\\\ =&wt_{H}\big{(}(\alpha t_{1})_{t_{i}\in
D_{i}}\big{)}+wt_{H}\big{(}(\alpha t_{1})_{t_{i}\in D_{i}}\big{)}\\\
=&2wt_{H}\big{(}(\alpha t_{1})_{t_{i}\in D_{i}}\big{)}.\end{split}$

Now if $v\in\mathbb{F}_{2}^{m}$, then $wt_{H}(v)=0\iff
v=\textbf{0}\in\mathbb{F}_{2}^{m}$. Hence,

$\begin{split}wt_{Lee}(c_{D}(x))&=2\big{[}|D|-\frac{1}{2}\sum\limits_{t_{1}\in
D_{1}}\sum\limits_{t_{2}\in D_{2}}\big{(}1+(-1)^{\alpha
t_{1}}\big{)}\big{]}\\\ &=|D|-\sum\limits_{t_{1}\in D_{1}}(-1)^{\alpha
t_{1}}\sum\limits_{t_{2}\in D_{2}}(1).\end{split}$ (3.3)

For $P\subseteq\mathbb{F}_{2}^{m}$ and $\alpha\in\mathbb{F}_{2}^{m}$, we
define

$\chi_{\alpha}(P)=\sum\limits_{t\in P}(-1)^{\alpha t}.$ (3.4)

Note that, for $\alpha\in\mathbb{F}_{2}^{m}$ and $\emptyset\neq
M\subseteq[m]$, we have

$\begin{split}\chi_{\alpha}(\Delta_{M})&=\sum\limits_{t\in\Delta_{M}}(-1)^{\alpha
t}\\\
&=\mathcal{H}_{\Delta_{M}}\big{(}(-1)^{\alpha_{1}},(-1)^{\alpha_{2}},\dots,(-1)^{\alpha_{m}}\big{)}\\\
&=\prod_{i\in M}(1+(-1)^{\alpha_{i}})=\prod_{i\in
M}(2-2\alpha_{i})\textnormal{ (By Lemma }\ref{generatinglemma})\\\
&=2^{|M|}\prod_{i\in M}(1-\alpha_{i})=2^{|M|}\Psi(\alpha|M),\end{split}$ (3.5)

where $\Psi(\cdot|M)$ is the Boolean function defined by Equation (2.6).
The following result describes Lee weight distributions of $C_{D}$ for various
choices of $D$.

######  Theorem 3.1.

Suppose that $m\in\mathbb{N}$ and $M,N\subseteq[m]$.

  1. 1.

Let $D=a\Delta_{M}+b\Delta_{N}\subseteq I^{m}$. Then $C_{D}$ is a linear
$I$-code of length $|D|=2^{|M|+|N|}$ and size $2^{|M|}$. The Lee weight
distribution of $C_{D}$ is displayed in Table 3.

Lee weight | Frequency
---|---
$2^{|M|+|N|}$ | $2^{2m-|M|}\times(2^{|M|}-1)$
$0$ | $2^{2m-|M|}$
Table 3: Lee weight distribution in Theorem 3.1 (1)

  2. 2.

Let $D=a\Delta^{\textnormal{c}}_{M}+b\Delta_{N}\subseteq I^{m}$. Then $C_{D}$
is a linear $I$-code of length $|D|=(2^{m}-2^{|M|})\times 2^{|N|}$ and size
$2^{m}$. The Lee weight distribution of $C_{D}$ is displayed in Table 4.

Lee weight | Frequency
---|---
$2^{m+|N|}$ | $2^{m}\times(2^{m-|M|}-1)$
$(2^{m}-2^{|M|})\times 2^{|N|}$ | $2^{2m-|M|}\times(2^{|M|}-1)$
$0$ | $2^{m}$
Table 4: Lee weight distribution in Theorem 3.1 (2)

  3. 3.

