∎

11institutetext: A. Kutsenko 22institutetext: Sobolev Institute of
Mathematics, Novosibirsk
Novosibirsk State University, Novosibirsk, Russia
22email<EMAIL_ADDRESS>
# On constructions and properties of self-dual generalized bent functions
††thanks: The work is supported by Mathematical Center in Akademgorodok under
agreement No. 075-15-2019-1613 with the Ministry of Science and Higher
Education of the Russian Federation and Laboratory of Cryptography JetBrains
Research.

Aleksandr Kutsenko

(Received: date / Accepted: date)

###### Abstract

Bent functions of the form $\mathbb{F}_{2}^{n}\rightarrow\mathbb{Z}_{q}$,
where $q\geqslant 2$ is a positive integer, are known as generalized bent
(gbent) functions. Gbent functions for which it is possible to define a dual
gbent function are called regular. A regular gbent function is said to be
self-dual if it coincides with its dual. In this paper we explore self-dual
gbent functions for even $q$. We consider several primary and secondary
constructions of such functions. It is proved that the numbers of self-dual
and anti-self dual gbent functions coincide. We give necessary and sufficient
conditions for the self-duality of Maiorana–McFarland gbent functions and find
Hamming and Lee distances spectrums between them. We find all self-dual gbent
functions symmetric with respect to two variables and prove that self-dual
gbent function can not be affine. The properties of sign functions of self-
dual gbent functions are considered. Symmetries that preserve self-duality are
also discussed.

###### Keywords:

Generalized Boolean functions Self-dual bent Maiorana–McFarland generalized
bent function Lee distance

##  1 Introduction

The study of Boolean functions having strong cryptographic properties is the
domain of current interest, see monographies Carlet_book ; Cusick  for detail.
Boolean bent functions were introduced by Rothaus Rothaus  in $1976$. Due to
maximal nonlinearity they have a number of applications in cryptography and
coding theory. They were used as building blocks of stream (Grain, $2004$) and
block (CAST, $1997$) ciphers and, for instance, in $2000$ T. Wada Wada
established a connection between bent functions and binary constant-amplitude
codewords. But despite the long history of research in this area there are
still many open problems. Among them the exact number of bent functions as
well as their complete classification seems elusive to be solved for now. One
can find more details on bent functions in books Tokareva_book ; Mesnager_book
.

Bent functions were initially generalized by P. V. Kumar in $1985$ by
considering functions of the form
$\mathbb{Z}_{q}^{n}\rightarrow\mathbb{Z}_{q}$ with corresponding definition of
bentness, see Kumar . Bent functions from a finite Abelian group into a finite
Abelian group were studied in Solodovnikov  by V. I. Solodovnikov. Having
applications of functions from $\mathbb{F}_{2}^{n}$ to $\mathbb{Z}_{4}$ in
code-division multiple access (CDMA) systems, K.-U Schmidt in Schmidt
(initially appeared in preprint from $2006$) generalized the notion of
bentness for functions of the form
$\mathbb{F}_{2}^{n}\rightarrow\mathbb{Z}_{q}$, where $q\geqslant 2$ is a
positive integer and studied these functions for the case $q=4$. The
considered functions are named generalized bent (gbent) functions. Note that
this generalization deals with the mappings of the form
$\mathbb{F}_{2}^{n}\rightarrow\mathbb{Z}_{q}$ called generalized Boolean
functions, that are also studied from the view of obtaining linear codes with
special properties, see Paterson . In recent years generalized bent functions
obtained much attention, in particular, for the case $q=2^{k}$. In
HodzicConstructions ; Martinsen ; Hodzic2018 ; Generalized_bent  different
constructions and properties of generalized bent functions were obtained. The
connection between concepts of strong regularity of (edge-weighted) Cayley
graph associated to a generalized Boolean function and gbent functions was
pointed in Riera . The complete characterization of generalized bent functions
from different perspectives was recently presented in Hodzic ; Tang ;
Mesnager2018 . A comprehensive survey on existing generalizations of bent
functions can be found in Tokareva_survey .

For every bent Boolean function its dual bent functions is uniquely defined.
It is important to note that the duality mapping is the unique known isometric
mapping of the set of bent functions into itself that cannot be extended to a
isometry of the whole set of all Boolean functions that preserves bentness.
Self-dual bent functions form a remarkable class of bent functions since they
have the direct relation to their dual bent functions and in terms of mappings
can be considered as fixed points of the duality mapping. These functions were
explored by C. Carlet et al. in $2010$ in work Self-dual , where a number of
constructions and properties were given and the classification for small
number of variables was provided. Next steps for the classification of qubic
self-dual bent functions in $8$ variables were made in Feulner , while
quadratic self-dual bent functions were completely characterized in Hou .
Constructions and properties of self-dual Boolean bent functions were studied
in Hyun ; Luo ; Mesnager . The overview of the known metrical properties of
self-dual bent functions can be found in Kutsenko_PDM .

The action of the duality mapping on bent functions within generalizations is
increasingly nontrivial since it is typically defined only for the part of
bent functions from corresponding generalization which are called regular,
while more accurate work with them also demands for intermediate notation like
weak regularity that also appears in this scope. The extension of the concept
of self-duality for different generalizations of bent functions was made in
several papers. The classification of quadratic self-dual bent functions of
the form $\mathbb{F}_{p}^{n}\rightarrow\mathbb{F}_{p}$, $p$ odd prime, was
made by X.-D. Hou in Hou_generalized . In paper Cesmelioglu  the self-duality
for bent functions within the same generalization type was studied by A.
Çeşmelioğlu et al. In $2018$, L. Sok. et al. Quaternary  studied quaternary
self-dual bent functions of the form
$\mathbb{F}_{2}^{n}\rightarrow\mathbb{Z}_{4}$ from the viewpoints of
existence, construction, and symmetry. The relation between sign functions of
quaternary self-dual bent function in $n$ variables and two self-dual bent
functions in $n$ variables was found. Based on this it was proved that there
are no quaternary self-dual bent functions in odd number of variables.

In current work we investigate constructions, symmetries and other properties
of self-dual generalized bent functions
$\mathbb{F}_{2}^{n}\rightarrow\mathbb{Z}_{q}$, when $q$ is even. The paper is
organized as follows. The next section contains the necessary notation. In
section 3 several primary and secondary constructions are given. The metrical
properties of self-dual gbent functions from Maiorana–McFarland class are
characterized in section 4. In section 5 sign functions of self-dual gbent
functions are studied. Section 6 deals with the properties of self-dual gbent
functions including upper bound for the set of self-dual gbent functions, the
existence of affine functions within self-dual gbent ones and characterization
of self-dual gbent functions symmetric with respect to two variables. In
Section 7 the construction of mappings preserving self-duality of gbent
function is given.

##  2 Notation

Let $\mathbb{F}_{2}^{n}$ be a set of binary vectors of length $n$. For
$x,y\in\mathbb{F}_{2}^{n}$ denote $\langle
x,y\rangle=\bigoplus\limits_{i=1}^{n}{x_{i}y_{i}}$, where the sign $\oplus$
denotes a sum modulo $2$. Denote, following Janusz , the orthogonal group of
index $n$ over the field $\mathbb{F}_{2}$ as

$\mathcal{O}_{n}=\left\\{L\in\text{GL}\left(n,\mathbb{F}_{2}\right):LL^{T}=I_{n}\right\\},$

where $L^{\mathrm{T}}$ denotes the transpose of $L$ and $I_{n}$ is an
identical matrix of order $n$ over the field $\mathbb{F}_{2}$.

