#  Construction of Minimal Binary Linear Codes of dimension $n+3$

Wajid M. Shaikh  Research Scholar, School of Mathematical Sciences, SRTMU
Nanded  Rupali S. Jain  Associate Professor, School of Mathematical Sciences,
SRTMU Nanded  B. Surendranath Reddy  Assistant Professor, School of
Mathematical Sciences, SRTMU Nanded  Bhagyashri S. Patil  Research Scholar,
School of Mathematical Sciences, SRTMU Nanded

###### Abstract

In this paper, we will give the generic construction of a binary linear code
of dimension $n+3$ and derive the necessary and sufficient conditions for the
constructed code to be minimal. Using generic construction, a new family of
minimal binary linear code will be constructed from a special class of Boolean
functions violating the Ashikhmin-Barg condition. We also obtain the weight
distribution of the constructed minimal binary linear code.

Keywords Linear code, minimal code, weight distribution, Ashikhmin-Barg
condition.
Mathematics Subject Classification(2000) 94B05, 94C10, 94A60

##  1 Introduction

Linear codes serve as a cornerstone of modern communication systems, providing
robust error control mechanisms essential for reliable data transmission in
various applications ranging from telecommunications to digital broadcasting.
Minimal linear codes is a special class of linear codes widely used in secret
sharing, data storage, and information processing systems in modern technology
[10][14]. So, the construction of minimal linear codes is one of the thrust
area for many researchers in coding theory.

Minimal codes are characterized by the property that none of the codeword is
covered by some other linearly independent codeword. There are two approaches
to studying minimal linear codes one is the algebraic method and another is
geometric method. The algebraic method is based on the Hamming weights of the
codeword. In 1998, Ashikhmin and Barg[1] gave a sufficient condition for
linear code to be minimal, which is stated below.

######  Lemma 1.1.

[1][Ashikhmin-Barg] Let $p$ be a prime. Then a linear code $\mathscr{C}$ over
$\mathbb{F}_{p}$ is minimal if

$\frac{{wt}_{min}}{{wt}_{max}}>\frac{p-1}{p}$

where, ${wt}_{min}$ and ${wt}_{max}$ denote the minimum and maximum non-zero
Hamming weights for $\mathscr{C}$, respectively.

From the above lemma, many minimal linear codes constructed but constructing
minimal linear codes violating Ashikhmin-Barg condition is an interesting
research topic. In the year 2018, Chang and Hyun[2] constructed an infinite
family of minimal binary linear codes, which are not satisfying Ashikhmin-Barg
condition with the help of mathematical objects called simplicial complexes
and generic construction

$\mathscr{C}_{f}=\\{(\alpha f({\bf x})+{\bf w}\cdot{\bf x})_{\bf
x\in\mathbb{F}_{p}^{n*}}\ |\ \alpha\in\mathbb{F}_{p},\ {\bf
w}\in\mathbb{F}_{p}^{n}\\}$ (1.1)

The necessary and sufficient condition for binary linear code to be minimal
was given by Ding, Heng, and Zhou [3] in 2018, and using special Boolean
functions, they constructed three classes of minimal binary linear codes. Heng
et al. [4] discovered a necessary and sufficient condition for linear codes
over finite fields to be minimum. This is a generalised version of result
Ding, C., Heng, Z., & Zhou, Z. [3]. Also, several researchers constructed
minimal linear code from different kinds of functions (see [9],[12],[5],[11],
and [13]).

Till date, several minimum linear codes have been constructed that do not meet
the Ashikhmin-Barg condition, but the dimension of these codes is either $n$
or $n+1$, which reduces the information rate of a code. Recently, Liu and Liao
[7] constructed a minimal binary linear code using a Boolean function with
dimension $n+2$.

In this paper, we extend the results of [7] to dimension $n+3$. This paper is
organised as follows: In Section 2, we give some basic definitions and
results, which are prerequisites. Section 3 contains the generic construction
of minimal binary linear code with dimension $n+3$ and give a necessary and
sufficient conditions for this code to be minimal. In Section 4, we obtain a
family of minimal linear codes and one special class of minimal linear codes,
which violets Ashikhmin-Barg condition.

##  2 Preliminaries

In this section, we give some basic definitions and results that are
prerequisites to this paper. We denote a finite field with $p$ elements by
$\mathbb{F}_{p}$. A $k$-dimensional subspace of $\mathbb{F}_{p}^{n}$ is a
$[n,k,d]$ linear code denoted by $\mathscr{C}$, where $d$ is the minimum
Hamming distance. Let $A_{i}$ denote the count of the codeword having weight
$i$ in $\mathscr{C}$ of length $n$. The weight enumerator of a code
$\mathscr{C}$ is defined as $1+A_{1}z+A_{2}z+\cdots+A_{n}z^{n}$, and the
sequence $(1,A_{1},A_{2},\ldots,A_{n})$ is called the weight distribution of a
code $\mathscr{C}$. If the number of non-zero $A_{i}$’s is $t$, then
$\mathscr{C}$ is called the $t$-weight code.

######  Definition 2.1 ($Support$).

[6] The $support$ of a codeword ${\bf c}$ is denoted by $\text{Supp}({\bf
c})$, is the set of the coordinates at which ${c}$ is non-zero. i.e.

$\text{Supp}({\bf c})=\\{1\leq i\leq n\ |\ c_{i}\neq 0\\}$

Note that, $\text{wt}({\bf c})=|\text{Supp}({\bf c})|$, where $\text{wt}({\bf
x})$ is the Hamming weight of ${\bf c}$. For ${\bf y}\in\mathbb{F}_{p}^{n}$,
if $\text{Supp}({\bf y})\subseteq\text{Supp}({\bf x})$, then we call ${\bf x}$
covers ${\bf y}$ and denoted by ${\bf y}\preceq{\bf x}$.

######  Definition 2.2.

A codeword ${\bf x}$ is minimal if ${\bf x}$ covers only the codeword of the
form $\alpha{\bf x}$, where $\alpha\in\mathbb{F}_{p}^{*}$. A code
$\mathscr{C}$ is called minimal if every codeword in $\mathscr{C}$ is minimal.

A Boolean function $f$ is a function from $\mathbb{F}_{2}^{n}$ to
$\mathbb{F}_{2}$. The Walsh transform for a Boolean function is defined as
follows

$\widehat{f}({\bf w})=\sum_{{\bf x}\in\mathbb{F}_{2}^{n}}(-1)^{f({\bf x})+{\bf
w}\cdot{\bf x}}$

where, ${\bf w}\in\mathbb{F}_{2}^{n}$ and ${\bf w}\cdot{\bf x}$ is the
standard inner product of ${\bf w}$ and ${\bf x}$. Another related walsh
transform of $f$ is given by

$\tilde{f}({\bf w})=\sum_{{\bf x}\in\mathbb{F}_{2}^{n}}f({\bf x})(-1)^{{\bf
w}\cdot{\bf x}}$

where, $f({\bf x})$ is a real valued function which takes only values $0$ and
$1$. Following lemma describe the relation between above defined Walsh
transforms.

######  Lemma 2.3.

[3] Let $f:\mathbb{F}^{n}_{2}\rightarrow\mathbb{F}_{2}$ be a Boolean function.
Then

$\widehat{f}({\bf w})=\begin{cases}2^{n}-2\tilde{f}({\bf 0}),&\text{ if }{\bf
w}={\bf 0}\\\ -2\tilde{f}({\bf w}),&\text{ if },{\bf w}\neq{\bf 0}\end{cases}$

######  Lemma 2.4.

[3] Let ${\bf x},{\bf y}\in\mathbb{F}_{2}^{n}$. Then ${\bf y}\preceq{\bf x}$
if and only if

$\text{wt}({\bf x}+{\bf y})=\text{wt}({\bf x})-\text{wt}({\bf y})$

##  3 Construction of Minimal Binary Linear Codes of dimension $n+3$ from a
class of Boolean Functions

In this section, we extend and generalise the construction of minimal binary
linear codes from the class of Boolean functions. Let $f,g$ and $h$ be Boolean
functions with domain $\mathbb{F}_{2}^{n}$ and denote the collection of the
functions and their linear combinations as $\mathcal{F}=\\{\alpha f+\beta
g+\gamma h\ |\ \alpha,\beta,\gamma\in\mathbb{F}_{2}\\}\setminus\\{0\\}$ such
that the following conditions hold:

  1. 1.

$\phi$ is non-zero function, for all $\phi\in\mathcal{F}$.

  2. 2.

$\phi({\bf 0})=0$, for all $\phi\in\mathcal{F}$.

  3. 3.

$\phi_{1}\neq\phi_{2}$, for all $\phi_{1},\phi_{2}\in\mathcal{F}$.

In this paper, we define a linear code $\mathscr{C}_{f,g,h}$, which is a
generalization of the linear code defined in [7] as follows:

$\mathscr{C}_{f,g,h}=\\{c_{f,g,h}({\bf w})=(\alpha f({\bf x})+\beta g({\bf
x})+\gamma h({\bf x})+{\bf w}\cdot{\bf x})_{{\bf x}\in\mathbb{F}_{2}^{n*}}\ |\
\alpha,\beta,\gamma\in\mathbb{F}_{2},\ {\bf w}\in\mathbb{F}_{2}^{n}\\}$ (3.1)

######  Theorem 3.1.

Let $\mathscr{C}_{f,g,h}$ be a binary code as defined above. Then, if
$\phi({\bf x})\neq{\bf v}\cdot{\bf x}\ \forall\ {\bf v}\in\mathbb{F}_{2}^{n}$
and $\forall\ \phi\in\mathcal{F}$, then $\mathscr{C}_{f,g,h}$ has length
$2^{n}-1$ and dimension $n+3$. Moreover, the weight distribution of
$\mathscr{C}_{f,g,h}$ is given by the following multiset union:
$\displaystyle\bigcup_{\phi\in\mathcal{F}}\left\\{\frac{2^{n}-\widehat{\phi}({\bf
w})}{2}|\ {\bf w}\in\mathbb{F}_{2}^{n}\right\\}\bigcup\\{2^{n-1}\ |\ {\bf
w}\in\mathbb{F}_{2}^{n}\\}\bigcup\\{0\\}$.

###### Proof.

Clearly, the length of the code $\mathscr{C}_{f,g,h}$ is $2^{n}-1$. Let
$\\{{\bf u}_{1},{\bf u}_{2},\ldots,{\bf u}_{n}\\}$ be a basis of
$\mathbb{F}_{2}^{n}$. Then, we prove that $\\{c_{f,g,h}({\bf
u_{1}}),c_{f,g,h}({\bf u_{2}}),\ldots,c_{f,g,h}({\bf u_{n}}),c_{f}({\bf
0}),c_{g}({\bf 0}),c_{h}({\bf 0})\\}$ is the largest linear independent set of
$\mathscr{C}_{f,g,h}$. Consider,

$\displaystyle\sum_{i=1}^{n}\delta_{i}c_{f,g,h}({\bf
u_{i}})+\delta_{n+1}c_{f}({\bf 0})+\delta_{n+2}c_{g}({\bf
0})+\delta_{n+3}c_{h}({\bf 0})$ $\displaystyle={\bf 0}$
$\displaystyle\sum_{i=1}^{n}\delta_{i}(f({\bf x})+g({\bf x})+h({\bf x})+{\bf
u_{i}}\cdot{\bf x})_{{\bf x}\in\mathbb{F}_{2}^{n}}+\delta_{n+1}(f({\bf
x}))_{{\bf x}\in\mathbb{F}_{2}^{n}}$ $\displaystyle+\delta_{n+2}(g({\bf
x}))_{{\bf x}\in\mathbb{F}_{2}^{n}}+\delta_{n+3}(h({\bf x}))_{{\bf
x}\in\mathbb{F}_{2}^{n}}$ $\displaystyle={\bf 0}$
$\displaystyle\sum_{i=1}^{n}\delta_{i}({\bf u_{i}}\cdot{\bf x})_{{\bf
x}\in\mathbb{F}_{2}^{n}}+(\delta_{1}+\delta_{2}+\cdots+\delta_{n}+\delta_{n+1})(f({\bf
x}))_{{\bf x}\in\mathbb{F}_{2}^{n}}+$
$\displaystyle(\delta_{1}+\delta_{2}+\cdots+\delta_{n}+\delta_{n+2})(g({\bf
x}))_{{\bf
x}\in\mathbb{F}_{2}^{n}}+(\delta_{1}+\delta_{2}+\cdots+\delta_{n}+\delta_{n+3})(h({\bf
x}))_{{\bf x}\in\mathbb{F}_{2}^{n}}$ $\displaystyle={\bf 0}$
$\displaystyle\sum_{i=1}^{n}\delta_{i}({\bf u_{i}}\cdot{\bf x})_{{\bf
x}\in\mathbb{F}_{2}^{n}}+\alpha_{1}f({\bf x})+\alpha_{2}g({\bf
x})+\alpha_{3}h({\bf x})$ $\displaystyle=0$