Let $D=a\Delta_{M}+b\Delta^{\textnormal{c}}_{N}\subseteq I^{m}$. Then $C_{D}$
is a linear $I$-code of length $|D|=2^{|M|}\times(2^{m}-2^{|N|})$ and size
$2^{|M|}$. The Lee weight distribution of $C_{D}$ is displayed in Table 5.

Lee weight | Frequency
---|---
$(2^{m}-2^{|N|})\times 2^{|M|}$ | $2^{2m-|M|}\times(2^{|M|}-1)$
$0$ | $2^{2m-|M|}$
Table 5: Lee weight distribution in Theorem 3.1 (3)

  4. 4.

Let $D=a\Delta^{\textnormal{c}}_{M}+b\Delta_{N}^{\textnormal{c}}\subseteq
I^{m}$. Then $C_{D}$ is a linear $I$-code of length
$|D|=(2^{m}-2^{|M|})(2^{m}-2^{|N|})$ and size $2^{m}$. The Lee weight
distribution of $C_{D}$ is displayed in Table 6.

Lee weight | Frequency
---|---
$2^{m}\times(2^{m}-2^{|N|})$ | $2^{m}\times(2^{m-|M|}-1)$
$(2^{m}-2^{|M|})(2^{m}-2^{|N|})$ | $2^{2m-|M|}\times(2^{|M|}-1)$
$0$ | $2^{m}$
Table 6: Lee weight distribution in Theorem 3.1 (4)

  5. 5.

Let $D=a\Delta_{M}+b\Delta_{N}$ so that
$D^{\textnormal{c}}=\big{(}a\Delta_{M}^{\textnormal{c}}+b\mathbb{F}_{2}^{m}\big{)}\bigsqcup\big{(}a\Delta_{M}+b\Delta_{N}^{\textnormal{c}}\big{)}$,
where $\bigsqcup$ denotes disjoint union. Then $C_{D^{\textnormal{c}}}$ is a
linear $I$-code of length $|D^{\textnormal{c}}|=2^{2m}-2^{|M|+|N|}$ and size
$2^{m}$. The Lee weight distribution of $C_{D^{\textnormal{c}}}$ is displayed
in Table 7.

Lee weight | Frequency
---|---
$2^{2m}$ | $2^{m}\times(2^{m-|M|}-1)$
$2^{2m}-2^{|M|+|N|}$ | $2^{2m-|M|}\times(2^{|M|}-1)$
$0$ | $2^{m}$
Table 7: Lee weight distribution in Theorem 3.1 (5)

Proof. We discuss the proof of part (4). The other parts can be proved in a
similar way.
Observe the length of $C_{D}$ is $|D|=(2^{m}-2^{|M|})(2^{m}-2^{|N|})$. Let
$x=a\alpha+b\beta\in I^{m}$. By Equation (3.3) and (3.5), we have

$\begin{split}wt_{Lee}(c_{D}(x))&=(2^{m}-2^{|M|})(2^{m}-2^{|N|})-\big{[}2^{m}\delta_{0,\alpha}-2^{|M|}\Psi(\alpha|M)\big{]}\times(2^{m}-2^{|N|}).\end{split}$
(3.6)

Here we discuss the following cases.

  1. 1.

If $\alpha=0$, then $wt_{Lee}(c_{D}(x))=0.$
In this case, $\\#\alpha=1,\\#\beta=2^{m}$.
Therefore, $\\#x=2^{m}$.

  2. 2.

If $\alpha\neq 0$, then
$wt_{Lee}(c_{D}(x))=(2^{m}-2^{|M|})(2^{m}-2^{|N|})+2^{|M|}(2^{m}-2^{|N|})\Psi(\alpha|M)$.

    1. (a)

If $\Psi(\alpha|M)=0$ then
$wt_{Lee}(c_{D}(x))=(2^{m}-2^{|M|})(2^{m}-2^{|N|}).$
In this case, by using Lemma 2.10, we get
$\\#\alpha=(2^{|M|}-1)\times 2^{m-|M|}$,
$\\#\beta=2^{m}$.
Therefore, $\\#x=2^{2m-|M|}(2^{|M|}-1)$.