A generalized Boolean function $f$ in $n$ variables is any map from
$\mathbb{F}_{2}^{n}$ to $\mathbb{Z}_{q}$, the integers modulo $q$. The set of
generalized Boolean functions in $n$ variables is denoted by
$\mathcal{GF}^{q}_{n}$, for the case $q=2$ we use $\mathcal{F}_{n}$. Let
$\omega=e^{2\pi i/q}$. A sign function of $f\in\mathcal{GF}_{n}^{q}$ is a
complex valued function $F=\omega^{f}$, we will also refer to it as to a
complex vector
$\left(\omega^{f_{0}},\omega^{f_{1}},...,\omega^{f_{2^{n}-1}}\right)$ of
length $2^{n}$, where $\left(f_{0},f_{1},...,f_{2^{n}-1}\right)$ is a vector
of values of the function $f$.

The Hamming weight $\mathrm{wt}_{H}(x)$ of the vector $x\in\mathbb{F}_{2}^{n}$
is the number of nonzero coordinates of $x$. The Hamming distance
$\mathrm{dist}_{H}(f,g)$ between generalized Boolean functions $f,g$ in $n$
variables is the cardinality of the set
$\left\\{x\in\mathbb{F}_{2}^{n}|f(x)\neq g(x)\right\\}$. The Lee weight of the
element $x\in\mathbb{Z}_{q}$ is
$\mathrm{wt}_{L}(x)=\min\left\\{x,q-x\right\\}$. The Lee distance
$\mathrm{dist}_{L}(f,g)$ between $f,g\in\mathcal{GF}^{q}_{n}$ is

$\mathrm{dist}_{L}(f,g)=\sum\limits_{x\in\mathbb{F}_{2}^{n}}\mathrm{wt}_{L}\left(\delta(x)\right),$

where $\delta\in\mathcal{GF}_{n}^{q}$ and $\delta(x)=f(x)+(q-1)g(x)$ for any
$x\in\mathbb{F}_{2}^{n}$. For Boolean case $q=2$ the Hamming distance
coincides with the Lee distance.

The (generalized) Walsh–Hadamard transform of $f\in\mathcal{GF}_{n}^{q}$ is
the complex valued function:

$H_{f}(y)=\sum\limits_{x\in\mathbb{F}_{2}^{n}}\omega^{f(x)}(-1)^{\langle
x,y\rangle}.$

A generalized Boolean function $f$ in $n$ variables is said to be generalized
bent (gbent) if

$\left\lvert H_{f}(y)\right\rvert=2^{n/2},$

for all $y\in\mathbb{F}_{2}^{n}$ Schmidt . If there exists such
$\widetilde{f}\in\mathcal{GF}^{q}_{n}$ that
$H_{f}(y)=\omega^{\widetilde{f}(y)}2^{n/2}$ for any $y\in\mathbb{F}_{2}^{n}$,
the gbent function $f$ is said to be regular and $\widetilde{f}$ is called its
dual. Note that $\widetilde{f}$ is generalized bent as well. A regular gbent
function $f$ is said to be self-dual if $f=\widetilde{f}$, and anti-self-dual
if $f=\widetilde{f}+q/2$. Consequently, it is the case only for even $q$. So
throughout this paper we assume that $q$ is a positive even integer.
Corresponding sets of gbent functions are denoted by $\mathrm{SB}_{q}^{+}(n)$
and $\mathrm{SB}_{q}^{-}(n)$, respectively.

##  3 Constructions

In this section we present several primary and seconady constructions of self-
dual gbent functions.

###  3.1 Direct sum

Suppose $n=n_{1}+n_{2}+\ldots+n_{r}$, where $n_{k}$ are positive integers for
$k=1,2,\ldots,r$. Let $f\in\mathcal{GF}_{n}^{q}$, consider gbent functions
$f_{k}\in\mathcal{GF}_{n_{k}}^{q}$, $k=1,2,\ldots,r$. The function

$f(x)=f_{1}\big{(}x^{(1)}\big{)}+f_{2}\big{(}x^{(2)}\big{)}+\ldots+f_{r}\big{(}x^{(r)}\big{)},$

where $x^{(k)}\in\mathbb{F}_{2}^{n_{k}}$ and
$x=\big{(}x^{(1)},x^{(2)},\ldots,x^{(r)}\big{)}\in\mathbb{F}_{2}^{n}$, is a
direct sum of generalized Boolean functions $f_{k}$. Gbent functions obtained
by a direct sum of generalized Boolean functions were studied in paper
HodzicConstructions , it was proved that the function $f$ is gbent if and only
if all $f_{k}$ are gbent functions. Here we consider self-dual bent functions
obtained by this construction.

###### Proposition 1

Assume all numbers $p_{k}$ are even and $f_{k}\in\mathcal{GF}_{n_{k}}^{q}$ are
gbent functions such that $\widetilde{f}_{k}=f_{k}+c_{k}\left(q/2\right)$,
where $c_{k}\in\mathbb{F}_{2}$, $k=1,2,\ldots,r$. If there is an even number
of nonzero coefficients $c_{k}$, then the function $f$ is a self-dual gbent
function in $n$ variables.

###### Proof

The Walsh–Hadamard transform of $f$ which is a direct sum of
$f_{k}\in\mathcal{GF}_{n_{k}}^{q}$, $k=1,2,\ldots,r$, is given by

$\displaystyle H_{f}(y)$
$\displaystyle=\sum\limits_{x\in\mathbb{F}_{2}^{n}}\omega^{f(x)}(-1)^{\langle
x,y\rangle}=H_{f_{1}}\big{(}y^{(1)}\big{)}H_{f_{2}}\big{(}y^{(2)}\big{)}\cdot\ldots\cdot
H_{f_{r}}\big{(}y^{(r)}\big{)}$
$\displaystyle=(-1)^{c_{1}+c_{2}+\ldots+c_{r}}2^{\big{(}n_{1}+n_{2}+\ldots+n_{r}\big{)}/2}\omega^{\widetilde{f}_{1}\big{(}y^{(1)}\big{)}+\widetilde{f}_{2}\big{(}y^{(2)}\big{)}+\ldots+\widetilde{f}_{r}\big{(}y^{(r)}\big{)}}$
$\displaystyle=(-1)^{c_{1}+c_{2}+\ldots+c_{r}}2^{n/2}\omega^{f_{1}\big{(}y^{(1)}\big{)}+f_{2}\big{(}y^{(2)}\big{)}+\ldots+f_{r}\big{(}y^{(r)}\big{)}}$
$\displaystyle=2^{n/2}\omega^{f(y)+(q/2)\big{(}c_{1}+c_{2}+\ldots+c_{r}\big{)}}$

for all $y^{(k)}\in\mathbb{F}_{2}^{n_{k}}$ and
$y=\big{(}y^{(1)},y^{(2)},\ldots,y^{(r)}\big{)}\in\mathbb{F}_{2}^{n}$.

###  3.2 Maiorana–McFarland class

Bent functions in $2k$ variables which have a representation

$f(x,y)=\langle x,\pi(y)\rangle\oplus g(y),\ \ x,y\in\mathbb{F}_{2}^{k},$

where $\pi:\mathbb{F}_{2}^{k}\rightarrow\mathbb{F}_{2}^{k}$ is a permutation
and $g$ is a Boolean function in $k$ variables, form the well known
Maiorana–McFarland class of bent functions McFarland . It is known Carlet
that the dual of a Maiorana–McFarland bent function $f(x,y)$ is equal to

$\widetilde{f}(x,y)=\big{\langle}\pi^{-1}(x),y\big{\rangle}\oplus
g\big{(}\pi^{-1}(x)\big{)},\ \ x,y\in\mathbb{F}_{2}^{k}.$

A generalization of this construction for the case $q=4$ was given by Schmidt
in Schmidt . In paper Generalized_bent  this construction was given for any
even $q$, thus, forming the following construction

$f(x,y)=\frac{q}{2}\langle x,\pi(y)\rangle+g(y),\ \ x,y\in\mathbb{F}_{2}^{k},$

where $\pi:\mathbb{F}_{2}^{k}\rightarrow\mathbb{F}_{2}^{k}$ is a permutation
and $g$ is a generalized Boolean function in $k$ variables. Its dual is

$\widetilde{f}(x,y)=\frac{q}{2}\big{\langle}\pi^{-1}(x),y\big{\rangle}+g\big{(}\pi^{-1}(x)\big{)},\
\ x,y\in\mathbb{F}_{2}^{k}.$

In the article Self-dual  necessary and sufficient conditions of (anti-)self-
duality of Maiorana–McFarland bent functions, denoted by
$\mathrm{SB}_{\mathcal{M}}^{+}(n)$ ($\mathrm{SB}_{\mathcal{M}}^{-}(n)$), were
given. In Quaternary  quaternary self-dual Maiorana–McFarland bent functions
were studied and necessary and sufficient conditions of self-duality were
obtained for them.