If $\alpha_{i}=0$, for $i\in\\{1,2,3\\}$, then $\delta_{j}=0$, for all $j$. If
$\alpha_{i}\neq 0$, for some $i$, then $\phi({\bf x})={\bf v}\cdot{\bf x}$,
for some $\phi\in\mathcal{F}$.
From this, we get $\\{c_{f,g,h}({\bf u_{1}}),c_{f,g,h}({\bf
u_{2}}),\ldots,c_{f,g,h}({\bf u_{n}}),c_{f}({\bf 0}),c_{g}({\bf 0}),c_{h}({\bf
0})\\}$ is linearly independent. From the dimension of $\mathbb{F}_{2}^{n}$,
we can prove that any subset of $\mathscr{C}_{f,g,h}$ containing $n+4$
elements is linearly dependent.
Now we will give the weight enumeration for the case $\alpha=\beta=\gamma=1$,
we have

$\displaystyle\widehat{f+g+h}({\bf w})$ $\displaystyle=\sum_{{\bf
x}\in\mathbb{F}_{2}^{n}}(-1)^{f({\bf x})+g({\bf x})+h({\bf x})+{\bf
w}\cdot{\bf x}}$ $\displaystyle=2^{n}-2|\\{{\bf x}\in\mathbb{F}_{2}^{n}\ |\
f({\bf x})+g({\bf x})+h({\bf x})+{\bf w}\cdot{\bf x}=0\\}|$
$\displaystyle=2^{n}-2\text{wt}({\bf w})$ $\displaystyle\text{wt}({\bf w})$
$\displaystyle=\frac{2^{n}-\widehat{f+g+h}({\bf w})}{2}$

All other cases follow in the same manner, which is similar to Theorem 3.1 in
[7]. Hence, we conclude the result. ∎

######  Theorem 3.2.

Let $\mathscr{C}_{f,g,h}$ be a binary code defined in (3.1). Then
$\mathscr{C}_{f,g,h}$ is minimal if and only if following two conditions hold
simultaneously for any ${\bf x},{\bf y}\in\mathbb{F}_{2}^{n}$

  1. 1.

For any $\phi_{1},\phi_{2}\in\mathcal{F}$ and ${\bf x}\neq{\bf y}$, we have

$\widehat{\phi_{1}}({\bf x})+\widehat{\phi_{2}}({\bf y})\neq
2^{n}\quad\text{and}\quad\widehat{\phi_{1}}({\bf x})-\widehat{\phi_{2}}({\bf
y})\neq 2^{n}.$

  2. 2.

For any $\phi_{1},\phi_{2}\in\mathcal{F}$ with $\phi_{1}\neq\phi_{2}$, we have

$\widehat{\phi_{1}}({\bf x}+{\bf y})+\widehat{\phi_{2}}({\bf
x})-\widehat{\phi_{1}+\phi_{2}}({\bf y})\neq 2^{n}$

###### Proof.

First, we rewrite each codeword in $\mathscr{C}_{f,g,h}$ as follows,
$c_{f,g,h}({\bf w})=\alpha{\bf f}+\beta{\bf g}+\gamma{\bf h}+c({\bf w})$,
where ${\bf f}=(f({\bf x}))_{{\bf x}\in\mathbb{F}_{2}^{n*}}$, ${\bf g}=(g({\bf
x}))_{{\bf x}\in\mathbb{F}_{2}^{n*}}$, ${\bf h}=(h({\bf x}))_{{\bf
x}\in\mathbb{F}_{2}^{n*}}$ and $c({\bf w})\in\mathscr{C}=\\{c({\bf w})=({\bf
w}\cdot{\bf x})_{{\bf x}\in\mathbb{F}_{2}^{n*}}\ |\ {\bf
w}\in\mathbb{F}_{2}^{n}\\}$. Note that $\mathscr{C}$ is a simplex code with
parameters $[2^{n}-1,n,2^{n-1}]$. Clearly, each codeword in $\mathscr{C}$ has
weight $2^{n-1}$. Also, from the Theorem 3.1, we have, $\text{wt}({\bf f}+{\bf
g}+c({\bf w}))=2^{n-1}-\frac{\widehat{f+g}({\bf w})}{2}$. We next consider the
following cases for codewords in $\mathscr{C}_{f,g,h}$
Case I: If $c_{1},c_{2}\in\\{c({\bf w}),{\bf f}+c({\bf w}),{\bf g}+c({\bf
w}),{\bf h}+c({\bf w}),{\bf f+g}+c({\bf w}),{\bf f+h}+c({\bf w}),{\bf
g+h}+c({\bf w}),{\bf f+g+h}+c({\bf w})\\}$, then consider the following
subcases,
(a) For $c_{1}=c({\bf w})$ and $c_{2}\in\\{{\bf f}+c({\bf w}),{\bf g}+c({\bf
w}),{\bf h}+c({\bf w}),{\bf f+g}+c({\bf w}),{\bf f+h}+c({\bf w}),{\bf
g+h}+c({\bf w}),{\bf f+g+h}+c({\bf w})\\}$. We give the proof in the case
$c_{2}={\bf f+g+h}+c({\bf w})$, all other cases also follow from this. From
Lemma 2.4 and Theorem 3.1, we get

$\displaystyle c_{1}\preceq c_{2}\Longleftrightarrow$
$\displaystyle\text{wt}({\bf f+g+h})=\text{wt}({\bf f+g+h}+c({\bf
w}))-2^{n-1}$ $\displaystyle\Longleftrightarrow$
$\displaystyle\widehat{f+g+h}({\bf 0})-\widehat{f+g+h}({\bf w})=2^{n}$

and

$\displaystyle c_{2}\preceq c_{1}\Longleftrightarrow$
$\displaystyle\text{wt}({\bf f+g+h})=2^{n-1}-\text{wt}({\bf f+g+h}+c({\bf
w}))$ $\displaystyle\Longleftrightarrow$ $\displaystyle\widehat{f+g+h}({\bf
0})+\widehat{f+g+h}({\bf w})=2^{n}$

(b) For $c_{1},c_{2}\in\\{{\bf f}+c({\bf w}),{\bf g}+c({\bf w}),{\bf h}+c({\bf
w}),{\bf f+g}+c({\bf w}),{\bf f+h}+c({\bf w}),{\bf g+h}+c({\bf w}),{\bf
f+g+h}+c({\bf w})\\}$ with $c_{1}\neq c_{2}$. In this case, we give the proof
for the following subcases all other subcases follow in the same way
(i) If $c_{1}={\bf f}+c({\bf w})$ and $c_{2}={\bf f+g}+c({\bf w})$, then

$\displaystyle c_{1}\preceq c_{2}\Longleftrightarrow$
$\displaystyle\text{wt}({\bf g})=\text{wt}({\bf f+g}+c({\bf
w}))-\text{wt}({\bf f}+c({\bf w}))$ $\displaystyle\Longleftrightarrow$
$\displaystyle\widehat{g}({\bf 0})+\widehat{f}({\bf w})-\widehat{f+g}({\bf
w})=2^{n}$

and

$\displaystyle c_{2}\preceq c_{1}\Longleftrightarrow$
$\displaystyle\text{wt}({\bf g})=\text{wt}({\bf f}+c({\bf w}))-\text{wt}({\bf
f+g}+c({\bf w}))$ $\displaystyle\Longleftrightarrow$
$\displaystyle\widehat{g}({\bf 0})+\widehat{f+g}({\bf w})-\widehat{f}({\bf
w})=2^{n}$

(ii) If $c_{1}={\bf f}+c({\bf w})$ and $c_{2}={\bf g+h}+c({\bf w})$, then

$\displaystyle c_{1}\preceq c_{2}\Longleftrightarrow$
$\displaystyle\text{wt}({\bf f+g+h})=\text{wt}({\bf g+h}+c({\bf
w}))-\text{wt}({\bf f}+c({\bf w}))$ $\displaystyle\Longleftrightarrow$
$\displaystyle\widehat{f+g+h}({\bf 0})+\widehat{f}({\bf w})-\widehat{g+h}({\bf
w})=2^{n}$

and

$\displaystyle c_{2}\preceq c_{1}\Longleftrightarrow$
$\displaystyle\text{wt}({\bf f+g+h})=\text{wt}({\bf f}+c({\bf
w}))-\text{wt}({\bf g+h}+c({\bf w}))$ $\displaystyle\Longleftrightarrow$
$\displaystyle\widehat{f+g+h}({\bf 0})+\widehat{g+h}({\bf w})-\widehat{f}({\bf
w})=2^{n}$

Case II: If $c_{1}\in\\{c({\bf w_{1}}),{\bf f}+c({\bf w_{1}}),{\bf g}+c({\bf
w_{1}}),{\bf h}+c({\bf w_{1}}),{\bf f+g}+c({\bf w_{1}}),{\bf f+h}+c({\bf
w_{1}}),{\bf g+h}+c({\bf w_{1}}),{\bf f+g+h}+c({\bf w_{1}})\\}$ and
$c_{2}\in\\{c({\bf w_{2}}),{\bf f}+c({\bf w_{2}}),{\bf g}+c({\bf w_{2}}),{\bf
h}+c({\bf w_{2}}),{\bf f+g}+c({\bf w_{2}}),{\bf f+h}+c({\bf w_{2}}),{\bf
g+h}+c({\bf w_{2}}),{\bf f+g+h}+c({\bf w_{2}})\\}$, where ${\bf w_{1}}\neq{\bf
w_{2}}$ and both are non-zero, then consider the following subcases:
(a) For $c_{1}=c({\bf w_{1}})$ and $c_{2}\in\\{{\bf f}+c({\bf w_{2}}),{\bf
g}+c({\bf w_{2}}),{\bf h}+c({\bf w_{2}}),{\bf f+g}+c({\bf w_{2}}),{\bf
f+h}+c({\bf w_{2}}),{\bf g+h}+c({\bf w_{2}}),{\bf f+g+h}+c({\bf w_{2}})\\}$,
we give the proof in the case $c_{2}={\bf f+g+h}+c({\bf w_{2}})$, all other
cases follow in the same manner. Now, we have

$\displaystyle c_{1}\preceq c_{2}\Longleftrightarrow$
$\displaystyle\text{wt}({\bf f+g+h}+c({\bf w_{1}}+{\bf w_{2}}))=\text{wt}({\bf
f+g+h}+c({\bf w_{2}}))-2^{n-1}$ $\displaystyle\Longleftrightarrow$
$\displaystyle\widehat{f+g+h}({\bf w_{1}}+{\bf w_{2}})-\widehat{f+g+h}({\bf
w_{2}})=2^{n}$

and

$\displaystyle c_{2}\preceq c_{1}\Longleftrightarrow$
$\displaystyle\text{wt}({\bf f+g+h+c({\bf w_{1}}+{\bf
w_{2}})})=2^{n-1}-\text{wt}({\bf f+g+h}+c({\bf w_{2}}))$
$\displaystyle\Longleftrightarrow$ $\displaystyle\widehat{f+g+h}({\bf
w_{1}}+{\bf w_{2}})+\widehat{f+g+h}({\bf w_{2}})=2^{n}$

(b) For $c_{1}={\bm{\phi}}+c({\bf w_{1}})$ and $c_{2}={\bm{\phi}}+c({\bf
w_{2}})$, where ${\bm{\phi}}\in\\{{\bf f},{\bf g},{\bf h},{\bf f+g},{\bf
f+h},{\bf g+h},{\bf f+g+h}\\}$, we give the proof in the case
${\bm{\phi}}={\bf g+h}$, all other cases follow in the same manner. Now, we
have

$\displaystyle c_{1}\preceq c_{2}\Longleftrightarrow$
$\displaystyle\text{wt}(c({\bf w_{1}}+{\bf w_{2}}))=\text{wt}({\bf g+h}+c({\bf
w_{2}}))-\text{wt}({\bf g+h}+c({\bf w_{1}}))$
$\displaystyle\Longleftrightarrow$ $\displaystyle\widehat{g+h}({\bf
w_{1}})-\widehat{g+h}({\bf w_{2}})=2^{n}$

and

$\displaystyle c_{2}\preceq c_{1}\Longleftrightarrow$
$\displaystyle\text{wt}(c({\bf w_{1}}+{\bf w_{2}}))=\text{wt}({\bf g+h}+c({\bf
w_{1}}))-\text{wt}({\bf g+h}+c({\bf w_{2}}))$
$\displaystyle\Longleftrightarrow$ $\displaystyle\widehat{g+h}({\bf
w_{2}})-\widehat{g+h}({\bf w_{1}})=2^{n}$