    2. (b)

If $\Psi(\alpha|M)=1$ then $wt_{Lee}(c_{D}(x))=2^{m}(2^{m}-2^{|N|}).$
In this case, by using Lemma 2.10, we get
$\\#\alpha=(2^{m-|M|}-1)$,
$\\#\beta=2^{m}$.
Therefore, $\\#x=2^{m}(2^{m-|M|}-1)$.

Based on the above computations, we obtain Table 6.
By using Table 6, we have $|\ker(c_{D})|=|\\{v\in I^{m}:v\cdot
d=0~{}\forall~{}d\in D\\}|=2^{m}$. By the first isomorphism theorem, we have
$|C_{D}|=\frac{|I^{m}|}{|\ker(c_{D})|}=2^{m}$. $\square$

##  4 Gray images of linear $I$-codes

In this section, we study the Gray images of linear $I$-codes which are
discussed in Section 3.
Now we recall a result (see Theorem $1.4.8~{}(ii)$ of [24]) for self-
orthogonality of binary linear codes.

######  Theorem 4.1.

[24] If the Hamming weight of every non-zero element of a binary linear code
$C$ is a multiple of $4$, then $C$ is self-orthogonal.

By using the above result, we give a sufficient condition for the binary Gray
images $\Phi(C_{D})$ to be self-orthogonal for each $C_{D}$ discussed in
Theorem 3.1.

######  Proposition 4.2.

Assume that $C_{D}$ is as in Theorem 3.1. Then its Gray image $\Phi(C_{D})$ is
a binary self-orthogonal code provided $|M|+|N|\geq 2$.

By using Lemma 2.6 and Lemma 2.4 respectively, we find minimal and optimal
linear codes among the Gray images of the codes studied in Theorem 3.1.

######  Theorem 4.3.

Let $m\in\mathbb{N}$ and $M,N\subseteq[m]$. Suppose $\Phi$ is the map defined
by Equation (2.1).

  1. 1.

Let $D=a\Delta_{M}+b\Delta_{N}\subseteq I^{m}$. Then $\Phi(C_{D})$ is a binary
$[2^{|M|+|N|+1},|M|,2^{|M|+|N|}]$-linear $1$-weight code.

  2. 2.

Let $D=a\Delta^{\textnormal{c}}_{M}+b\Delta_{N}\subseteq I^{m}$. Then
$\Phi(C_{D})$ is a binary
$[(2^{m}-2^{|M|})2^{|N|+1},m,(2^{m}-2^{|M|})2^{|N|}]$-linear $2$-weight code.
It is minimal provided $|M|\leq m-2$.

    1. (a)

Let $1\leq|M|+|N|\leq m-1$ and $\theta_{1}=2^{|N|+1}-1$. If
$0<\theta_{1}<|M|+|N|+1$ then $\Phi(C_{D})$ is optimal with respect to the
Griesmer bound.

    2. (b)

Let $m\leq|M|+|N|\leq 2m-1$ and $\theta_{2}=2^{|M|+|N|+1-m}(2^{m-|M|}-1)$. If
$0<\theta_{2}<m$ then $\Phi(C_{D})$ is optimal with respect to the Griesmer
bound.

  3. 3.

Let $D=a\Delta_{M}+b\Delta^{\textnormal{c}}_{N}\subseteq I^{m}$. Then
$\Phi(C_{D})$ is a binary
$[2^{|M|+1}(2^{m}-2^{|N|}),|M|,2^{|M|}(2^{m}-2^{|N|})]$-linear $1$-weight
code.

  4. 4.

Let $D=a\Delta^{\textnormal{c}}_{M}+b\Delta^{\textnormal{c}}_{N}\subseteq
I^{m}$. Then $\Phi(C_{D})$ is a binary
$[2(2^{m}-2^{|M|})(2^{m}-2^{|N|}),m,(2^{m}-2^{|M|})(2^{m}-2^{|N|})]$-linear
$2$-weight code. If $|M|\leq m-2$ then $C_{D}$ is a minimal code.