In the current work we generalize these results for any even $q$. Denote the
sets of (anti-)self-dual generalized Maiorana–McFarland bent functions by
$\mathrm{SB}_{\mathcal{M}^{q}}^{+}(n)$ ($\mathrm{SB}_{\mathcal{M}^{q}}^{-}(n)$
correspondingly).

###### Theorem 3.1

A generalized Maiorana–McFarland bent function

$f(x,y)=\frac{q}{2}\left\langle x,\pi(y)\right\rangle+g(y),\
x,y\in\mathbb{F}_{2}^{n/2},$

is (anti-)self-dual bent if and only if for any $y\in\mathbb{F}_{2}^{n/2}$

$\pi(y)=L\left(y\oplus b\right),\ \ \ g(y)=\frac{q}{2}\left\langle
b,y\right\rangle+d,$

where $L\in\mathcal{O}_{n/2},b\in\mathbb{F}_{2}^{n/2}$,
$\mathrm{wt}\left(b\right)$ is even (odd), $d\in\mathbb{Z}_{q}$.

###### Proof

Let $f(x,y)=\frac{q}{2}\left\langle x,\pi(y)\right\rangle\oplus g(y)$, where
$\pi$ is a permutation on $\mathbb{F}_{2}^{n/2}$,
$g\in\mathcal{GF}_{n/2}^{q}$, $x,y\in\mathbb{F}_{2}^{n/2}$. By the definition
of (anti-)self-duality a generalized bent function is (anti-)self-dual if it
coincides with (the complement of) its dual. Then for all
$x,y\in\mathbb{F}_{2}^{n}$ it must hold

$\frac{q}{2}\left\langle
x,\pi(y)\right\rangle+g(y)=\frac{q}{2}\big{\langle}\pi^{-1}(x),y\big{\rangle}+g\big{(}\pi^{-1}(x)\big{)}+\frac{q}{2}c,$
(1)

where $c\in\mathbb{F}_{2}$: $c=0$ if $f=\widetilde{f}$ and $c=1$ if
$f=\widetilde{f}+\frac{q}{2}$.

Put zero vector $x=\bm{0}\in\mathbb{F}_{2}^{n/2}$ in $(\ref{MM function equals
its dual})$, then for any $y\in\mathbb{F}_{2}^{n}$ we have

$g(y)=\frac{q}{2}\big{\langle}\pi^{-1}\left(\bm{0}\right),y\big{\rangle}+g\big{(}\pi^{-1}\left(\bm{0}\big{)}\right)+\frac{q}{2}c.$

The condition $(\ref{MM function equals its dual})$ can be transformed to

$\frac{q}{2}\langle
x,\pi(y)\rangle+\frac{q}{2}\big{\langle}\pi^{-1}\left(\bm{0}\right),y\big{\rangle}+g\big{(}\pi^{-1}\left(\bm{0}\right)\big{)}\\\
=\frac{q}{2}\big{\langle}\pi^{-1}(x),y\big{\rangle}+\frac{q}{2}\big{\langle}\pi^{-1}\left(\bm{0}\right),\pi^{-1}(x)\big{\rangle}+g\big{(}\pi^{-1}\left(\bm{0}\right)\big{)}+\frac{q}{2}c,$

or, equivalently,

$\frac{q}{2}\langle
x,\pi(y)\rangle+\frac{q}{2}\big{\langle}\pi^{-1}\left(\bm{0}\right),y\big{\rangle}=\frac{q}{2}\big{\langle}\pi^{-1}(x),y\big{\rangle}+\frac{q}{2}\big{\langle}\pi^{-1}\left(\bm{0}\right),\pi^{-1}(x)\big{\rangle}+\frac{q}{2}c.$
(2)

In both sides of $(\ref{MM function equals its dual1})$ monomials of algebraic
degree more than $2$ can not occur, since the left part has algebraic degree
at most $1$ with respect to $x$ provided that $y$ is fixed and the right part
has algebraic degree at most $1$ with respect to $y$ provided that $x$ is
fixed. Therefore, the mapping $\pi$ is an affine permutation defined by
$\pi(x)=L\left(x\oplus b\right)$ for any $x\in\mathbb{F}_{2}^{n}$, where $L$
is a $\left({n/2}\right)\times\left({n/2}\right)$ nonsingular binary matrix,
$b\in\mathbb{F}_{2}^{n/2}$.

Since the equality $(\ref{MM function equals its dual1})$ should be considered
by modulo $q$, we only care about the parity of components of both sides,
thus, for any $x,y\in\mathbb{F}_{2}^{n/2}$ having the following equality

$\left\langle x,L\left(y\oplus b\right)\right\rangle\oplus\left\langle
b,y\right\rangle=\big{\langle}L^{-1}x\oplus
b,y\big{\rangle}\oplus\big{\langle}b,L^{-1}x\oplus b\big{\rangle}\oplus c.$
(3)

Putting $x\in\mathbb{F}_{2}^{n/2}$ to be zero vector in $(\ref{MM function
equals its dual2})$, then for any $y\in\mathbb{F}_{2}^{n/2}$ it must hold
$\left\langle b,b\right\rangle=c$. Rewrite $(\ref{MM function equals its
dual2})$ in the form

$\big{\langle}x,Ly\oplus\big{(}L^{-1}\big{)}^{T}y\big{\rangle}=\big{\langle}x,Lb\oplus\big{(}L^{-1}\big{)}^{T}b\big{\rangle},$

and consider it for a zero vector $y=\bm{0}=\in\mathbb{F}_{2}^{n/2}$:

$\big{\langle}x,Lb\oplus\big{(}L^{-1}\big{)}^{T}b\big{\rangle}=0,$

that is $Lb\oplus\left(L^{-1}\right)^{T}b=\bm{0}$ or, equivalently,
$Lb=\left(L^{-1}\right)^{T}b$. It means that

$\big{\langle}x,\big{(}L\oplus\big{(}L^{-1}\big{)}^{T}\big{)}y\big{\rangle}=0,$

for any $x,y\in\mathbb{F}_{2}^{n/2}$. From this it follows that
$L^{-1}=L^{T}$, that is $L\in\mathcal{O}_{n/2}$.

It follows that the number of such functions is a function of $q$ and the
cardinality of the orthogonal group.

###### Corollary 1

It holds

$\big{|}\mathrm{SB}^{+}_{\mathcal{GM}^{q}}(n)\big{|}=\big{|}\mathrm{SB}^{-}_{\mathcal{GM}^{q}}(n)\big{|}=q\cdot
2^{n/2-1}\left|\mathcal{O}\left(n/2,\mathbb{F}_{2}\right)\right|.$

###  3.3 Dillon functions type

In paper Martinsen  an explicit representation of functions in a
generalization of Dillon’s $\mathcal{PS}_{ap}$ class to gbent functions with
$q=2^{k}$ was presented. By comparing the function from $\mathcal{PS}_{ap}$ in
a bivariate form with its dual we obtain the following result.