(c) For $c_{1}={\bm{\phi}_{1}}+c({\bf w_{1}})$ and
$c_{2}={\bm{\phi}_{2}}+c({\bf w_{2}})$,
where ${\bm{\phi}_{1}},{\bm{\phi}_{2}}\in\\{{\bf f},{\bf g},{\bf h},{\bf
f+g},{\bf f+h},{\bf g+h},{\bf f+g+h}\\}$, we give the proof in the case
${\bm{\phi}_{1}}={\bf f}$ and ${\bm{\phi}_{2}}={\bf g+h}$, all other cases
follow in the same manner. Now, we have

$\displaystyle c_{1}\preceq c_{2}\Longleftrightarrow$
$\displaystyle\text{wt}({\bf f+g+h}+c({\bf w_{1}}+{\bf w_{2}}))=\text{wt}({\bf
g+h}+c({\bf w_{2}}))-\text{wt}({\bf f}+c({\bf w_{1}}))$
$\displaystyle\Longleftrightarrow$ $\displaystyle\widehat{f+g+h}({\bf
w_{1}}+{\bf w_{2}})+\widehat{f}({\bf w_{1}})-\widehat{g+h}({\bf w_{2}})=2^{n}$

and

$\displaystyle c_{2}\preceq c_{1}\Longleftrightarrow$
$\displaystyle\text{wt}({\bf f+g+h}+c({\bf w_{1}}+{\bf w_{2}}))=\text{wt}({\bf
f}+c({\bf w_{1}}))-\text{wt}({\bf g+h}+c({\bf w_{2}}))$
$\displaystyle\Longleftrightarrow$ $\displaystyle\widehat{f+g+h}({\bf
w_{1}}+{\bf w_{2}})+\widehat{g+h}({\bf w_{2}})-\widehat{f}({\bf w_{1}})=2^{n}$

∎

##  4 A family of Minimal Binary Linear Codes of dimension $n+3$

In this section, with the help of code $\mathscr{C}_{f,g,h}$, we construct a
family of minimal binary linear codes using a class of special Boolean
functions. In particular, we provide a class of the codes, which doesn’t obey
Ashikhmin-Barg condition.
Throughout this section, we assume that $n$ is a positive even integer and
$t=\frac{n}{2}$. A partial spread of order $\nu$ in $\mathbb{F}_{2}^{n}$ is a
collection of $\nu$ pairwise disjoint t-dimensional subspaces
$W_{1},W_{2},\ldots,W_{\nu}$. Also, observe that $\nu\leq 2^{t}+1$. Partial
spread can be used to construct the Bent function[3]. In a similar manner, we
will use partial spread to obtain a class of special Boolean functions as
defined below.
Let $f_{i}:\mathbb{F}_{2}^{n}\rightarrow\mathbb{F}_{2}$ be a Boolean function,
and for $1\leq i\leq 2^{t}+1$, defined as

$f_{i}({\bf x})=\begin{cases}1,\text{ if }{\bf x}\in W_{i}\setminus\\{{\bf
0}\\}\\\ 0,\text{ Otherwise}\end{cases}$

For $1\leq s_{1},s_{2},s_{3}\leq 2^{t}+1$, let
$\mathcal{A}_{1}=\\{i_{1,}i_{2},\ldots,i_{s_{1}}\\},\mathcal{A}_{2}=\\{j_{1},j_{2},\ldots,j_{s_{2}}\\}$
and $\mathcal{A}_{3}=\\{k_{1},k_{2},\ldots,k_{s_{3}}\\}$. Define $f,g$ and $h$
as follows

$f=\sum_{i\in\mathcal{A}_{1}}f_{i},\quad
g=\sum_{i\in\mathcal{A}_{2}}f_{i}\quad\text{and}\quad\
h=\sum_{i\in\mathcal{A}_{3}}f_{i}$ (4.1)

We define $f+g,f+h,g+h$ and $f+g+h$ as follows

$f+g=\sum_{i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}}f_{i},\quad
f+h=\sum_{i\in\mathcal{A}_{1}\Delta\mathcal{A}_{3}}f_{i}\quad\quad
g+h=\sum_{i\in\mathcal{A}_{2}\Delta\mathcal{A}_{3}}f_{i}\quad\text{and}\quad
f+g+h=\sum_{i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3}}f_{i}$
(4.2)

We use the following notation to denote the cardinality of the above defined
set:
$|\mathcal{A}_{1}\Delta\mathcal{A}_{2}|=\chi_{12}=s_{1}+s_{2}-2s_{12}$
$|\mathcal{A}_{1}\Delta\mathcal{A}_{3}|=\chi_{13}=s_{1}+s_{3}-2s_{13},$
$|\mathcal{A}_{2}\Delta\mathcal{A}_{3}|=\chi_{23}=s_{2}+s_{3}-2s_{23}$
$|\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3}|=\chi_{123}=s_{1}+s_{2}+s_{3}-2s_{12}-2s_{13}-2s_{23}+4s_{123}$
With the above notations, we give the weight distribution of
$\mathscr{C}_{f,g,h}$

######  Theorem 4.1.

Let the notations defined above, if $\mathcal{A}_{i}$ are distinct sets for
$i\in\\{1,2,3\\}$ and
$\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3}\neq\emptyset$, then
$\mathscr{C}_{f,g,h}$ is a $[2^{n}-1,n+3]$ binary linear code whose weight
distribution is given in Table 1

Weight $w$ |  Multiplicity $A_{W}$ |  Weight $w$ |  Multiplicity $A_{W}$
---|---|---|---
0 | 1 | $2^{n-1}-s_{3}$ | $(2^{t}+1-s_{3})(2^{t}-1)$
$s_{1}(2^{t}-1)$ | 1 | $2^{n-1}-\chi_{12}$ | $(2^{t}+1-\chi_{12})(2^{t}-1)$
$s_{2}(2^{t}-1)$ | 1 | $2^{n-1}-\chi_{13}$ | $(2^{t}+1-\chi_{13})(2^{t}-1)$
$s_{3}(2^{t}-1)$ | 1 | $2^{n-1}-\chi_{23}$ | $(2^{t}+1-\chi_{23})(2^{t}-1)$
$\chi_{12}(2^{t}-1)$ | 1 | $2^{n-1}-\chi_{123}$ | $(2^{t}+1-\chi_{123})(2^{t}-1)$
$\chi_{13}(2^{t}-1)$ | 1 | $2^{n-1}+2^{t}-s_{1}$ | $s_{1}(2^{t}-1)$
$\chi_{23}(2^{t}-1)$ | 1 | $2^{n-1}+2^{t}-s_{2}$ | $s_{2}(2^{t}-1)$
|  | $2^{n-1}+2^{t}-s_{3}$ | $s_{3}(2^{t}-1)$
$\chi_{123}(2^{t}-1)$ | 1 | $2^{n-1}+2^{t}-\chi_{12}$ | $\chi_{12}(2^{t}-1)$
$2^{n-1}$ | $2^{n}-1$ | $2^{n-1}+2^{t}-\chi_{13}$ | $\chi_{13}(2^{t}-1)$
$2^{n-1}-s_{1}$ | $(2^{t}+1-s_{1})(2^{t}-1)$ | $2^{n-1}+2^{t}-\chi_{23}$ | $\chi_{23}(2^{t}-1)$
$2^{n-1}-s_{2}$ | $(2^{t}+1-s_{2})(2^{t}-1)$ | $2^{n-1}+2^{t}-\chi_{123}$ | $\chi_{123}(2^{t}-1)$
Table 1: Weight Distribution of $\mathscr{C}_{f,g,h}$

###### Proof.

For any ${\bf w}\in\mathbb{F}_{2}^{n}$, we have three cases ${\bf w}={\bf 0}$
or ${\bf w}\neq{\bf 0}$ and ${\bf w}\notin W_{i}^{\perp}\ \forall\ i$ or ${\bf
w}\in W^{\perp}_{i}$ for some $i$. From the definition of $f$ and $f_{i}$, we
have

$\displaystyle\tilde{f}({\bf 0})$ $\displaystyle=\sum_{{\bf
x}\in\mathbb{F}^{n}_{2}}\sum_{i\in\mathcal{A}_{i}}f_{i}({\bf x})(-1)^{{\bf
0}\cdot{\bf x}}$ $\displaystyle=(|W_{i}|-1)+(|W_{i}|-1)+\cdots+(|W_{i}|-1)$
$\displaystyle=s_{1}(2^{t}-1)$

For ${\bf w}\notin W_{i}^{\perp}\ \forall i\in\mathcal{A}_{1}$. Note that, if
${\bf w}\notin W_{i}^{\perp}$, then ${\bf w}.{\bf x}=0$ for exactly half of
the elements in $W_{i}$ and hence

$\displaystyle\sum_{{\bf x}\in\mathbb{F}_{2}^{n}}f_{i}({\bf x})(-1)^{{\bf
w}\cdot{\bf x}}$ $\displaystyle=-1$ $\displaystyle\Rightarrow\tilde{f}({\bf
w})$ $\displaystyle=\sum_{{\bf
x}\in\mathbb{F}_{2}^{n}}\sum_{i\in\mathcal{A}_{1}}f_{i}({\bf x})(-1)^{{\bf
w}\cdot{\bf x}}$ $\displaystyle=-s_{1}$

Similarly, we get $\tilde{f}({\bf w})=2^{t}-s_{1}$, if ${\bf w}\in W_{i}$, for
some $i\in\mathcal{A}_{1}$. From Lemma 2.3, we have

$\widehat{f}({\bf w})=\begin{cases}2^{n}-2s_{1}(2^{t}-1)&\text{if }{\bf
w}=0\\\ 2s_{1}&\text{if }{\bf w}\notin W_{i}^{\perp}\ \forall
i\in\mathcal{A}_{1}\\\ -2^{t+1}+2s_{1}&\text{if }{\bf w}\neq{\bf 0}\text{ and
}{\bf w}\in W_{i}^{\perp}\text{ for some }i\in\mathcal{A}_{1}\end{cases}$

In a similar way, we find $\widehat{\phi}$, for all $\phi\in\mathcal{F}$ and
the remaining proof follows from Theorem 3.1. ∎

According to Theorem 3.1, for the code $\mathscr{C}_{f,g,h}$ to be minimal, we
impose the following conditions on $\mathcal{A}_{1},\mathcal{A}_{2}$ and
$\mathcal{A}_{3}$.

  1. 1.

$\mathcal{A}_{k}\not\subset\mathcal{A}_{i}\Delta\mathcal{A}_{j}\quad\text{and}\quad\mathcal{A}_{i}\Delta\mathcal{A}_{j}\not\subset
A_{k},\quad\forall i,j,k\in\\{1,2,3\\}$

  2. 2.

For atleast two set in
$\\{\mathcal{A}_{1}\cap\mathcal{A}_{2},\mathcal{A}_{1}\cap\mathcal{A}_{3},\mathcal{A}_{2}\cap\mathcal{A}_{3}\\}$,
we have
$\mathcal{A}_{1}\cap\mathcal{A}_{2}\cap\mathcal{A}_{3}\neq\mathcal{A}_{i}\cap\mathcal{A}_{j}$
and $\mathcal{A}_{1}\cap\mathcal{A}_{2}\cap\mathcal{A}_{3}\neq\phi$

  3. 3.

$\chi_{ij}=|\mathcal{A}_{i}\Delta\mathcal{A}_{j}|\geq 2$ and
$\chi_{123}=|\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3}|\geq 2$

Observe that these conditions are generalized version of the special case
given in [7].
Following lemmas are immediate consequences of the above condition, which will
be useful in proving $\mathscr{C}_{f,g,h}$ is a minimal linear code.

######  Lemma 4.2.

Let $\mathcal{A}_{i}$ are non-empty sets, for $i\in\\{1,2,3\\}$. If condition
1 is true for all $\mathcal{A}_{i}$, then following hold:

  1. 1.

$\mathcal{A}_{i}\not\subset\mathcal{A}_{j}$ and
$\mathcal{A}_{i}\cap\mathcal{A}_{j}\neq\phi$, for all $i,j$.

  2. 2.

$(\mathcal{A}_{i}\cap\mathcal{A}_{j})\Delta(\mathcal{A}_{i}\cap\mathcal{A}_{k})\subsetneq\mathcal{A}_{j}\Delta\mathcal{A}_{k}$.

###### Proof.

  1. 1.