  5. 5.

Let $D=a\Delta_{M}+b\Delta_{N}\subseteq I^{m}$ so that
$D^{\textnormal{c}}=(a\Delta^{\textnormal{c}}_{M}+b\mathbb{F}_{2}^{m})\bigsqcup(a\Delta_{M}+b\Delta^{\textnormal{c}}_{N})$.
Then $\Phi(C_{D^{\textnormal{c}}})$ is a binary
$[2(2^{2m}-2^{|M|+|N|}),m,(2^{2m}-2^{|M|+|N|})]$-linear $2$-weight code. If
$|M|+|N|\leq 2m-2$ then $\Phi(C_{D^{\textnormal{c}}})$ is a minimal code.

Proof. We discuss case (2) and omit others.
From Table (4), we have $wt_{0}=(2^{m}-2^{|M|})\times 2^{|N|}$ and
$wt_{\infty}=2^{m+|N|}$.
Now, $\frac{w_{0}}{w_{\infty}}=\frac{(2^{m}-2^{|M|})\times
2^{|N|}}{2^{m+|N|}}>\frac{1}{2}\iff|M|\leq m-2$. Therefore, by Lemma 2.6,
$\Phi(C_{D})$ is minimal if $|M|\leq m-2$.
(2a) Suppose $1\leq|M|+|N|\leq m-1$ and $\theta_{1}=2^{|N|+1}-1$. The
parameters of the code $\Phi(C_{D})$ are given by
$n=(2^{m}-2^{|M|})2^{|N|+1},k=m,d=(2^{m}-2^{|M|})2^{|N|}$.
Note that

$\begin{split}\sum_{i=0}^{k-1}\left\lceil\frac{d}{2^{i}}\right\rceil&=\sum_{i=0}^{m-1}\left\lceil\frac{2^{m+|N|}-2^{|M|+|N|}}{2^{i}}\right\rceil\\\
&=\sum_{i=0}^{m-1}\frac{2^{m+|N|}}{2^{i}}-\sum_{i=0}^{m-1}\left\lfloor\frac{2^{|M|+|N|}}{2^{i}}\right\rfloor\\\
&=(2^{m+|N|+1}-2^{|N|+1})-(2^{|M|+|N|+1}-1)\\\
&=(2^{m+|N|+1}-2^{|M|+|N|+1})-(2^{|N|+1}-1)\\\ &=n-\theta_{1}.\end{split}$

By Griesmer bound, $\sum\limits_{i=0}^{k-1}\lceil\frac{d}{2^{i}}\rceil\leq
n\iff\theta_{1}>0$.
On the other hand,

$\begin{split}\sum_{i=0}^{m-1}\left\lceil\frac{2^{m+|N|}-2^{|M|+|N|}+1}{2^{i}}\right\rceil&=\sum_{i=0}^{m-1}\frac{2^{m+|N|}}{2^{i}}-\sum_{i=0}^{|M|+|N|}\left\lfloor\frac{2^{|M|+|N|}}{2^{i}}-\frac{1}{2^{i}}\right\rfloor\\\
&-\sum_{i=|M|+|N|+1}^{m-1}\left\lfloor\frac{2^{|M|+|N|}}{2^{i}}-\frac{1}{2^{i}}\right\rfloor\\\
&=(2^{m+|N|+1}-2^{|N|+1})-\big{[}(2^{|M|+|N|+1}-1)\\\
&~{}-(|M|+|N|+1)\big{]}-0\\\
&=(2^{m+|N|+1}-2^{|M|+|N|+1})-(2^{|N|+1}-1)+|M|+|N|+1\\\
&=n-\theta_{1}+|M|+|N|+1.\end{split}$

Therefore
$\sum\limits_{i=0}^{k-1}\lceil\frac{d+1}{2^{i}}\rceil>n\iff\theta_{1}<|M|+|N|+1$.
This completes the proof of (2a).
(2b) Suppose $m\leq|M|+|N|\leq 2m-1$ and
$\theta_{2}=2^{|M|+|N|-m+1}(2^{m-|M|}-1)$. Now we have