###### Proposition 2

Assume $G_{j}$, $j=0,1,...,k-1$, be balanced Boolean functions in $m$
variables with $G_{j}(0)=0$ and
$\sum\limits_{t\in\mathbb{F}_{2^{m}}}\omega^{\sum\limits_{j=0}^{k-1}2^{j}G_{j}(t)}=0$.
Then, if $G_{j}(u)=G_{j}(1/u)$ for any $u\in\mathbb{F}_{2^{m}}$ (with the
convention $1/0=0$), then the function
$f:\mathbb{F}_{2^{m}}\times\mathbb{F}_{2^{m}}\rightarrow\mathbb{Z}_{2^{k}}$
given by

$f(x,y)=\sum\limits_{j=0}^{k-1}2^{j}G_{j}(x/y),\ \
x,y\in\mathbb{F}_{2}^{n/2},$ (4)

is self-dual gbent in $2m$ variables.

###### Proof

It is enough to mention that, as was shown in Martinsen , the dual gbent
function of (4) has the form

$\widetilde{f}(x,y)=\sum\limits_{j=0}^{k-1}2^{j}G_{j}(y/x),\ \
x,y\in\mathbb{F}_{2}^{n/2}.$

###  3.4 Iterative construction

Let $f_{0},f_{1},f_{2},f_{3}$ be Boolean functions in $n$ variables. Consider
a Boolean function $g$ in $n+2$ variables which is defined as

$g(00,x)=f_{0}(x),\ \ g(01,x)=f_{1}(x),\ \ g(10,x)=f_{2}(x),\ \
g(11,x)=f_{3}(x),\ \ x\in\mathbb{F}_{2}^{n}.$

It is known (Preneel et al. Preneel ; see also Tokareva_iterative ) that under
condition that all functions $f_{0},f_{1},f_{2},f_{3}$ are Boolean bent
functions in $n$ variables, the mentioned function $g$ is a bent function in
$n+2$ variables if and only if

$\widetilde{f_{0}}\oplus\widetilde{f_{1}}\oplus\widetilde{f_{2}}\oplus\widetilde{f_{3}}=1,$

that gives the construction of a bent function in $n+2$ variables through the
concatenation of vectors of values of four bent functions in $n$ variables
Preneel .

Following N. Tokareva Tokareva_iterative , we will refer to Boolean bent
functions obtained by this construction as bent iterative functions
$\left(\mathcal{BI}\right)$. A construction of generalized bent functions in
$n+2$ variables obtained by a concatenation of four generalized Boolean
functions on $n$ variables was studied in Singh .

Bent iterative constructions of self-dual Boolean bent functions in $n+2$
variables, based on concatenation of $4$ Boolean bent functions in $n$
variables, were presented in Self-dual ; Kutsenko_DCC . In current work we
give two constructions of generalized bent iterative functions that generalize
the constructions for Boolean case:

###### Proposition 3

  * 1)

Let $f$ be a regular gbent function in $n$ variables, then the sign function

$\big{(}F,\widetilde{F},\widetilde{F},-F\big{)},$

where $F=\omega^{f}$ and $\widetilde{F}=\omega^{\widetilde{f}}$, is the sign
function of a self-dual gbent function in $n+2$ variables;

  * 2)

Let $f$ be a self-dual gbent function in $n$ variables with the sign function
$F$, and $g$ be an anti-self-dual gbent function in $n$ variables with the
sign function $G$, then the sign function

$\big{(}F,G,-G,F\big{)},$

where $F=\omega^{f}$ and $G=\omega^{g}$, is the sign function of a gbent
function in $n+2$ variables.

###### Proof

Let $F=\omega^{f}$ be a sign function of regular gbent function $f$ in $n$
variables. It is clear that the function $h$ is self-dual gbent if and only if

$\displaystyle\mathcal{H}_{n+2}\begin{pmatrix}F\\\ \widetilde{F}\\\
\widetilde{F}\\\ -F\end{pmatrix}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}\mathcal{H}_{n}&\mathcal{H}_{n}&\mathcal{H}_{n}&\mathcal{H}_{n}\\\
\mathcal{H}_{n}&-\mathcal{H}_{n}&\mathcal{H}_{n}&-\mathcal{H}_{n}\\\
\mathcal{H}_{n}&\mathcal{H}_{n}&-\mathcal{H}_{n}&-\mathcal{H}_{n}\\\
\mathcal{H}_{n}&-\mathcal{H}_{n}&-\mathcal{H}_{n}&\mathcal{H}_{n}\end{pmatrix}\begin{pmatrix}F\\\
\widetilde{F}\\\ \widetilde{F}\\\ -F\end{pmatrix}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}\widetilde{F}+F+F-\widetilde{F}\\\
\widetilde{F}-F+F+\widetilde{F}\\\ \widetilde{F}+F-F+\widetilde{F}\\\
\widetilde{F}-F-F-\widetilde{F}\end{pmatrix}=\begin{pmatrix}F\\\
\widetilde{F}\\\ \widetilde{F}\\\ -F\end{pmatrix},$

Let $f$ be a self-dual gbent function in $n$ variables with the sign function
$F=\omega^{f}$, and $g$ be an anti-self-dual gbent function in $n$ variables
with the sign function $G=\omega^{g}$, then

$\displaystyle\mathcal{H}_{n+2}\begin{pmatrix}F\\\ G\\\ -G\\\ F\end{pmatrix}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}\mathcal{H}_{n}&\mathcal{H}_{n}&\mathcal{H}_{n}&\mathcal{H}_{n}\\\
\mathcal{H}_{n}&-\mathcal{H}_{n}&\mathcal{H}_{n}&-\mathcal{H}_{n}\\\
\mathcal{H}_{n}&\mathcal{H}_{n}&-\mathcal{H}_{n}&-\mathcal{H}_{n}\\\
\mathcal{H}_{n}&-\mathcal{H}_{n}&-\mathcal{H}_{n}&\mathcal{H}_{n}\end{pmatrix}\begin{pmatrix}F\\\
G\\\ -G\\\ F\end{pmatrix}$ $\displaystyle=\frac{1}{2}\begin{pmatrix}F+G-G+F\\\
F-G-G-F\\\ F+G+G-F\\\ F-G+G+F\end{pmatrix}=\begin{pmatrix}-F\\\ -G\\\ G\\\
-F\end{pmatrix},$

##  4 Hamming and Lee distance spectrums

The spectrum of Hamming distances between self-dual Maiorana–McFraland Boolean
bent functions was studied in Kutsenko_MM . It was proved that

$\mathrm{Sp}_{H}\left(\mathrm{SB}^{+}_{\mathcal{M}}(n)\cup\mathrm{SB}^{-}_{\mathcal{M}}(n)\right)=\left\\{2^{n-1}\right\\}\cup\bigcup\limits_{r=0}^{n/2-1}\left\\{2^{n-1}\left(1\pm\frac{1}{2^{r}}\right)\right\\},$

and, if either $f,g\in\mathrm{SB}_{\mathcal{M}}^{+}(n)$ or
$f,g\in\mathrm{SB}_{\mathcal{M}}^{-}(n)$, then all distances except $2^{n-1}$
are attainable, and for any pair $f\in\mathrm{SB}_{\mathcal{M}}^{+}(n)$ and
$g\in\mathrm{SB}_{\mathcal{M}}^{-}(n)$ it holds $\mathrm{dist}(f,g)=2^{n-1}$.

###  4.1 Hamming distance spectrum

For generalized case we have

###### Proposition 4

It holds

$\mathrm{Sp}_{H}\left(\mathrm{SB}^{+}_{\mathcal{GM}^{q}}(n)\cup\mathrm{SB}^{-}_{\mathcal{GM}^{q}}(n)\right)=\mathrm{Sp}_{H}\left(\mathrm{SB}^{+}_{\mathcal{M}}(n)\cup\mathrm{SB}^{-}_{\mathcal{M}}(n)\right).$

Moreover, all given distances are attainable.