Suppose, $\mathcal{A}_{i}\subseteq\mathcal{A}_{j}$, then
$\mathcal{A}_{i}\cup\mathcal{A}_{j}=\mathcal{A}_{j}$ and
$\mathcal{A}_{i}\cap\mathcal{A}_{j}=\mathcal{A}_{i}$, this give,
$\mathcal{A}_{i}\Delta\mathcal{A}_{j}\subseteq\mathcal{A}_{j}$, which is a
contradiction.
If $\mathcal{A}_{i}\cap\mathcal{A}_{j}=\phi$, then
$\mathcal{A}_{i}\subseteq\mathcal{A}_{i}\Delta\mathcal{A}_{j}$, which is also
not possible.

  2. 2.

If ${\bf
x}\in(\mathcal{A}_{i}\cap\mathcal{A}_{j})\Delta(\mathcal{A}_{i}\cap\mathcal{A}_{k})$,
which implies ${\bf
x}\in\mathcal{A}_{i}\cap(\mathcal{A}_{j}\cup\mathcal{A}_{k})$ and ${\bf
x}\notin\mathcal{A}_{i}\cap\mathcal{A}_{j}\cap\mathcal{A}_{k}$ , we have
$(\mathcal{A}_{i}\cap\mathcal{A}_{j})\Delta(\mathcal{A}_{i}\cap\mathcal{A}_{k})\subseteq\mathcal{A}_{j}\Delta\mathcal{A}_{k}$.
Now, assume that,
$(\mathcal{A}_{i}\cap\mathcal{A}_{j})\Delta(\mathcal{A}_{i}\cap\mathcal{A}_{k})=\mathcal{A}_{j}\Delta\mathcal{A}_{k}$.
Note that,
$(\mathcal{A}_{i}\cap\mathcal{A}_{j})\Delta(\mathcal{A}_{i}\cap\mathcal{A}_{k})\subseteq\mathcal{A}_{i}$,
which implies, $\mathcal{A}_{j}\Delta\mathcal{A}_{k}\subseteq A_{i}$. Which is
a contradiction.

∎

######  Lemma 4.3.

Let $\mathcal{A}_{i}$ are non-empty sets, for $i\in\\{1,2,3\\}$ and $2\leq
s_{i}\leq 2^{t-1}$. If conditions 1 and 2 hold simultaneously, then
$\chi_{ij}\leq 2^{t}-2$ and $\chi_{123}\leq 2^{t}-2$.

###### Proof.

Since, $\chi_{ij}=s_{i}+s_{j}-2s_{ij}\leq 2^{t-1}+2^{t-1}-2=2^{t}-2$.
Note that, $|\mathcal{A}_{1}\cup\mathcal{A}_{2}\cup\mathcal{A}_{3}|\leq
2^{t}+1$. We know that,

$\displaystyle\chi_{123}$
$\displaystyle=s_{1}+s_{2}+s_{3}-2s_{12}-2s_{13}-2s_{23}+4s_{123}$
$\displaystyle=s_{1}+s_{2}+s_{3}-s_{12}-s_{13}-s_{23}+2s_{123}-(s_{12}+s_{13}+s_{23}-2s_{123})$
$\displaystyle\leq 2^{t}+1-(s_{12}+s_{13}+s_{23}-2s_{123})$ $\displaystyle\leq
2^{t}+1-3=2^{t}-2$

∎

From onwards, we assume that sets $\mathcal{A}_{1},\mathcal{A}_{2}$ and
$\mathcal{A}_{3}$ satisfy the above stated conditions, we also give the values
of Walsh transforms for $\phi\in\mathcal{F}$ in Table 2.

Walsh Transform/ Condition | ${\bf w}={\bf 0}$ | ${\bf w}\notin W^{\perp}_{i},\ \forall i\in\mathcal{A}$ |  ${\bf w}\in W^{\perp}_{i}$ for some $i\in\mathcal{A}$
---|---|---|---
$\widehat{f}({\bf w})$ | $2^{n}-2s_{1}(2^{t}-1)$ | $2s_{1}$ | $-2^{t+1}+2s_{1}$
$\widehat{g}({\bf w})$ | $2^{n}-2s_{2}(2^{t}-1)$ | $2s_{2}$ | $-2^{t+1}+2s_{2}$
$\widehat{h}({\bf w})$ | $2^{n}-2s_{3}(2^{t}-1)$ | $2s_{3}$ | $-2^{t+1}+2s_{3}$
$\widehat{f+g}({\bf w})$ | $2^{n}-2\chi_{12}(2^{t}-1)$ | $2\chi_{12}$ | $-2^{t+1}+2\chi_{12}$
$\widehat{f+h}({\bf w})$ | $2^{n}-2\chi_{13}(2^{t}-1)$ | $2\chi_{13}$ | $-2^{t+1}+2\chi_{13}$
$\widehat{g+h}({\bf w})$ | $2^{n}-2\chi_{23}(2^{t}-1)$ | $2\chi_{23}$ | $-2^{t+1}+2\chi_{23}$
$\widehat{f+g+h}({\bf w})$ | $2^{n}-2\chi_{123}(2^{t}-1)$ | $2\chi_{123}$ | $-2^{t+1}+2\chi_{123}$
Table 2: The values of Walsh Transforms

######  Proposition 4.4.

Let $n\geq 6$ and $2\leq s\leq 2^{t-1}$, for $s\in\\{s_{1},s_{2},s_{3}\\}$ and
$\mathcal{F}$ be a collection of Boolean functions defined in 4.1 and 4.2.
Then, for ${\bf x},{\bf y}\in\mathbb{F}_{2}^{n}$ with ${\bf x}\neq{\bf y}$, we
have

$\widehat{\phi}_{1}({\bf x})+\widehat{\phi}_{2}({\bf y})\neq 2^{n}$

for all, $\phi_{1},\phi_{2}\in\mathcal{F}$.

###### Proof.

We can observe various cases by considering different values of $\phi_{1}$ and
$\phi_{2}$. Within each case, there are seven subcases that arise from
different choices of ${\bf x}$ and ${\bf y}$. We discuss the proof for two
subcases of each possibility.
Case I: For $\phi_{1}=\phi_{2}=f$, we have
(a) If ${\bf x}={\bf 0}$ and ${\bf y}\notin W_{i}^{\perp}$, for all
$i\in\mathcal{A}_{1}$.

$\displaystyle\widehat{f}({\bf 0})+\widehat{f}({\bf y})\neq
2^{n}\Leftrightarrow 2^{n}-2s_{1}(2^{t}-1)+2s_{1}\neq 2^{n}\Leftrightarrow
s_{1}(2^{t}-2)\neq 0$

(b) If ${\bf x}=0$ and ${\bf y}\in W_{i}^{\perp}$, for some
$i\in\mathcal{A}_{1}$.

$\displaystyle\widehat{f}({\bf 0})+\widehat{f}({\bf y})\neq
2^{n}\Leftrightarrow 2^{n}-2s_{1}(2^{t}-1)-2^{t+1}+2s_{1}\neq
2^{n}\Leftrightarrow s_{1}(2^{t}-2)\neq-2^{t}$

Proof of $\phi_{1}=\phi_{2}=f+g$ and $\phi_{1}=\phi_{2}=f+g+h$, follows from
Case I and Lemma 4.3.
Case II: For $\phi_{1}=f$ and $\phi_{2}=g$, we have
(a) If ${\bf x}={\bf 0}$ and ${\bf y}\notin W_{i}^{\perp}$, for all
$i\in\mathcal{A}_{2}$.

$\displaystyle\widehat{f}({\bf 0})+\widehat{g}({\bf y})\neq
2^{n}\Leftrightarrow 2^{n}-2s_{1}(2^{t}-1)+2s_{2}\neq 2^{n}\Leftrightarrow
s_{2}\neq s_{1}(2^{t}-1)$

(b) If ${\bf x}=0$ and ${\bf y}\in W_{i}^{\perp}$, for some
$i\in\mathcal{A}_{2}$.

$\displaystyle\widehat{f}({\bf 0})+\widehat{g}({\bf y})\neq
2^{n}\Leftrightarrow 2^{n}-2s_{1}(2^{t}-1)-2^{t+1}+2s_{2}\neq
2^{n}\Leftrightarrow s_{1}+s_{2}\neq 2^{t}s_{1}+2^{t}$

Case III: For $\phi_{1}=f$ and $\phi_{2}=f+g$, we have
(a) If ${\bf x}={\bf 0}$ and ${\bf y}\notin W_{i}^{\perp}$, for all
$i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}$.

$\displaystyle\widehat{f}({\bf 0})+\widehat{f+g}({\bf y})\neq
2^{n}\Leftrightarrow 2^{n}-2s_{1}(2^{t}-1)+2\chi_{12}\neq
2^{n}\Leftrightarrow\chi_{12}\neq s_{1}(2^{t}-1)$

(b) If ${\bf y}={\bf 0}$ and ${\bf x}\notin W_{i}^{\perp}$, for all
$i\in\mathcal{A}_{1}$.

$\displaystyle\widehat{f}({\bf x})+\widehat{f+g}({\bf 0})\neq
2^{n}\Leftrightarrow 2s_{1}+2^{n}-2\chi_{12}(2^{t}-1)\neq 2^{n}\Leftrightarrow
s_{1}\neq\chi_{12}(2^{t}-1)$

Case IV: If $\phi_{1}=f$ and $\phi_{2}=g+h$, then this case follows from Case
III.
Case V: For $\phi_{1}=f$ and $\phi_{2}=f+g+h$, we have
(a) If ${\bf x}={\bf 0}$ and ${\bf y}\notin W_{i}^{\perp}$, for all
$i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3}$.

$\displaystyle\widehat{f}({\bf 0})+\widehat{f+g+h}({\bf y})\neq
2^{n}\Leftrightarrow 2\chi_{123}+2^{n}-2s_{1}(2^{t}-1)\neq
2^{n}\Leftrightarrow\chi_{123}\neq s_{1}(2^{t}-1)$

(b) If ${\bf y}={\bf 0}$ and ${\bf x}\notin W_{i}^{\perp}$, for all
$i\in\mathcal{A}_{1}$.

$\displaystyle\widehat{f}({\bf x})+\widehat{f+g+h}({\bf 0})\neq
2^{n}\Leftrightarrow 2s_{1}+2^{n}-2\chi_{123}(2^{t}-1)\neq
2^{n}\Leftrightarrow s_{1}\neq\chi_{123}(2^{t}-1)$

Case VI: For $\phi_{1}=f+g$ and $\phi_{2}=f+g+h$, we have
(a) If ${\bf x}={\bf 0}$ and ${\bf y}\notin W_{i}^{\perp}$, for all
$i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3}$.

$\displaystyle\widehat{f+g}({\bf 0})+\widehat{f+g+h}({\bf y})\neq
2^{n}\Leftrightarrow 2\chi_{123}+2^{n}-2\chi_{12}(2^{t}-1)\neq
2^{n}\Leftrightarrow\chi_{123}\neq\chi_{12}(2^{t}-1)$

(b) If ${\bf y}={\bf 0}$ and ${\bf x}\in W_{i}^{\perp}$, for some
$i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}$.

$\displaystyle\widehat{f+g}({\bf x})+\widehat{f+g+h}({\bf 0})\neq
2^{n}\Leftrightarrow 2\chi_{12}+2^{n}-2\chi_{123}(2^{t}-1)\neq
2^{n}\Leftrightarrow\chi_{12}\neq\chi_{123}(2^{t}-1)$

All other cases follow in the similar manner. Hence, $\widehat{\phi}_{1}({\bf
x})+\widehat{\phi}_{2}({\bf y})\neq 2^{n}$, for all
$\phi_{1},\phi_{2}\in\mathcal{F}$ and ${\bf x},{\bf y}\in\mathbb{F}_{2}^{n}$,
with ${\bf x}\neq{\bf y}$. ∎

######  Proposition 4.5.

Let $n\geq 6$ and $2\leq s\leq 2^{t-1}$, for $s\in\\{s_{1},s_{2},s_{3}\\}$ and
$\mathcal{F}$ be a collection of Boolean functions defined in 4.1 and 4.2.
Then, for ${\bf x},{\bf y}\in\mathbb{F}_{2}^{n}$ with ${\bf x}\neq{\bf y}$, we
have

$\widehat{\phi}_{1}({\bf x})-\widehat{\phi}_{2}({\bf y})\neq 2^{n}$

for all, $\phi_{1},\phi_{2}\in\mathcal{F}$.