$\begin{split}\sum_{i=0}^{k-1}\left\lceil\frac{d}{2^{i}}\right\rceil&=\sum_{i=0}^{m-1}\left\lceil\frac{2^{m+|N|}-2^{|M|+|N|}}{2^{i}}\right\rceil\\\
&=\sum_{i=0}^{m-1}\frac{2^{m+|N|}}{2^{i}}-\sum_{i=0}^{m-1}\left\lfloor\frac{2^{|M|+|N|}}{2^{i}}\right\rfloor\\\
&=(2^{m+|N|+1}-2^{|N|+1})-(2^{|M|+|N|+1}-2^{|M|+|N|-m+1})\\\
&=(2^{m+|N|+1}-2^{|M|+|N|+1})-(2^{|N|+1}-2^{|M|+|N|-m+1})\\\
&=n-\theta_{2}.\end{split}$

By Griesmer bound, $\sum\limits_{i=0}^{k-1}\lceil\frac{d}{2^{i}}\rceil\leq
n\iff\theta_{2}>0$.
On the other hand,

$\begin{split}\sum_{i=0}^{m-1}\left\lceil\frac{2^{m+|N|}-2^{|M|+|N|}+1}{2^{i}}\right\rceil&=\sum_{i=0}^{m-1}\frac{2^{m+|N|}}{2^{i}}-\sum_{i=0}^{m-1}\left\lfloor\frac{2^{|M|+|N|}}{2^{i}}-\frac{1}{2^{i}}\right\rfloor\\\
&=(2^{m+|N|+1}-2^{|N|+1})-\big{[}(2^{|M|+|N|+1}-2^{|M|+|N|-m+1})-m\big{]}\\\
&=(2^{m+|N|+1}-2^{|M|+|N|+1})-(2^{|N|+1}-2^{|M|+|N|-m+1})+m\\\
&=n-\theta_{2}+m.\end{split}$

Therefore
$\sum\limits_{i=0}^{k-1}\lceil\frac{d+1}{2^{i}}\rceil>n\iff\theta_{2}<m$. This
completes the proof of (2b). $\square$

The following examples illustrate Theorem 3.1, Proposition 4.2 and Theorem
4.3.

######  Example 4.4.

  1. 1.

Let $m=6$, $M=\\{2,3\\},N=\\{4,5\\}$. If $D=a\Delta_{M}+b\Delta_{N}$, then by
using Theorem 3.1(1) $C_{D}$ is a linear $I$-code with Lee weight enumerator
$X^{32}+3X^{16}Y^{16}$. By Proposition 4.2 and Theorem 4.3(1), $\Phi(C_{D})$
is a $[32,2,16]$ binary minimal and self-orthogonal code.

  2. 2.

    1. (a)

Let $m=5$, $M=\\{1,2,3\\},N=\\{4\\}$. If
$D=a\Delta_{M}^{\textnormal{c}}+b\Delta_{N}$, then by using Theorem 3.1(2)
$C_{D}$ is a linear $I$-code with Lee weight enumerator
$X^{96}+28X^{48}Y^{48}+3X^{32}Y^{64}$. By Proposition 4.2 and Theorem 4.3(2a),
$\Phi(C_{D})$ is a $[96,5,48]$ binary minimal, self-orthogonal and distance
optimal code. Note that optimality of $\Phi(C_{D})$ can be verified using the
Database [19].

    2. (b)

Let $m=9$, $M=\\{1,2,3,4,7,8,9\\},N=\\{5,6\\}$. If
$D=a\Delta_{M}^{\textnormal{c}}+b\Delta_{N}$, then by using Theorem 3.1(2)
$C_{D}$ is a linear $I$-code with Lee weight enumerator
$X^{3072}+508X^{1536}Y^{1536}+3X^{1024}Y^{2048}$. By Proposition 4.2 and
Theorem 4.3(2b), $\Phi(C_{D})$ is a $[3072,9,1536]$ binary minimal, self-
orthogonal and distance optimal code. Note that one can directly verify that
there is no $[3072,9,1537]$-linear code over $\mathbb{F}_{2}$.