###### Proof

Let
$f_{1},f_{2}\in\mathrm{SB}^{+}_{\mathcal{GM}^{q}}(n)\cup\mathrm{SB}^{-}_{\mathcal{GM}^{q}}(n)$.
We have

$f_{1}(x,y)=\frac{q}{2}h_{1}(x,y)+d_{1},\ \ x,y\in\mathbb{F}_{2}^{n/2},$
$f_{2}(x,y)=\frac{q}{2}h_{2}(x,y)+d_{2},\ \ x,y\in\mathbb{F}_{2}^{n/2},$

for some
$h_{1},h_{2}\in\mathrm{SB}^{+}_{\mathcal{M}}(n)\cup\mathrm{SB}^{-}_{\mathcal{M}}(n)$
and $d_{1},d_{2}\in\mathbb{Z}_{q}$. If
$\mathrm{wt}\left(d_{1}-d_{2}\right)\notin\left\\{0,q/2\right\\}$, then
$\mathrm{dist}_{L}\left(f_{1},f_{2}\right)=2^{n}$. Otherwise, the distance
coincides with some value from the spectrum for binary case so by taking
$d_{1},d_{2}=0$ and varying $h_{1},h_{2}$ this spectrum can be entirely
covered.

###  4.2 Lee distance spectrum

For binary case the Hamming distance coincides with the Lee distance, so for
this case the Lee distance spectrum follows. For $q>2$ the spectrum can be
obtained by using the set of attainable Hamming distances from binary case.

###### Theorem 4.1

It holds

$\mathrm{Sp}_{L}\left(\mathrm{SB}^{+}_{\mathcal{GM}^{q}}(n)\cup\mathrm{SB}^{-}_{\mathcal{GM}^{q}}(n)\right)\\\
=\left\\{q\cdot
2^{n-2}\right\\}\cup\bigcup\limits_{w=0}^{q/2}\bigcup\limits_{r=0}^{n/2-1}\left\\{q\cdot
2^{n-2}\left(1\pm\frac{1}{2^{r}}\right)\mp w\cdot 2^{n-r}\right\\}.$

Moreover, all given distances are attainable.

###### Proof

Let
$f_{1},f_{2}\in\mathrm{SB}^{+}_{\mathcal{GM}^{q}}(n)\cup\mathrm{SB}^{-}_{\mathcal{GM}^{q}}(n)$.
Again, we have

$f_{1}(x,y)=\frac{q}{2}h_{1}(x,y)+d_{1},\ \ x,y\in\mathbb{F}_{2}^{n/2},$
$f_{2}(x,y)=\frac{q}{2}h_{2}(x,y)+d_{2},\ \ x,y\in\mathbb{F}_{2}^{n/2},$

for some
$h_{1},h_{2}\in\mathrm{SB}^{+}_{\mathcal{M}}(n)\cup\mathrm{SB}^{-}_{\mathcal{M}}(n)$
and $d_{1},d_{2}\in\mathbb{Z}_{q}$. Denote
$\mathrm{wt}_{L}\left(d_{1}-d_{2}\right)$ by $w$ and the Hamming distance
$\mathrm{dist}_{H}\left(h_{1},h_{2}\right)$ between Boolean functions
$h_{1},h_{2}$ by $d$. Under this notation the Lee distance between $f_{1}$ and
$f_{2}$ is the function of $w$, $d$ and number of variables $n$. Indeed,

$\displaystyle\mathrm{dist}_{L}\left(f_{1},f_{2}\right)$
$\displaystyle=\sum\limits_{x,y\in\mathbb{F}_{2}^{n/2}}\mathrm{wt}_{L}\left(f_{1}(x,y)-f_{2}(x,y)\right)$
$\displaystyle=\sum\limits_{\begin{subarray}{c}x,y\in\mathbb{F}_{2}^{n/2}\\\
h_{1}(x,y)=h_{2}(x,y)\end{subarray}}\mathrm{wt}_{L}\left(f_{1}(x,y)-f_{2}(x,y)\right)$
$\displaystyle+\sum\limits_{\begin{subarray}{c}x,y\in\mathbb{F}_{2}^{n/2}\\\
h_{1}(x,y)\neq
h_{2}(x,y)\end{subarray}}\mathrm{wt}_{L}\left(f_{1}(x,y)-f_{2}(x,y)\right)$
$\displaystyle=\sum\limits_{\begin{subarray}{c}x,y\in\mathbb{F}_{2}^{n/2}\\\
h_{1}(x,y)=h_{2}(x,y)\end{subarray}}\mathrm{wt}_{L}\left(d_{1}-d_{2}\right)$
$\displaystyle+\sum\limits_{\begin{subarray}{c}x,y\in\mathbb{F}_{2}^{n/2}\\\
h_{1}(x,y)\neq
h_{2}(x,y)\end{subarray}}\mathrm{wt}_{L}\left(d_{1}-d_{2}+\frac{q}{2}\right)$
$\displaystyle=\left(2^{n}-d\right)w+d\left(\frac{q}{2}-w\right)=2^{n}w-2dw+\frac{q}{2}d.$

If $h_{1}\in\mathrm{SB}^{+}_{\mathcal{M}}(n)$ and
$h_{2}\in\mathrm{SB}^{-}_{\mathcal{M}}(n)$, then

$\mathrm{dist}_{L}\left(f_{1},f_{2}\right)=2^{n}w-2\cdot
2^{n-1}w+\frac{q}{2}\cdot 2^{n-1}=q\cdot 2^{n-2}.$

From the aforementioned Hamming distance spectrum for binary case it follows
that for any $r\in\\{0,1,...,n/2-1\\}$ there exists at least one pair of
(anti-)self-dual Boolean Maiorana–McFarland bent functions in $n$ variables at
the Hamming distance $d=2^{n-1}+2^{n-r-1}$ as well as
$2^{n}-d=2^{n-1}-2^{n-r-1}$. Assume $r$ is fixed, put $d=2^{n-1}\left(1\pm
2^{-r}\right)$ in the expression for
$\mathrm{dist}_{L}\left(f_{1},f_{2}\right)$:

$\displaystyle\mathrm{dist}_{L}\left(f_{1},f_{2}\right)$
$\displaystyle=2^{n}w-2w\cdot
2^{n-1}\left(1\pm\frac{1}{2^{r}}\right)+\frac{q}{2}\cdot
2^{n-1}\left(1\pm\frac{1}{2^{r}}\right)$ $\displaystyle=2^{n}w-2^{n}w\mp
w\cdot 2^{n-r}+q\cdot 2^{n-2}\left(1\pm\frac{1}{2^{r}}\right)$
$\displaystyle=q\cdot 2^{n-2}\left(1\pm\frac{1}{2^{r}}\right)\mp w\cdot
2^{n-r}.$

Observation that $r$ runs $\\{0,1,...,n/2-1\\}$ and $w$ varies within the set
$\\{0,1,...,q/2\\}$ yields the result.

###### Proposition 5

The minimal Lee distance between generalized (anti-)self-dual
Maiorana–McFarland bent functions in $n$ variables is equal to $q\cdot
2^{n-3}$.

###### Proof

Estimate the minimal value of the term

$D=q\cdot 2^{n-2}\left(1\pm\frac{1}{2^{r}}\right)\mp w\cdot 2^{n-r},$

with $r\in\\{1,2,...,n/2-1\\}$ and $w\in\\{0,1,...,q/2\\}$. Here we exclude
the case $r=0$ since then the Lee distance is equal to either $D=w\cdot
2^{n}\geqslant 2^{n}$ or $D=2^{n-1}\left(q-2w\right)\geqslant 2^{n}$, provided
that $f_{1},f_{2}$ are distinct. Indeed, $r=0$ implies
$d\in\left\\{0,2^{n}\right\\}$, and the first aforementioned expression for
$D$ corresponds to $h_{1}=h_{2}$, while the second one to $h_{1}\oplus
h_{2}=1$.

Consider two cases depending on sequence of the signs.