###### Proof.

We can observe various cases by considering different values of $\phi_{1}$ and
$\phi_{2}$. Within each case, there are seven subcases that arise from
different choices of ${\bf x}$ and ${\bf y}$. We discuss the proof for few
subcases for each possinilities.
Case I: For $\phi_{1}=\phi_{2}=f$, consider following subcases:
(a) If ${\bf x}={\bf 0}$ and ${\bf y}\notin W_{i}^{\perp}$, for all
$i\in\mathcal{A}_{1}$, then

$\displaystyle\widehat{f}({\bf 0})-\widehat{f}({\bf y})\neq
2^{n}\Leftrightarrow 2^{n}-2s_{1}(2^{t}-1)-2s_{1}\neq 2^{n}\Leftrightarrow
2^{t+1}s_{1}\neq 0$ and $\displaystyle\widehat{f}({\bf y})-\widehat{f}({\bf
0})\neq 2^{n}\Leftrightarrow 2s_{1}-2^{n}+2s_{1}(2^{t}-1)\neq
2^{n}\Leftrightarrow 2^{t+1}s_{1}\neq 2^{n+1}\Leftrightarrow s_{1}\neq 2^{t}$

(b) If ${\bf x}={\bf 0}$ and ${\bf y}\in W_{i}^{\perp}$, for some
$i\in\mathcal{A}_{1}$, then

$\displaystyle\widehat{f}({\bf 0})-\widehat{f}({\bf y})\neq
2^{n}\Leftrightarrow 2^{n}-2s_{1}(2^{t}-1)+2^{t+1}-2s_{1}\neq
2^{n}\Leftrightarrow-2s_{1}2^{t}+2^{t+1}\neq 0\Leftrightarrow s_{1}\neq 1$ and
$\displaystyle\widehat{f}({\bf y})-\widehat{f}({\bf 0})\neq
2^{n}\Leftrightarrow-2^{t+1}+2s_{1}-2^{n}+2s_{1}(2^{t}-1)\neq
2^{n}\Leftrightarrow s_{1}\neq 2^{t}+1$

Proof of $\phi_{1}=\phi_{2}=f+g$ and $\phi_{1}=\phi_{2}=f+g+h$, follows from
Case I and Lemma 4.3.
Case II: For $\phi_{1}=f$ and $\phi_{2}=g$, consider following subcases:
(a) If ${\bf x}={\bf 0}$ and ${\bf y}\notin W_{i}^{\perp}$, for all
$i\in\mathcal{A}_{2}$, then

$\displaystyle\widehat{f}({\bf 0})-\widehat{g}({\bf y})\neq
2^{n}\Leftrightarrow 2^{n}-2s_{1}(2^{t}-1)-2s_{2}\neq 2^{n}\Leftrightarrow
s_{2}\neq-s_{1}(2^{t}-1)$

(b) If ${\bf x}={\bf 0}$ and ${\bf y}\in W_{i}^{\perp}$, for some
$i\in\mathcal{A}_{2}$, then

$\displaystyle\widehat{f}({\bf 0})-\widehat{g}({\bf y})\neq
2^{n}\Leftrightarrow 2^{n}-2s_{1}(2^{t}-1)+2^{t+1}-2s_{2}\neq
2^{n}\Leftrightarrow s_{1}-s_{2}\neq 2^{t}(s_{1}-1)$

(c) If ${\bf x}\notin W_{i}$, for all $i\in\mathcal{A}_{1}$ and ${\bf y}={\bf
0}$, then

$\displaystyle\widehat{f}({\bf x})-\widehat{g}({\bf 0})\neq
2^{n}\Leftrightarrow 2s_{1}-2^{n}+2s_{2}(2^{t}-1)\neq 2^{n}\Leftrightarrow
s_{1}+s_{2}(2^{t}-1)\neq 2^{n}$

(d) If ${\bf x}\in W_{i}$, for some $i\in\mathcal{A}_{1}$ and ${\bf y}={\bf
0}$, then

$\displaystyle\widehat{f}({\bf x})-\widehat{g}({\bf 0})\neq
2^{n}\Leftrightarrow-2^{t+1}+2s_{1}-2^{n}+2s_{2}(2^{t}-1)\neq
2^{n}\Leftrightarrow s_{1}+s_{2}(2^{t}-1)\neq 2^{n}+2^{t}$

Case III: If $\phi_{1}=g$ and $\phi_{2}=f$, then this case follows from Case
II.
Case IV: For $\phi_{1}=f$ and $\phi_{2}=f+g$, consider following subcases:
(a) If ${\bf x}={\bf 0}$ and ${\bf y}\notin W_{i}^{\perp}$, for all
$i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}$, then

$\displaystyle\widehat{f}({\bf 0})-\widehat{f+g}({\bf y})\neq
2^{n}\Leftrightarrow 2^{n}-2s_{1}(2^{t}-1)-2\chi_{12}\neq
2^{n}\Leftrightarrow\chi_{12}\neq-s_{1}(2^{t}-1)$

(b) If ${\bf x}={\bf 0}$ and ${\bf y}\in W_{i}^{\perp}$, for some
$i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}$, then

$\displaystyle\widehat{f}({\bf 0})-\widehat{f+g}({\bf y})\neq
2^{n}\Leftrightarrow 2^{n}-2s_{1}(2^{t}-1)+2^{t+1}-2\chi_{12}\neq
2^{n}\Leftrightarrow s_{1}(2^{t}-1)+\chi_{12}\neq 2^{t}$

(c) If ${\bf y}={\bf 0}$ and ${\bf x}\notin W_{i}^{\perp}$, for all
$i\in\mathcal{A}_{1}$, then

$\displaystyle\widehat{f}({\bf x})-\widehat{f+g}({\bf 0})\neq
2^{n}\Leftrightarrow 2s_{1}-2^{n}+2\chi_{12}(2^{t}-1)\neq 2^{n}\Leftrightarrow
s_{1}+\chi_{12}(2^{t}-1)\neq 2^{n}$

(d) If ${\bf y}={\bf 0}$ and ${\bf x}\in W_{i}^{\perp}$, for some
$i\in\mathcal{A}_{1}$, then

$\displaystyle\widehat{f}({\bf x})-\widehat{f+g}({\bf 0})\neq
2^{n}\Leftrightarrow-2^{t+1}+2s_{1}-2^{n}+2\chi_{12}(2^{t}-1)\neq
2^{n}\Leftrightarrow s_{1}+\chi_{12}(2^{t}-1)\neq 2^{n}+2^{t}$

Case V: If $\phi_{1}=f+g$ and $\phi_{2}=f$, then this case follow from above.
Case VI: For $\phi_{1}=f$ and $\phi_{2}=g+h$, consider following subcases:
(a) If ${\bf x}={\bf 0}$ and ${\bf y}\notin W_{i}^{\perp}$, for all
$i\in\mathcal{A}_{2}\Delta\mathcal{A}_{3}$, then

$\displaystyle\widehat{f}({\bf 0})-\widehat{g+h}({\bf y})\neq
2^{n}\Leftrightarrow 2^{n}-2s_{1}(2^{t}-1)-2\chi_{23}\neq
2^{n}\Leftrightarrow\chi_{23}\neq-s_{1}(2^{t}-1)$

(b) If ${\bf x}={\bf 0}$ and ${\bf y}\in W_{i}^{\perp}$, for some
$i\in\mathcal{A}_{2}\Delta\mathcal{A}_{3}$, then

$\displaystyle\widehat{f}({\bf 0})-\widehat{g+h}({\bf y})\neq
2^{n}\Leftrightarrow 2^{n}-2s_{1}(2^{t}-1)+2^{t+1}-2\chi_{23}\neq
2^{n}\Leftrightarrow s_{1}(2^{t}-1)+\chi_{23}\neq 2^{t}$

Case VII: If $\phi_{1}=g+h$ and $\phi_{2}=f$, then this case follow from
above.
Case VIII: For $\phi_{1}=f$ and $\phi_{2}=f+g+h$, consider following subcases:
(a) If ${\bf x}={\bf 0}$ and ${\bf y}\notin W_{i}^{\perp}$, for all
$i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3}$, then

$\displaystyle\widehat{f}({\bf 0})-\widehat{f+g+h}({\bf y})\neq
2^{n}\Leftrightarrow 2^{n}-s_{1}(2^{t}-1)-2\chi_{123}\neq
2^{n}\Leftrightarrow\chi_{123}\neq-s_{1}(2^{t}-1)$

(b) If ${\bf x}={\bf 0}$ and ${\bf y}\in W_{i}^{\perp}$, for some
$i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3}$, then

$\displaystyle\widehat{f}({\bf 0})-\widehat{f+g+h}({\bf y})\neq
2^{n}\Leftrightarrow 2^{n}-s_{1}(2^{t}-1)+2^{t+1}-2\chi_{123}\neq
2^{n}\Leftrightarrow s_{1}(2^{t}-1)+\chi_{123}\neq 2^{t}$

Case IX: For $\phi_{1}=f+g$ and $\phi_{2}=f+g+h$, consider following subcases:
(a) If ${\bf x}={\bf 0}$ and ${\bf y}\notin W_{i}^{\perp}$, for all
$i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3}$, then

$\displaystyle\widehat{f+g}({\bf 0})-\widehat{f+g+h}({\bf y})\neq
2^{n}\Leftrightarrow 2^{n}-\chi_{12}(2^{t}-1)-2\chi_{123}\neq
2^{n}\Leftrightarrow\chi_{123}\neq-\chi_{12}(2^{t}-1)$

(b) If ${\bf x}={\bf 0}$ and ${\bf y}\in W_{i}^{\perp}$, for some
$i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3}$, then

$\displaystyle\widehat{f+g}({\bf 0})-\widehat{f+g+h}({\bf y})\neq
2^{n}\Leftrightarrow 2^{n}-\chi_{12}(2^{t}-1)+2^{t+1}-2\chi_{123}\neq
2^{n}\Leftrightarrow\chi_{12}(2^{t}-1)+\chi_{123}\neq 2^{t}$

All other cases follow in the similar manner. Hence, $\widehat{\phi_{1}}({\bf
x})-\widehat{\phi_{2}}({\bf y})\neq 2^{n}$, for all
$\phi_{1},\phi_{2}\in\mathcal{F}$ and ${\bf x},{\bf y}\in\mathbb{F}_{2}^{n}$
with ${\bf x}\neq{\bf y}$. ∎

######  Proposition 4.6.

Let $n\geq 6$ and $2\leq s\leq 2^{t-1}$, for $s\in\\{s_{1},s_{2},s_{3}\\}$ and
$\mathcal{F}$ be a collection of Boolean functions defined in 4.1 and 4.2.
Then, for ${\bf x},{\bf y}\in\mathbb{F}_{2}^{n}$ with ${\bf x}\neq{\bf y}$, we
have

$\widehat{\phi}_{1}({\bf 0})+\widehat{\phi}_{2}({\bf
x})-\widehat{(\phi_{1}+\phi_{2})}({\bf x})\neq 2^{n}$

for all, $\phi_{1},\phi_{2}\in\mathcal{F}$ with $\phi_{1}\neq\phi_{2}$.

###### Proof.

For different choices of functions, we have following cases:
Case I: If $\phi_{1}=f$ and $\phi_{2}=g$, then for ${\bf x}$, we have
following subcases:
(a) If ${\bf x}={\bf 0}$, then

$\displaystyle\widehat{f}({\bf 0})+\widehat{g}({\bf 0})-\widehat{f+g}({\bf
0})\neq 2^{n}$ $\displaystyle\Leftrightarrow
2^{n}-2s_{1}(2^{t}-1)+2^{n}-2s_{2}(2^{t}-1)-2^{n}+2\chi_{12}(2^{t}-1)\neq
2^{n}\Leftrightarrow s_{12}\neq 0$

(b) If ${\bf x}\notin W_{i}^{\perp}$, for all
$i\in\mathcal{A}_{2}\cup(\mathcal{A}_{1}\Delta\mathcal{A}_{2})$, then

$\displaystyle\widehat{f}({\bf 0})+\widehat{g}({\bf x})-\widehat{f+g}({\bf
x})\neq 2^{n}\Leftrightarrow 2^{n}-2s_{1}(2^{t}-1)+2s_{2}-2\chi_{12}\neq
2^{n}\Leftrightarrow s_{12}\neq 2^{t-1}s_{1}$

(c) If ${\bf x}\in W_{i}^{\perp}$ for some
$i\in\mathcal{A}_{2}\cup(\mathcal{A}_{1}\Delta\mathcal{A}_{2})$, then there
are three subcases, we give proof for only one subcase and the other two cases
follow from this.
If $i\notin\mathcal{A}_{2}$ and $i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}$,
then

$\displaystyle\widehat{f}({\bf 0})+\widehat{g}({\bf x})-\widehat{f+g}({\bf
x})\neq 2^{n}\Leftrightarrow
2^{n}-2s_{1}(2^{t}-1)+2s_{2}+2^{t+1}-2\chi_{12}\neq 2^{n}$ $\displaystyle-
s_{1}2^{t}+2^{t}+2s_{12}\neq 0\Leftrightarrow s_{12}\neq 2^{t-1}(s_{1}-1)$