  3. 3.

Let $m=3$, $M=\\{1,2,3\\},N=\\{1,2\\}$. If
$D=a\Delta_{M}+b\Delta_{N}^{\textnormal{c}}$, then by using Theorem 3.1(3)
$C_{D}$ is a linear $I$-code with Lee weight enumerator
$X^{64}+7X^{32}Y^{32}$. By Proposition 4.2 and Theorem 4.3(3), $\Phi(C_{D})$
is a $[64,3,32]$ binary minimal and self-orthogonal code.

  4. 4.

Let $m=5$, $M=\\{2,3,4\\},N=\\{1,2,4,5\\}$. If
$D=a\Delta_{M}^{\textnormal{c}}+b\Delta_{N}^{\textnormal{c}}$, then by using
Theorem 3.1(4) $C_{D}$ is a linear $I$-code with Lee weight enumerator
$X^{768}+28X^{384}Y^{384}+3X^{256}Y^{512}$. By Proposition 4.2 and Theorem
4.3(4), $\Phi(C_{D})$ is a $[768,5,384]$ binary minimal and self-orthogonal
code.

  5. 5.

Let $m=4$, $M=\\{2,3,4\\},N=\\{1,2,4\\}$. If $D=a\Delta_{M}+b\Delta_{N}$, then
by using Theorem 3.1(5) $C_{D^{\textnormal{c}}}$ is a linear $I$-code with Lee
weight enumerator $X^{384}+14X^{192}Y^{192}+X^{128}Y^{256}$. By Proposition
4.2 and Theorem 4.3(5), $\Phi(C_{D^{\textnormal{c}}})$ is a $[384,4,192]$
binary minimal and self-orthogonal code.

######  Remark 4.5.

By Theorem 2.3 all the binary $1$-weight codes of this article are simplex
codes and hence they are distance optimal.

Result $[n,k,d]$-code $\\#$Weight Distance optimal Minimal Theorem 4.3(1)
$[2^{|M|+|N|+1},|M|,2^{|M|+|N|}]$ 1 Yes Yes Theorem 4.3(2a)
$[(2^{m}-2^{|M|})2^{|N|+1},m,(2^{m}-2^{|M|})2^{|N|}]$ 2 Yes, if
$1\leq\theta_{1}<|M|+|N|+1\leq m$ Yes, if $|M|\leq m-2$ Theorem 4.3(2b) Yes,
$0<\theta_{2}<m\leq|M|+|N|$ Theorem 4.3(3)
$[2^{|M|+1}(2^{m}-2^{|N|}),|M|,2^{|M|}(2^{m}-2^{|N|})]$ 1 Yes Yes Theorem
4.3(4) $[2(2^{m}-2^{|M|})(2^{m}-2^{|N|}),m,(2^{m}-2^{|M|})(2^{m}-2^{|N|})]$ 2
$-$ Yes, if $|M|\leq m-2$ Theorem 4.3(5)
$[2(2^{2m}-2^{|M|+|N|}),m,(2^{2m}-2^{|M|+|N|})]$ 2 $-$ Yes, if $|M|+|N|\leq
2m-2$

Table 8: Binary linear codes from simplicial complexes obtained this paper

##  5 Conclusion

In this article, we study few-weight linear $I$-codes via simplicial complexes
having one maximal element. A Gray map is utilized to study the corresponding
binary linear codes. We obtain their weight distributions by using Boolean
functions. As a consequence, we achieve a class of distance optimal codes with
respect to the Griesmer bound. Besides, we obtain many classes of binary
minimal codes. Every family of the binary codes obtained here are self-
orthogonal under certain mild conditions. For the reader’s convenience, codes
obtained in this article that have good parameters, are listed in in Table 8.

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