Case 1:

$D=q\cdot 2^{n-2}\left(1+\frac{1}{2^{r}}\right)-w\cdot 2^{n-r}=q\cdot
2^{n-2}+2^{n-r}\left(\frac{q}{4}-w\right).$

Since $w\in\\{0,1,...,q/2\\}$ it follows that

$-\frac{q}{4}\leqslant\frac{q}{4}-w\leqslant\frac{q}{4},$

hence

$-\frac{q}{4}\cdot 2^{n-r}\leqslant
2^{n-r}\left(\frac{q}{4}-w\right)\leqslant\frac{q}{4}\cdot 2^{n-r}.$

Then

$D\geqslant q\cdot 2^{n-2}-\frac{q}{4}\cdot 2^{n-r}=q\cdot
2^{n-2}\left(1-\frac{1}{2^{r}}\right)\geqslant q\cdot 2^{n-3},$

and $D_{\min}=q\cdot 2^{n-3}$, that is attainable for $r=1$ and $w=q/2$.

Case 2:

$D=q\cdot 2^{n-2}\left(1-\frac{1}{2^{r}}\right)+w\cdot 2^{n-r}=q\cdot
2^{n-2}+2^{n-r}\left(w-\frac{q}{4}\right).$

From $w\in\\{0,1,...,q/2\\}$ it follows that

$-\frac{q}{4}\leqslant w-\frac{q}{4}\leqslant\frac{q}{4},$

hence

$-\frac{q}{4}\cdot 2^{n-r}\leqslant
2^{n-r}\left(w-\frac{q}{4}\right)\leqslant\frac{q}{4}\cdot 2^{n-r}.$

Then

$D\geqslant q\cdot 2^{n-2}-\frac{q}{4}\cdot 2^{n-r}=q\cdot
2^{n-2}\left(1-\frac{1}{2^{r}}\right)\geqslant q\cdot 2^{n-3},$

and again $D_{\min}=q\cdot 2^{n-3}$, that is attainable for $r=1$ and $w=0$.

Thus the minimal Lee distance is equal to $q\cdot 2^{n-3}$.

In Paterson  it was shown that both minimal Hamming and Lee distances of
generalized Reed–Muller codes $\mathrm{RM}_{q}(r,n)$ are equal to $2^{n-r}$
for any positive integer $q$. Therefore, it immediately follows that

###### Corollary 2

The minimal Hamming distance $2^{n-2}$ between quadratic (generalized) bent
functions is attainable on the sets of self-dual and anti-self-dual
Maiorana–McFarland bent functions from $\mathcal{GM}^{q}_{n}$ only for $q=2$.

##  5 Sign functions of (anti-)self-dual gbent functions

Let $I_{n}$ be the identity matrix of size $n$ and $H_{n}=H_{1}^{\otimes n}$
be the $n$-fold tensor product of the matrix $H_{1}$ with itself, where

$H_{1}=\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix}.$

It is known the Hadamard property of this matrix

$H_{n}H_{n}^{T}=2^{n}I_{2^{n}},$

where $H_{n}^{T}$ is transpose of $H_{n}$ (it holds $H_{n}^{T}=H_{n}$ by
symmetricity of $H_{n}$). Denote $\mathcal{H}_{n}=2^{-n/2}H_{n}$.

Recall an orthogonal decomposition of $\mathbb{R}^{2^{n}}$ in eigenspaces of
$H_{n}$ from Self-dual  (Lemma 5.2):

$\mathbb{R}^{2^{n}}=\mathrm{Ker}\left(H_{n}+2^{n/2}I_{2^{n}}\right)\oplus\mathrm{Ker}\left(H_{n}-2^{n/2}I_{2^{n}}\right),$

where the symbol $\oplus$ denotes a direct sum of subspaces. Consider the same
decomposition

$\mathbb{C}^{2^{n}}=\mathrm{Ker}\left(H_{n}+2^{n/2}I_{2^{n}}\right)\oplus\mathrm{Ker}\left(H_{n}-2^{n/2}I_{2^{n}}\right),$

for a complex space $\mathbb{C}^{2^{n}}$.

As for the Boolean case (see Kutsenko_CC ), we note that sign function of any
self-dual gbent function is the eigenvector of $\mathcal{H}_{n}$ attached to
the eigenvalue $(+1)$, that is an element from the subspace
$\mathrm{Ker}\big{(}\mathcal{H}_{n}-I_{2^{n}}\big{)}=\mathrm{Ker}\big{(}H_{n}-2^{n/2}I_{2^{n}}\big{)}$.
The same holds for a sign function of any anti-self-dual gbent function, which
obviously is an eigenvector of $\mathcal{H}_{n}$ attached to the eigenvalue
$(-1)$, that is an element from the subspace
$\mathrm{Ker}\big{(}\mathcal{H}_{n}+I_{2^{n}}\big{)}=\mathrm{Ker}\big{(}H_{n}+2^{n/2}I_{2^{n}}\big{)}$.

It is known that

$\mathrm{dim}\big{(}\mathrm{Ker}\left(\mathcal{H}_{n}+I_{2^{n}}\right)\big{)}=\mathrm{dim}\big{(}\mathrm{Ker}\big{(}\mathcal{H}_{n}-I_{2^{n}}\big{)}\big{)}=2^{n-1},$

where $\mathrm{dim}(V)$ is the dimension of the subspace
$V\subseteq\mathbb{R}^{2^{n}}$. Moreover, since $\mathcal{H}_{n}$ is symmetric
(Hermitian), the subspaces
$\mathrm{Ker}\big{(}\mathcal{H}_{n}+I_{2^{n}}\big{)}$ and
$\mathrm{Ker}\big{(}\mathcal{H}_{n}-I_{2^{n}}\big{)}$ are mutually orthogonal.

In Kutsenko_DCC  it was proved that provided $n\geqslant 4$, the linear span
of sign functions of self-dual as well as anti-self-dual Boolean bent
functions Boolean bent functions in $n$ variables has dimension $2^{n-1}$. The
same result can be also given for gbent functions:

###### Theorem 5.1

Let $n\geqslant 4$, then the linear span of sign functions of (anti-)self-dual
gbent functions in $n$ variables has dimension $2^{n-1}$.

###### Proof

It is enough to mention that since $q$ is even it holds
$(-1)=\omega^{q/2}\in\big{\\{}\omega,\omega^{2},...,\omega^{q-1}\big{\\}}$,
therefore the set of sign fuctions of (anti-)self-dual Boolean bent functions
in $n$ variables is a subset of the set of sign functions of (anti-)self-dual
gbent functions in $n$ variables. Then from Kutsenko_DCC  (Theorem 2) the
dimension follows.

It is worth to note that the example of the basis of the subspace
$\mathrm{Ker}\big{(}\mathcal{H}_{n}-I_{2^{n}}\big{)}$ can be constructed by
using the functions obtained from the construction from Proposition 3.