Case II: If $\phi_{1}=f$ and $\phi_{2}=f+g$, then for ${\bf x}$, we have
following subcases:
(a) If ${\bf x}={\bf 0}$, then

$\displaystyle\widehat{f}({\bf 0})+\widehat{f+g}({\bf 0})-\widehat{g}({\bf
0})\neq 2^{n}$ $\displaystyle\Leftrightarrow
2^{n}-2s_{1}(2^{t}-1)+2^{n}-2\chi_{12}(2^{t}-1)-2^{n}+2s_{2}(2^{t}-1)\neq
2^{n}\Leftrightarrow s_{12}\neq s_{1}$

(b) If ${\bf x}\notin W_{i}^{\perp}$, for all
$i\in\mathcal{A}_{2}\cup(\mathcal{A}_{1}\Delta\mathcal{A}_{2})$, then

$\displaystyle\widehat{f}({\bf 0})+\widehat{f+g}({\bf x})-\widehat{g}({\bf
x})\neq 2^{n}\Leftrightarrow 2^{n}-2s_{1}(2^{t}-1)+2\chi_{12}-2s_{2}\neq
2^{n}\Leftrightarrow s_{12}\neq-s_{1}(2^{t-1}-1)$

(c) If ${\bf x}\in W_{i}^{\perp}$ for some
$i\in\mathcal{A}_{2}\cup(\mathcal{A}_{1}\Delta\mathcal{A}_{2})$, then there
are three subcases, we give proof for only one subcase and the other two cases
follow from this.
If $i\in\mathcal{A}_{2}$ and $i\notin\mathcal{A}_{1}\Delta\mathcal{A}_{2}$,
then

$\displaystyle\widehat{f}({\bf 0})+\widehat{f+g}({\bf x})-\widehat{g}({\bf
x})\neq 2^{n}\Leftrightarrow
2^{n}-2s_{1}(2^{t}-1)+2\chi_{12}+2^{t+1}-2s_{2}\neq 2^{n}$ $\displaystyle-
s_{1}2^{t-1}+s_{1}-s_{12}+2^{t-1}\neq 0\Leftrightarrow s_{1}-s_{12}\neq
2^{t-1}(s_{1}-1)$

Case III: If $\phi_{1}=f+g$ and $\phi_{2}=f$, then for ${\bf x}$, we have
following subcases:
(a) If ${\bf x}={\bf 0}$, then this follows from above.
(b) If ${\bf x}\notin W_{i}^{\perp}$, for all
$i\in\mathcal{A}_{1}\cup\mathcal{A}_{2}$, then

$\displaystyle\widehat{f+g}({\bf 0})+\widehat{f}({\bf x})-\widehat{g}({\bf
x})\neq 2^{n}\Leftrightarrow 2^{n}-2\chi_{12}(2^{t}-1)+2s_{1}-2s_{2}\neq
2^{n}\Leftrightarrow s_{1}-s_{12}\neq 2^{t-1}\chi_{12}$

(c)If ${\bf x}\in W_{i}^{\perp}$ for some
$i\in\mathcal{A}_{1}\cup\mathcal{A}_{2}$, then there are three subcases, we
give proof for only one subcase and the other two cases follow from this.
If $i\in\mathcal{A}_{2}$ and $i\notin\mathcal{A}_{1}$, then

$\displaystyle\widehat{f+g}({\bf 0})+\widehat{f}({\bf x})-\widehat{g}(x)\neq
2^{n}\Leftrightarrow 2^{n}-2\chi_{12}(2^{t}-1)+2s_{1}+2^{t+1}-2s_{2}\neq
2^{n}$ $\displaystyle\Leftrightarrow s_{1}-s_{12}\neq 2^{t-1}(\chi_{12}-1)$

Case IV: If $\phi_{1}=f$ and $\phi_{2}=g+h$, then for ${\bf x}$, we have
following possibilities.
(a) If ${\bf x}={\bf 0}$, then

$\displaystyle\widehat{f}({\bf 0})+\widehat{g+h}({\bf 0})-\widehat{f+g+h}({\bf
0})\neq 2^{n}\Leftrightarrow 2^{n}-2s_{1}(2^{t}-1)+2^{n}-2\chi_{23}(2^{t}-1)$
$\displaystyle-2^{n}+2\chi_{123}(2^{t}-1)\neq
2^{n}\Leftrightarrow(2^{t}-1)(-2s_{12}-2s_{13}+4s_{123})\neq 0\Leftrightarrow
2s_{123}\neq s_{12}+s_{13}$

(b) If ${\bf x}\notin W_{i}^{\perp}$, for all
$i\in(\mathcal{A}_{2}\Delta\mathcal{A}_{3})\cup(\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3})$,
then

$\displaystyle\widehat{f}({\bf 0})+\widehat{g+h}({\bf x})-\widehat{f+g+h}({\bf
x})\neq 2^{n}\Leftrightarrow 2^{n}-2s_{1}(2^{t}-1)+2\chi_{23}-2\chi_{123}\neq
2^{n}$ $\displaystyle\Leftrightarrow-
s_{1}(2^{t}-1)-(s_{1}-2s_{13}-2s_{13}+4s_{123})\neq 0\Leftrightarrow
s_{12}+s_{13}-2s_{123}\neq 2^{t-1}s_{1}$

(c) If ${\bf x}\in W_{i}^{\perp}$ for some
$i\in(\mathcal{A}_{2}\Delta\mathcal{A}_{3})\cup(\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3})$,
then there are three subcases, we give proof for only one subcase and the
other two cases follow from this.
If $i\notin\mathcal{A}_{2}\Delta\mathcal{A}_{3}$ and
$i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3}$, then

$\displaystyle\widehat{f}({\bf 0})+\widehat{g+h}({\bf x})-\widehat{f+g+h}({\bf
x})\neq 2^{n}\Leftrightarrow
2^{n}-2s_{1}(2^{t}-1)+2\chi_{23}+2^{t+1}-2\chi_{123}\neq 2^{n}$
$\displaystyle\Leftrightarrow-
s_{1}(2^{t}-1)+2^{t}-(s_{1}-2s_{13}-2s_{13}+4s_{123})\neq 0\Leftrightarrow
s_{12}+s_{13}-2s_{123}\neq 2^{t-1}(s_{1}-1)$

Case V: If $\phi_{1}=g+h$ and $\phi_{2}=f$, then for ${\bf x}$, we have
following possibilities.
(a) If ${\bf x}={\bf 0}$, then this follows from above.
(b) If ${\bf x}\notin E_{i}^{\perp}$, for all
$i\in\mathcal{A}_{1}\cup(\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3})$,
then

$\displaystyle\widehat{g+h}({\bf 0})+\widehat{f}({\bf x})-\widehat{f+g+h}({\bf
x})\neq 2^{n}\Leftrightarrow 2^{n}-2\chi_{23}(2^{t}-1)+2s_{1}-2\chi_{123}\neq
2^{n}$
$\displaystyle\Leftrightarrow-2^{t}\chi_{23}-(-2s_{12}-2s_{13}+4s_{123})\neq
0\Leftrightarrow 2^{t-1}\chi_{23}\neq(s_{12}+s_{13}-2s_{123})$

(c) If ${\bf x}\in W_{i}^{\perp}$ for some
$i\in\mathcal{A}_{1}\cup(\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3})$,
then there are three subcases, we give proof for only one subcase and the
other two cases follow from this.
If $i\notin\mathcal{A}_{1}$ and
$i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3}$, then

$\displaystyle\widehat{g+h}({\bf 0})+\widehat{f}({\bf
x})-\widehat{f+g+h}(x)\neq 2^{n}\Leftrightarrow
2^{n}-2\chi_{23}(2^{t}-1)+2s_{1}+2^{t+1}-2\chi_{123}\neq 2^{n}$
$\displaystyle\Leftrightarrow s_{12}+s_{13}-2s_{123}\neq 2^{t-1}(\chi_{23}-1)$

Case VI: If $\phi_{1}=f$ and $\phi_{2}=f+g+h$, then for ${\bf x}$, we have
following subcases:
(a) If ${\bf x}={\bf 0}$, then

$\displaystyle\widehat{f}({\bf 0})+\widehat{f+g+h}({\bf 0})-\widehat{g+h}({\bf
0})\neq 2^{n}\Leftrightarrow
2^{n}-2s_{1}(2^{t}-1)+2^{n}-2\chi_{123}(2^{t}-1)-2^{n}+2\chi_{23}(2^{t}-1)$
$\displaystyle\Leftrightarrow-2s_{1}+2s_{12}+2s_{13}-4s_{123}\neq
0\Leftrightarrow s_{1}\neq s_{12}+s_{13}-2s_{123}$

(b) If ${\bf x}\notin W_{i}^{\perp}$, for all
$i\in(\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3})\cup(\mathcal{A}_{2}\Delta\mathcal{A}_{3})$,
then

$\displaystyle\widehat{f}({\bf 0})+\widehat{f+g+h}({\bf x})-\widehat{g+h}({\bf
x})\neq 2^{n}\Leftrightarrow 2^{n}-2s_{1}(2^{t}-1)+2\chi_{123}-2\chi_{23}\neq
2^{n}$ $\displaystyle\Leftrightarrow
s_{1}(1-2^{t-1})\neq(s_{12}+s_{13}-2s_{123})$

(c) If ${\bf x}\in W_{i}^{\perp}$ for some
$i\in(\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3})\cup(\mathcal{A}_{2}\Delta\mathcal{A}_{3})$,
then there are three subcases, we give proof for only one subcase and the
other two cases follow from this.
If $i\notin\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3}$ and
$i\in\mathcal{A}_{2}\Delta\mathcal{A}_{3}$, then

$\displaystyle\widehat{f}({\bf 0})+\widehat{f+g+h}({\bf x})-\widehat{g+h}({\bf
x})\neq 2^{n}\Leftrightarrow
2^{n}-2s_{1}(2^{t}-1)+2\chi_{123}+2^{t+1}-2\chi_{23}\neq 2^{n}$
$\displaystyle\Leftrightarrow s_{1}-(s_{12}+s_{13}-2s_{123})\neq
2^{t-1}(s_{1}-1)$

Case VII: If $\phi_{1}=f+g+h$ and $\phi_{2}=f$, then for ${\bf x}$, we have
following subcases:
(a) If ${\bf x}={\bf 0}$, it is follows from above.
(b) If ${\bf x}\notin W_{i}^{\perp}$, for all
$i\in\mathcal{A}_{1}\cup(\mathcal{A}_{2}\Delta\mathcal{A}_{3})$, then

$\displaystyle\widehat{f+g+h}({\bf 0})+\widehat{f}({\bf x})-\widehat{g+h}({\bf
x})\neq 2^{n}\Leftrightarrow 2^{n}-2\chi_{123}(2^{t}-1)+2s_{1}-2\chi_{23}\neq
2^{n}$ $\displaystyle\Leftrightarrow s_{1}-(s_{12}+s_{13}-2s_{123})\neq
2^{t-1}\chi_{123}$

(c) If ${\bf x}\in W_{i}^{\perp}$ for some
$i\in\mathcal{A}_{1}\cup(\mathcal{A}_{2}\Delta\mathcal{A}_{3})$, then there
are three subcases, we give proof for only one subcase and the other two cases
follow from this.
If $i\notin\mathcal{A}_{1}$ and $i\in\mathcal{A}_{2}\Delta\mathcal{A}_{3}$,
then

$\displaystyle\widehat{f+g+h}({\bf 0})+\widehat{f}({\bf
x})-\widehat{g+h}(x)\neq 2^{n}\Leftrightarrow
2^{n}-2\chi_{123}(2^{t}-1)+2s_{1}+2^{t+1}-2\chi_{23}\neq 2^{n}$
$\displaystyle\Leftrightarrow s_{1}-(s_{12}+s_{13}-2s_{123})\neq
2^{t-1}(\chi_{123}-1)$

Case VIII: If $\phi_{1}=f+g$ and $\phi_{2}=f+g+h$, then for ${\bf x}$, we have
following subcases:
(a) If ${\bf x}={\bf 0}$

$\displaystyle\widehat{f+g}({\bf 0})+\widehat{f+g+h}({\bf 0})-\widehat{h}({\bf
0})\neq 2^{n}$ $\displaystyle\Leftrightarrow
2^{n}-2\chi_{12}(2^{t}-1)+2^{n}-2\chi_{123}(2^{t}-1)-2^{n}+2s_{3}(2^{t}-1)\neq
2^{n}\Leftrightarrow(s_{13}+s_{23}-2s_{123})\neq\chi_{12}$

(b) If ${\bf x}\notin W_{i}^{\perp}$, for all
$i\in(\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3})\cup\mathcal{A}_{3}$,
then