When $n=2$ there are two self-dual Boolean bent functions, namely $x_{1}x_{2}$
and $x_{1}x_{2}\oplus 1$, which have sign functions $(1,1,1,-1)$ and
$(-1,-1,-1,1)$ respectively. These sign functions are linearly dependent
vectors in $\mathbb{R}^{4}$. The set $\mathrm{SB}^{-}(2)$ consists of
functions $x_{1}x_{2}\oplus x_{1}\oplus x_{2}$ and $x_{1}x_{2}\oplus
x_{1}\oplus x_{2}\oplus 1$ with sign functions $(1,-1,-1,-1)$ and $(-1,1,1,1)$
respectively. These sign functions are linearly dependent vectors in
$\mathbb{R}^{4}$ as well. Generalization comprises solution of the system

$\frac{1}{2}\begin{pmatrix}1&1&1&1\\\ 1&-1&1&-1\\\ 1&1&-1&-1\\\
1&-1&-1&1\end{pmatrix}\begin{pmatrix}\omega^{d_{1}}\\\ \omega^{d_{2}}\\\
\omega^{d_{3}}\\\ \omega^{d_{4}}\end{pmatrix}=\begin{pmatrix}\omega^{d_{1}}\\\
\omega^{d_{2}}\\\ \omega^{d_{3}}\\\ \omega^{d_{4}}\end{pmatrix},$

where variables are numbers $d_{1},d_{2},d_{3},d_{4}\in\mathbb{Z}_{q}$ in
fact. It is clear that the only solution pattern is

$\big{(}\omega^{d},\omega^{d},\omega^{d},\omega^{d+q/2}\big{)}=\omega^{d}\cdot\left(1,1,1,-1\right)\in\mathbb{C}^{4},$

where $d\in\mathbb{Z}_{q}$. It means that any two sign functions of self-dual
gbent functions from $\mathrm{SB}^{+}_{q}(2)$ are linearly dependent over
$\mathbb{C}$ and $\big{\lvert}\mathrm{SB}^{+}_{q}(2)\big{\rvert}=q$.

The next result is a generalization of the similar one from Kutsenko_DCC .

###### Theorem 5.2

Let $n\geqslant 4$ and $f\in\mathrm{SB}_{q}^{+}(n)$. For sign function
$\omega^{f}=\big{(}F^{00},F^{01},F^{10},F^{11}\big{)}$, where
$F^{00},F^{01},F^{10},F^{11}\in\big{\\{}1,\omega,\omega^{2},\ldots,\omega^{q-1}\big{\\}}^{2^{n-2}}$,
it holds

$\displaystyle\big{\langle}F^{00},F^{01}\big{\rangle}+\big{\langle}F^{10},F^{11}\big{\rangle}$
$\displaystyle=0,$
$\displaystyle\big{\langle}F^{00},F^{10}\big{\rangle}+\big{\langle}F^{01},F^{11}\big{\rangle}$
$\displaystyle=0.$

###### Proof

Let $f\in\mathrm{SB}_{q}^{+}(n)$, then by Theorem 5.1 there exist vectors

$\displaystyle\alpha$
$\displaystyle=\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{2^{n-3}}\right)\in\mathbb{C}^{2^{n-3}},$
$\displaystyle\beta$
$\displaystyle=\left(\beta_{1},\beta_{2},\ldots,\beta_{2^{n-3}}\right)\in\mathbb{C}^{2^{n-3}},$
$\displaystyle\gamma$
$\displaystyle=\left(\gamma_{1},\gamma_{2},\ldots,\gamma_{2^{n-2}}\right)\in\mathbb{C}^{2^{n-2}},$

such that

$\omega^{f}=\sum\limits_{i=1}^{2^{n-3}}{\alpha_{i}{\mathbf{F}}_{i}^{n}}+\sum\limits_{j=1}^{2^{n-3}}{\beta_{j}{\mathbf{G}}_{j}^{n}}+\sum\limits_{k=1}^{2^{n-2}}{\gamma_{k}\left({\mathbf{FG}}\right)_{k}^{n}},$

where the sets
$S_{\mathbf{F}}=\left\\{\mathbf{F}_{i}^{n}\right\\}_{i=1}^{2^{n-3}}$,
$S_{\mathbf{G}}=\left\\{\mathbf{G}_{j}^{n}\right\\}_{j=1}^{2^{n-3}}$ and
$S_{\mathbf{FG}}=\left\\{\left(\mathbf{FG}\right)_{k}^{n}\right\\}_{k=1}^{2^{n-2}}$
are described in the proof of Theorem $2$ from Kutsenko_DCC . Consider the
sets $S_{\mathbf{F}}$, $S_{\mathbf{G}}$, $S_{\mathbf{FG}}$ and denote

$\displaystyle\mathbf{F}_{i}^{n}$
$\displaystyle=\left(F_{i},F_{i},F_{i},-F_{i}\right),$
$\displaystyle\mathbf{G}_{j}^{n}$
$\displaystyle=\left(G_{j},-G_{j},-G_{j},-G_{j}\right),$
$\displaystyle\left(\mathbf{FG}\right)_{k}^{n}$
$\displaystyle=\left(A_{k},-B_{k},B_{k},A_{k}\right),$

where $F_{i},A_{k}\in\mathrm{Ker}\big{(}\mathcal{H}_{n-2}-I_{2^{n-2}}\big{)}$,
$G_{j},B_{k}\in\mathrm{Ker}\big{(}\mathcal{H}_{n-2}+I_{2^{n-2}}\big{)}$,
$i,j=1,2,\ldots,2^{n-3}$, $k=1,2,\ldots,2^{n-2}$, and define the vectors

$\displaystyle\mathbf{F}$
$\displaystyle=\sum\limits_{i=1}^{2^{n-3}}\alpha_{i}F_{i},$
$\displaystyle\mathbf{G}$
$\displaystyle=\sum\limits_{j=1}^{2^{n-3}}\beta_{j}G_{j},$
$\displaystyle\mathbf{A}$
$\displaystyle=\sum\limits_{k=1}^{2^{n-2}}\gamma_{k}A_{k},$
$\displaystyle\mathbf{B}$
$\displaystyle=\sum\limits_{k=1}^{2^{n-2}}\gamma_{k}B_{k}.$

Under this notation the sign function $\omega^{f}$ has the form

$\omega^{f}=\begin{pmatrix}F^{00}\\\ F^{01}\\\ F^{10}\\\
F^{11}\end{pmatrix}=\begin{pmatrix}\mathbf{F}+\mathbf{G}+\mathbf{A}\\\
\mathbf{F}-\mathbf{G}-\mathbf{B}\\\ \mathbf{F}-\mathbf{G}+\mathbf{B}\\\
-\mathbf{F}-\mathbf{G}+\mathbf{A}\end{pmatrix}\in\big{\\{}1,\omega,\omega^{2},...,\omega^{q-1}\big{\\}}^{2^{n}}.$

For any $j=1,2,\ldots,2^{n-2}$ denote

$\displaystyle\big{(}\mathbf{F}+\mathbf{G}\big{)}_{j}+\mathbf{A}_{j}=\omega^{t_{j}},$
$\displaystyle\big{(}\mathbf{F}-\mathbf{G}\big{)}_{j}-\mathbf{B}_{j}=\omega^{r_{j}},$
$\displaystyle\big{(}\mathbf{F}-\mathbf{G}\big{)}_{j}+\mathbf{B}_{j}=\omega^{l_{j}},$
$\displaystyle-\big{(}\mathbf{F}+\mathbf{G}\big{)}_{j}+\mathbf{A}_{j}=\omega^{k_{j}},$

where $t_{j},r_{j},l_{j},k_{j}\in\mathbb{Z}_{q}$. Then

$\displaystyle\mathbf{A}_{j}$
$\displaystyle=\frac{1}{2}\big{(}\omega^{t_{j}}+\omega^{k_{j}}\big{)},$
$\displaystyle\mathbf{B}_{j}$
$\displaystyle=\frac{1}{2}\big{(}\omega^{l_{j}}-\omega^{r_{j}}\big{)},$
$\displaystyle\big{(}\mathbf{F}+\mathbf{G}\big{)}_{j}$
$\displaystyle=\frac{1}{2}\big{(}\omega^{t_{j}}-\omega^{k_{j}}\big{)},$
$\displaystyle\big{(}\mathbf{F}-\mathbf{G}\big{)}_{j}$
$\displaystyle=\frac{1}{2}\big{(}\omega^{r_{j}}+\omega^{l_{j}}\big{)}.$