$\displaystyle\widehat{f+g}({\bf 0})+\widehat{f+g+h}({\bf x})-\widehat{h}({\bf
x})\neq 2^{n}$ $\displaystyle\Leftrightarrow
2^{n}-2\chi_{12}(2^{t}-1)+2\chi_{123}-2s_{3}\neq
2^{n}\Leftrightarrow\chi_{12}-(s_{13}+s_{23}-2s_{123})\neq 2^{t-1}\chi_{12}$

(c) If ${\bf x}\in W_{i}^{\perp}$ for some
$i\in(\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3})\cup\mathcal{A}_{3}$,
then there are three subcases, we give proof for only one subcase and the
other two cases follow from this.
If $i\notin\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3}$ and
$i\in\mathcal{A}_{3}$, then

$\displaystyle\widehat{f+g}({\bf 0})+\widehat{f+g+h}({\bf x})-\widehat{h}({\bf
x})\neq 2^{n}$ $\displaystyle\Leftrightarrow
2^{n}-2\chi_{12}(2^{t}-1)+2\chi_{123}+2^{t+1}-2s_{3}\neq 2^{n}\Leftrightarrow
2^{t-1}(\chi_{12}-1)\neq\chi_{12}-(s_{13}+s_{23}-2s_{123})$

Case IX: If $\phi_{1}=f+g+h$ and $\phi_{2}=f+g$, then for ${\bf x}$, we have
following subcases:
(a) If ${\bf x}={\bf 0}$, then this case follows from above.
(b) If ${\bf x}\notin W_{i}^{\perp}$, for all
$i\in(\mathcal{A}_{1}\Delta\mathcal{A}_{2})\cup\mathcal{A}_{3}$, then

$\displaystyle\widehat{f+g+h}({\bf 0})+\widehat{f+g}({\bf x})-\widehat{h}({\bf
x})\neq 2^{n}$ $\displaystyle\Leftrightarrow
2^{n}-2\chi_{123}(2^{t}-1)+2\chi_{12}-2s_{3}\neq
2^{n}\Leftrightarrow\chi_{123}(2^{t}-1)+s_{3}\neq\chi_{12}$
$\displaystyle[\text{Since }\max{(\chi_{12})}=2^{t}-2\text{ and
}\min{(\chi_{123}(2^{t}-1)+s_{3})}=2^{t+1}-2+2]$

(c) If ${\bf x}\in W_{i}^{\perp}$, for some
$i\in(\mathcal{A}_{1}\Delta\mathcal{A}_{2})\cup\mathcal{A}_{3}$, then there
are three subcases, we give proof for only one subcase and the other two cases
follow from this.
If $i\notin\mathcal{A}_{1}\Delta\mathcal{A}_{2}$ and $i\in\mathcal{A}_{3}$,
then

$\displaystyle\widehat{f+g+h}({\bf 0})+\widehat{f+g}({\bf x})-\widehat{h}({\bf
x})\neq 2^{n}$ $\displaystyle\Leftrightarrow
2^{n}-2\chi_{123}(2^{t}-1)+2\chi_{12}+2^{t+1}-2s_{3}\neq
2^{n}\Leftrightarrow\chi_{123}(2^{t}-1)+s_{3}\neq\chi_{12}+2^{t}$
$\displaystyle[\text{Since }\max{(\chi_{12})}+2^{t}=2^{t}-2+2^{t}\text{ and
}\min{(\chi_{123}(2^{t}-1)+s_{3})}=2^{t+1}-2+2]$

All other cases follow in the similar manner.
Hence, $\widehat{\phi}_{1}({\bf 0})+\widehat{\phi}_{2}({\bf
x})-\widehat{(\phi_{1}+\phi_{2})}({\bf x})\neq 2^{n}$, for all
$\phi_{1},\phi_{2}\in\mathcal{F}$, with $\phi_{1}\neq\phi_{2}$ and ${\bf
x},{\bf y}\in\mathbb{F}_{2}^{n}$, for all ${\bf
x}\in\mathbb{F}_{2}^{n}\setminus\\{{\bf 0}\\}$. ∎

######  Proposition 4.7.

Let $n\geq 6$ and $2\leq s\leq 2^{t-1}$, for $s\in\\{s_{1},s_{2},s_{3}\\}$ and
$\mathcal{F}$ be a collection of Boolean functions defined in 4.1 and 4.2.
Then for any ${\bf x}\in\mathbb{F}_{2}^{n}\setminus\\{{\bf 0}\\}$, we have

$\widehat{\phi}_{1}({\bf x})+\widehat{\phi}_{2}({\bf
x})-\widehat{\phi_{1}+\phi_{2}}({\bf 0})\neq 2^{n}$

for all, $\phi_{1},\phi_{2}\in\mathcal{F}$ with $\phi_{1}\neq\phi_{2}$.

###### Proof.

For different choices of $\phi_{1}$ and $\phi_{2}$, we have following cases:
Case I: For $\phi_{1}=f$ and $\phi_{2}=g$. Now, according to choice of ${\bf
x}$, we have following possibilities:
(a) If ${\bf x}\notin W_{i}$ for all $i\in\mathcal{A}_{1}\cup\mathcal{A}_{2}$,
then

$\displaystyle\widehat{f}({\bf x})+\widehat{g}({\bf x})-\widehat{f+g}({\bf
0})\neq 2^{n}\Leftrightarrow 2s_{1}+2s_{2}-2^{n}+2\chi_{12}(2^{t}-1)\neq
2^{n}\Leftrightarrow s_{12}\neq-2^{t-1}(2^{t}-\chi_{12})$

(b) If ${\bf x}\in W_{i}^{\perp}$ for some
$i\in\mathcal{A}_{1}\cup\mathcal{A}_{2}$, then we divide this case into three
subcases
(i) If $i\in\mathcal{A}_{1}\cap\mathcal{A}_{2}$, then

$\displaystyle\widehat{f}({\bf x})+\widehat{g}({\bf x})-\widehat{f+g}({\bf
0})\neq
2^{n}\Leftrightarrow-2^{t+1}+2s_{1}-2^{t+1}+2s_{2}-2^{n}+2\chi_{12}(2^{t}-1)\neq
2^{n}$ $\displaystyle\Leftrightarrow 2^{t}\chi_{12}-2s_{12}\neq
2^{n}+2^{t+1}\Leftrightarrow 2^{t-1}\chi_{12}\neq 2^{2t-1}+2^{t}+s_{12}$

(ii) If $i\in\mathcal{A}_{1}/\mathcal{A}_{2}$, then

$\displaystyle\widehat{f}({\bf x})+\widehat{g}({\bf x})-\widehat{f+g}({\bf
0})\neq
2^{n}\Leftrightarrow-2^{t+1}+2s_{1}+2s_{2}-2^{n}+2\chi_{12}(2^{t}-1)\neq
2^{n}$ $\displaystyle\Leftrightarrow 2^{t}\chi_{12}-2s_{12}\neq
2^{n}\Leftrightarrow 2^{t-1}\chi_{12}\neq 2^{2t-1}+2^{t-1}+s_{12}$

(iii) If $i\in\mathcal{A}_{2}/\mathcal{A}_{1}$, then this is the same case as
above.
Case II: For $\phi_{1}=f$ and $\phi_{2}=f+g$. Now, according to choice of
${\bf x}$, we have following possibilities:
(a) If ${\bf x}\notin W_{i}$, for all
$i\in\mathcal{A}_{1}\cup(\mathcal{A}_{1}\Delta\mathcal{A}_{2})$, then

$\displaystyle\widehat{f}({\bf x})+\widehat{f+g}({\bf x})-\widehat{g}({\bf
0})\neq 2^{n}\Leftrightarrow 2s_{1}+2\chi_{12}-2^{n}+2s_{2}(2^{t}-1)\neq
2^{n}\Leftrightarrow s_{1}-s_{12}\neq 2^{t-1}(2^{t}-s_{2})$

(b) If ${\bf x}\in W_{i}^{\perp}$, for some
$i\in\mathcal{A}_{1}\cup(\mathcal{A}_{1}\Delta\mathcal{A}_{2})$, then we
divide this case into three subcases
(i) If $i\in\mathcal{A}_{1}\cap(\mathcal{A}_{1}\Delta\mathcal{A}_{2})$, then

$\displaystyle\widehat{f}({\bf x})+\widehat{f+g}({\bf x})-\widehat{g}({\bf
0})\neq
2^{n}\Leftrightarrow-2^{t+1}+2s_{1}-2^{t+1}+2\chi_{12}-2^{n}+2s_{2}(2^{t}-1)\neq
2^{n}$ $\displaystyle\Leftrightarrow s_{1}-s_{12}\neq 2^{t-1}(2^{t}-s_{2}+2)$

(ii) If $i\in\mathcal{A}_{1}$ and
$i\notin\mathcal{A}_{1}\Delta\mathcal{A}_{2}$, then

$\displaystyle\widehat{f}({\bf x})+\widehat{f+g}({\bf x})-\widehat{g}({\bf
0})\neq
2^{n}\Leftrightarrow-2^{t+1}+2s_{1}+2\chi_{12}-2^{n}+2s_{2}(2^{t}-1)\neq
2^{n}$ $\displaystyle\Leftrightarrow s_{1}-s_{12}\neq 2^{t-1}(2^{t}-s_{2}+1)$

(iii) If $i\notin\mathcal{A}_{1}$ and
$i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}$, then this is the same case as
above.
Case III: For $\phi_{1}=f$ and $\phi_{2}=g+h$. Now, according to choice of
${\bf x}$, we have following possibilities:
(a) If ${\bf x}\notin W_{i}$, for all
$i\in\mathcal{A}_{1}\cup(\mathcal{A}_{2}\Delta\mathcal{A}_{3})$, then

$\displaystyle\widehat{f}({\bf x})+\widehat{g+h}({\bf x})-\widehat{f+g+h}({\bf
0})\neq 2^{n}\Leftrightarrow 2s_{1}+2\chi_{23}-2^{n}+2\chi_{123}(2^{t}-1)\neq
2^{n}$ $\displaystyle\Leftrightarrow
2^{t-1}\chi_{123}+(s_{12}+s_{13}+2s_{123})\neq
2^{n-1}\Leftrightarrow(s_{12}+s_{13}+2s_{123})\neq 2^{t-1}(2^{t}-\chi_{123})$

(b) If ${\bf x}\in W_{i}^{\perp}$, for some
$i\in\mathcal{A}_{1}\cup(\mathcal{A}_{2}\Delta\mathcal{A}_{3})$, then we
divide this case into three subcases
(i) If $i\in\mathcal{A}_{1}\cap(\mathcal{A}_{2}\Delta\mathcal{A}_{3})$, then

$\displaystyle\widehat{f}({\bf x})+\widehat{g+h}({\bf x})-\widehat{f+g+h}({\bf
0})\neq
2^{n}\Leftrightarrow-2^{t+1}+2s_{1}-2^{t+1}+2\chi_{23}-2^{n}+2\chi_{123}(2^{t}-1)\neq
2^{n}$ $\displaystyle\Leftrightarrow
2^{t-1}\chi_{123}+(s_{12}+s_{13}-2s_{123})\neq
2^{n-1}+2^{t}\Leftrightarrow(s_{12}+s_{13}+2s_{123})\neq
2^{t-1}(2^{t}-\chi_{123})+2^{t}$

(ii) If $i\in\mathcal{A}_{1}$ and
$i\notin\mathcal{A}_{2}\Delta\mathcal{A}_{3}$, then

$\displaystyle\widehat{f}({\bf x})+\widehat{g+h}({\bf x})-\widehat{h}({\bf
0})\neq
2^{n}\Leftrightarrow-2^{t+1}+2s_{1}+2\chi_{23}-2^{n}+2\chi_{123}(2^{t}-1)\neq
2^{n}$ $\displaystyle\Leftrightarrow
2^{t-1}\chi_{123}+(s_{12}+s_{13}+2s_{123})\neq
2^{n-1}+2^{t-1}\Leftrightarrow(s_{12}+s_{13}+2s_{123})\neq
2^{t-1}(2^{t}-\chi_{123})+2^{t-1}$

(iii) If $i\notin\mathcal{A}_{1}$ and
$i\in\mathcal{A}_{2}\Delta\mathcal{A}_{3}$, then this is the same case as
above.
Case IV: For $\phi_{1}=f$ and $\phi_{2}=f+g+h$. Now, according to choice of
${\bf x}$, we have following possibilities:
(a) If ${\bf x}\notin W_{i}$ for all
$i\in\mathcal{A}_{1}\cup(\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3})$,
then

$\displaystyle\widehat{f}({\bf x})+\widehat{f+g+h}({\bf x})-\widehat{g+h}({\bf
0})\neq 2^{n}\Leftrightarrow 2s_{1}+2\chi_{123}-2^{n}+2\chi_{23}(2^{t}-1)\neq
2^{n}$ $\displaystyle\Leftrightarrow 2s_{1}-2s_{12}-2s_{13}+4s_{123}\neq
2^{n}-2^{t}\chi_{23}\Leftrightarrow s_{1}-(s_{12}+s_{13}-2s_{123})\neq
2^{t-1}(2^{t}-\chi_{23})$

(b) If ${\bf x}\in W_{i}^{\perp}$ for some
$i\in\mathcal{A}_{1}\cup(\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3})$,
then we divide this case into three subcases
(i) If
$i\in\mathcal{A}_{1}\cap(\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3})$,
then

$\displaystyle\widehat{f}({\bf x})+\widehat{f+g+h}({\bf x})-\widehat{g+h}({\bf
0})\neq
2^{n}\Leftrightarrow-2^{t+1}+2s_{1}-2^{t+1}+2\chi_{123}-2^{n}+2\chi_{23}(2^{t}-1)\neq
2^{n}$ $\displaystyle\Leftrightarrow 2s_{1}-2s_{12}-2s_{13}+4s_{123}\neq
2^{n}-2^{t}\chi_{23}+2^{t}\Leftrightarrow s_{1}-(s_{12}+s_{13}-2s_{123})\neq
2^{t-1}(2^{t}-\chi_{23})+2^{t}$

(ii) If $i\in\mathcal{A}_{1}$ and
$i\notin\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3}$, then

$\displaystyle\widehat{f}({\bf x})+\widehat{f+g+h}({\bf x})-\widehat{g+h}({\bf
0})\neq
2^{n}\Leftrightarrow-2^{t+1}+2s_{1}+2\chi_{123}-2^{n}+2\chi_{23}(2^{t}-1)\neq
2^{n}$ $\displaystyle\Leftrightarrow 2s_{1}-2s_{12}-2s_{13}+4s_{123}\neq
2^{n}-2^{t}\chi_{23}+2^{t}\Leftrightarrow s_{1}-(s_{12}+s_{13}-2s_{123})\neq
2^{t-1}(2^{t}-\chi_{23})+2^{t-1}$

(iii) If $i\notin\mathcal{A}_{1}$ and
$i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3}$, then this is
the same case as above.
Case V: For $\phi_{1}=f+g$ and $\phi_{2}=f+g+h$. Now, according to choice of
${\bf x}$, we have following possibilities:
(a) If ${\bf x}\notin W_{i}$ for all
$i\in(\mathcal{A}_{1}\Delta\mathcal{A}_{2})\cup(\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3})$,
then

$\displaystyle\widehat{f+g}({\bf x})+\widehat{f+g+h}({\bf x})-\widehat{h}({\bf
0})\neq 2^{n}\Leftrightarrow 2\chi_{12}+2\chi_{123}-2^{n}+2s_{3}(2^{t}-1)\neq
2^{n}$ $\displaystyle\Leftrightarrow 2\chi_{12}-2(s_{13}+s_{23}-2s_{123})\neq
2^{n}-2^{t}s_{3}\Leftrightarrow\chi_{12}-(s_{13}+s_{23}-2s_{123})\neq
2^{t-1}(2^{t}-s_{3})$

(b) If ${\bf x}\in W_{i}^{\perp}$ for some
$i\in(\mathcal{A}_{1}\Delta\mathcal{A}_{2})\cup(\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3})$,
then we divide this case into three subcases
(i) If
$i\in(\mathcal{A}_{1}\Delta\mathcal{A}_{2})\cap(\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3})$,
then

$\displaystyle\widehat{f+g}({\bf x})+\widehat{f+g+h}({\bf x})-\widehat{h}({\bf
0})\neq
2^{n}\Leftrightarrow-2^{t+1}+2\chi_{12}-2^{t+1}+2\chi_{123}-2^{n}+2s_{3}(2^{t}-1)\neq
2^{n}$ $\displaystyle\Leftrightarrow\chi_{12}-(s_{13}+s_{23}-2s_{123})\neq
2^{t-1}(2^{t}-s_{3})+2^{t}$

(ii) If $i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}$ and
$i\notin\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3}$, then

$\displaystyle\widehat{f+g}({\bf x})+\widehat{f+g+h}({\bf x})-\widehat{h}({\bf
0})\neq
2^{n}\Leftrightarrow-2^{t+1}+2\chi_{12}-2^{t+1}+2\chi_{123}-2^{n}+2s_{3}(2^{t}-1)\neq
2^{n}$ $\displaystyle\Leftrightarrow\chi_{12}-(s_{13}+s_{23}-2s_{123})\neq
2^{t-1}(2^{t}-s_{3})+2^{t-1}$

(iii) If $i\notin\mathcal{A}_{1}\Delta\mathcal{A}_{2}$ and
$i\in\mathcal{A}_{1}\Delta\mathcal{A}_{2}\Delta\mathcal{A}_{3}$, then this is
the same case as above. All other cases follow in the similar manner.
Henec, $\widehat{\phi}_{1}({\bf x})+\widehat{\phi}_{2}({\bf
x})-\widehat{\phi_{1}+\phi_{2}}({\bf 0})\neq 2^{n}$, for all
$\phi_{1},\phi_{2}\in\mathcal{F}$ with $\phi_{1}\neq\phi_{2}$ and ${\bf
x}\in\mathbb{F}_{2}^{n}\setminus\\{{\bf 0}\\}$. ∎

######  Proposition 4.8.

Let $n\geq 6$ and $2\leq s\leq 2^{t-1}$, for $s\in\\{s_{1},s_{2},s_{3}\\}$ and
$\mathcal{F}$ be an collection of Boolean functions defined in 4.1 and 4.2.
Then for any ${\bf x},{\bf y}\in\mathbb{F}_{2}^{n}\setminus\\{{\bf 0}\\}$ with
${\bf x}\neq{\bf y}$, we have

$\widehat{\phi}_{1}({\bf x}+{\bf y})+\widehat{\phi}_{2}({\bf
x})-\widehat{\phi_{1}+\phi_{2}}({\bf y})\neq 2^{n}$

for all, $\phi_{1},\phi_{2}\in\mathcal{F}$ with $\phi_{1}\neq\phi_{2}$.

###### Proof.

Denote $\psi_{1}=\widehat{\phi}_{1}({\bf x}+{\bf y})+\widehat{\phi}_{2}({\bf
x})-\widehat{\phi_{1}+\phi_{2}}({\bf y})$. As, ${\bf x}$ and ${\bf y}$ are
non-zero vectors with ${\bf x}\neq{\bf y}$, this implies ${\bf x}+{\bf y}\neq
0$. Now, for different choice of $\phi_{1}$ and $\phi_{2}$, we have following
cases:
Case I: If $\phi_{1}=f$ and $\phi_{2}=g$, then
$\psi_{1}\in\\{s-2^{t+2},s-2^{t+1},s,s+2^{t+1}\\}$, where $s=4s_{12}$. We know
that, $s_{12}<2^{t-1}$, from this, we get $\psi_{1}<2^{n}$.
Case II: If $\phi_{1}=f$ and $\phi_{2}=f+g$, then
$\psi_{1}\in\\{s-2^{t+2},s-2^{t-1},s,s+2^{t+1}\\}$, where $s=4s_{1}-4s_{12}$.
We know that, $s_{1}\leq 2^{t-1}$, from this, we get $\psi_{1}<2^{n}$.
Case III: If $\phi_{1}=f$ and $\phi_{2}=g+h$, then
$\psi_{1}\in\\{s-2^{t+2},s-2^{t+1},s,s+2^{t+1}\\}$, where
$s=4(s_{12}+s_{13}-2s_{123})$. We know that,
$(s_{12}+s_{13}-2s_{123})<2^{t-1}$, from this, we get $\psi_{1}<2^{n}$.
Case IV: If $\phi_{1}=f$ and $\phi_{2}=f+g+h$, then
$\psi_{1}\in\\{s-2^{t+2},s-2^{t+1},s,s+2^{t+1}\\}$, where
$s=4(s_{1}-(s_{12}+s_{13}-2s_{123}))$. We know that,
$(s_{1}-(s_{12}+s_{13}-2s_{123}))<2^{t-1}$, from this, we get
$\psi_{1}<2^{n}$.
Case V: If $\phi_{1}=f+g$ and $\phi_{2}=f+g+h$, then
$\psi_{1}\in\\{s-2^{t+2},s-2^{t+1},s,s+2^{t+1}\\}$, where
$s=4((s_{1}+s_{2}-2s_{12})-(s_{13}+s_{23}-2s_{123}))$. We know that,
$(s_{1}+s_{2}-2s_{12})\leq 2^{t}-2$, from this, we get $\psi_{1}<2^{n}$.
All other cases follow in the similar manner.
Hence, $\widehat{\phi}_{1}({\bf x}+{\bf y})+\widehat{\phi}_{2}({\bf
x})-\widehat{\phi_{1}+\phi_{2}}({\bf y})\neq 2^{n}$, for all
$\phi_{1},\phi_{2}\in\mathcal{F}$ with $\phi_{1}\neq\phi_{2}$ and ${\bf
x},{\bf y}\in\mathbb{F}_{2}^{n}\setminus\\{0\\}$, with ${\bf x}\neq{\bf y}$. ∎

The following result gives a family of minimal linear codes of dimension $n+3$
violating the Ashikhmin-Barg condition.

######  Theorem 4.9.

Let $n\geq 6$ and $2\leq s\leq 2^{t-1}$, for $s\in\\{s_{1},s_{2},s_{3}\\}$. If
$\epsilon\leq 2^{t-1}-1$, where
$\epsilon=\min{\\{s_{1},s_{2},s_{3},\chi_{12},\chi_{13},\chi_{23},\chi_{123}\\}}$,
then the code $\mathscr{C}_{f,g,h}$ is $[2^{n-1},n+3,\epsilon(2^{t}-1)]$ a
minimal linear code. Moreover, if $\epsilon\leq 2^{t-2}$, then
$\frac{{wt}_{min}}{{wt}_{max}}\leq\frac{1}{2}$.

###### Proof.

We first prove that $\mathscr{C}_{f,g,h}$ is minimal linear code by Theorem
3.2. From Proposition 4.4 and 4.5, Theorem 3.2(1) is true for
$\mathscr{C}_{f,g,h}$. For Theorem 3.1(2), we divide proof into three cases as
follows:
Case I: If ${\bf x}={\bf y}\in\mathbb{F}_{2}^{n}$, then by Proposition 4.6,
Theorem 3.1(2) is true for $\mathscr{C}_{f,g,h}$.
Case II: If one of ${\bf x}$ and ${\bf y}$ is zero, then from Proposition 4.6
and 4.7, we get Theorem 3.1(2) holds.
Case III: If ${\bf x}$ and ${\bf y}$ are different non-zero vectors, then
Theorem 3.1(2) holds by Proposition 4.8.
Hence, $\mathscr{C}_{f,g,h}$ is a minimal linear code. Note that, possible
values for minimum weight is $\epsilon(2^{t}-1)$ and $2^{m-1}-\mu$, where
$\mu=\max{\\{s_{1},s_{2},s_{3},\chi_{12},\chi_{13},\chi_{23},\chi_{123}\\}}$.
We know that, possibly large value of $\mu$ is $=2^{t}-2$ and
$\epsilon(2^{t}-1)\leq 2^{m-1}-\mu$. From this, we get
${wt}_{min}=\epsilon(2^{t}-1)$ and $wt_{max}=2^{m-1}+2^{t}-\epsilon(2^{t}-1)$.
We can easily prove that $\frac{wt_{min}}{wt_{max}}\leq\frac{1}{2}$.
This conclude the result. ∎

##  5 Conclusion

In [7] Liu, H., & Liao, Q. constructed a minimal binary linear code violating
the Ashikhmin-Barg condition of dimension $n+2$ and using the procedure from
this paper, we extend and generalize this construction of a minimal binary
linear code of dimension $n+3$. We can think of this as an algorithm to
increase the dimension of minimal linear code. In this paper, we gave a
generic construction of a binary linear code of dimension $n+3$ and stated the
necessary and sufficient condition for the constructed code to be minimal.
Using this condition, we constructed a class of minimal binary linear code
from a class of special Boolean functions violating the Ashikhmin-Barg
condition. Also, we obtained the weight distribution of the constructed
minimal binary linear code.

Finding a minimal linear code of dimension greater than $n+3$ and extending
the construction in 1.1 for non-binary fields to construct a minimal linear
code violating the Ashikhmin-Barg condition is the further scope of the study.

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