Note that

$\langle\mathbf{G},\mathbf{A}\rangle=\langle\mathbf{F},\mathbf{B}\rangle=0.$

By using this we obtain the expression for the first inner product

$\displaystyle\big{\langle}F^{00},F^{01}\big{\rangle}+\big{\langle}F^{10},F^{11}\big{\rangle}$
$\displaystyle=\langle\mathbf{F}+\mathbf{G}+\mathbf{A},\mathbf{F}-\mathbf{G}-\mathbf{B}\rangle$
$\displaystyle+\langle\mathbf{F}-\mathbf{G}+\mathbf{B},-\mathbf{F}-\mathbf{G}+\mathbf{A}\rangle$
$\displaystyle=\langle\mathbf{F},\mathbf{F}\rangle-\langle\mathbf{F},\mathbf{G}\rangle-\langle\mathbf{F},\mathbf{B}\rangle$
$\displaystyle+\langle\mathbf{G},\mathbf{F}\rangle-\langle\mathbf{G},\mathbf{G}\rangle-\langle\mathbf{G},\mathbf{B}\rangle$
$\displaystyle+\langle\mathbf{A},\mathbf{F}\rangle-\langle\mathbf{A},\mathbf{G}\rangle-\langle\mathbf{A},\mathbf{B}\rangle$
$\displaystyle-\langle\mathbf{F},\mathbf{F}\rangle-\langle\mathbf{F},\mathbf{G}\rangle+\langle\mathbf{F},\mathbf{A}\rangle$
$\displaystyle+\langle\mathbf{G},\mathbf{F}\rangle+\langle\mathbf{G},\mathbf{G}\rangle-\langle\mathbf{G},\mathbf{A}\rangle$
$\displaystyle-\langle\mathbf{B},\mathbf{F}\rangle-\langle\mathbf{B},\mathbf{G}\rangle+\langle\mathbf{B},\mathbf{A}\rangle$
$\displaystyle=\langle\mathbf{A},\mathbf{F}\rangle+\langle\mathbf{F},\mathbf{A}\rangle-\langle\mathbf{G},\mathbf{B}\rangle-\langle\mathbf{B},\mathbf{G}\rangle$
$\displaystyle=\langle\mathbf{A},\mathbf{F}+\mathbf{G}\rangle+\overline{\langle\mathbf{A},\mathbf{F}+\mathbf{G}\rangle}+\langle\mathbf{B},\mathbf{F}-\mathbf{G}\rangle+\overline{\langle\mathbf{B},\mathbf{F}-\mathbf{G}\rangle}$
(5)

while the second one has the form

$\displaystyle\big{\langle}F^{00},F^{10}\big{\rangle}+\big{\langle}F^{01},F^{11}\big{\rangle}$
$\displaystyle=\langle\mathbf{F}+\mathbf{G}+\mathbf{A},\mathbf{F}-\mathbf{G}+\mathbf{B}\rangle$
$\displaystyle+\langle\mathbf{F}-\mathbf{G}-\mathbf{B},-\mathbf{F}-\mathbf{G}+\mathbf{A}\rangle$
$\displaystyle=\langle\mathbf{F},\mathbf{F}\rangle-\langle\mathbf{F},\mathbf{G}\rangle+\langle\mathbf{F},\mathbf{B}\rangle$
$\displaystyle+\langle\mathbf{G},\mathbf{F}\rangle-\langle\mathbf{G},\mathbf{G}\rangle+\langle\mathbf{G},\mathbf{B}\rangle$
$\displaystyle+\langle\mathbf{A},\mathbf{F}\rangle-\langle\mathbf{A},\mathbf{G}\rangle+\langle\mathbf{A},\mathbf{B}\rangle$
$\displaystyle-\langle\mathbf{F},\mathbf{F}\rangle-\langle\mathbf{F},\mathbf{G}\rangle+\langle\mathbf{F},\mathbf{A}\rangle$
$\displaystyle+\langle\mathbf{G},\mathbf{F}\rangle+\langle\mathbf{G},\mathbf{G}\rangle-\langle\mathbf{G},\mathbf{A}\rangle$
$\displaystyle+\langle\mathbf{B},\mathbf{F}\rangle+\langle\mathbf{B},\mathbf{G}\rangle-\langle\mathbf{B},\mathbf{A}\rangle$
$\displaystyle=\langle\mathbf{A},\mathbf{F}\rangle+\langle\mathbf{F},\mathbf{A}\rangle+\langle\mathbf{G},\mathbf{B}\rangle+\langle\mathbf{B},\mathbf{G}\rangle$
$\displaystyle=\langle\mathbf{A},\mathbf{F}+\mathbf{G}\rangle+\overline{\langle\mathbf{A},\mathbf{F}+\mathbf{G}\rangle}-\langle\mathbf{B},\mathbf{F}-\mathbf{G}\rangle-\overline{\langle\mathbf{B},\mathbf{F}-\mathbf{G}\rangle}$
(6)

Consider inner in details the following inner products

$\displaystyle\langle\mathbf{A},\mathbf{F}+\mathbf{G}\rangle$
$\displaystyle=\sum\limits_{j=1}^{2^{n}}\mathbf{A}_{j}\overline{\big{(}\mathbf{F}+\mathbf{G}\big{)}}_{j}=\frac{1}{4}\sum\limits_{j=1}^{2^{n}}\big{(}\omega^{t_{j}}+\omega^{k_{j}}\big{)}\big{(}\overline{\omega^{t_{j}}}-\overline{\omega^{k_{j}}}\big{)}$
$\displaystyle=\frac{1}{4}\sum\limits_{j=1}^{2^{n}}\big{(}1-1+\omega^{k_{j}}\overline{\omega^{t_{j}}}-\omega^{t_{j}}\overline{\omega^{k_{j}}}\big{)}=\frac{1}{2}\mathrm{Im}\left(\sum\limits_{j=1}^{2^{n}}\omega^{k_{j}}\overline{\omega^{t_{j}}}\right)i,$
$\displaystyle\overline{\langle\mathbf{A},\mathbf{F}+\mathbf{G}\rangle}$
$\displaystyle=-\frac{1}{2}\mathrm{Im}\left(\sum\limits_{j=1}^{2^{n}}\omega^{k_{j}}\overline{\omega^{t_{j}}}\right)i,$
$\displaystyle\langle\mathbf{B},\mathbf{F}-\mathbf{G}\rangle$
$\displaystyle=\sum\limits_{j=1}^{2^{n}}\mathbf{B}_{j}\overline{\big{(}\mathbf{F}-\mathbf{G}\big{)}}_{j}=\frac{1}{4}\sum\limits_{j=1}^{2^{n}}\big{(}\omega^{l_{j}}-\omega^{r_{j}}\big{)}\big{(}\overline{\omega^{l_{j}}}+\overline{\omega^{r_{j}}}\big{)}$
$\displaystyle=\frac{1}{4}\sum\limits_{j=1}^{2^{n}}\big{(}1-1+\omega^{l_{j}}\overline{\omega^{r_{j}}}-\omega^{r_{j}}\overline{\omega^{l_{j}}}\big{)}=\frac{1}{2}\mathrm{Im}\left(\sum\limits_{j=1}^{2^{n}}\omega^{l_{j}}\overline{\omega^{r_{j}}}\right)i,$
$\displaystyle\overline{\langle\mathbf{B},\mathbf{F}-\mathbf{G}\rangle}$
$\displaystyle=-\frac{1}{2}\mathrm{Im}\left(\sum\limits_{j=1}^{2^{n}}\omega^{l_{j}}\overline{\omega^{r_{j}}}\right)i,$

therefore, both $(\ref{equation:the first inner product})$ and
$(\ref{equation:the second inner product})$ are zero numbers.

##  6 Properties of self-dual gbent function

###  6.1 Upper bound for the number of self-dual gbent functions

Let $2^{h-1}<q\leqslant 2^{h}$. For any $f\in\mathcal{GF}_{n}^{q}$ it is
possible to associate a unique sequence of Boolean functions
$a_{0},a_{1},\ldots,a_{h-1}\in\mathcal{F}_{n}$ such that Generalized_bent

$f(x)=a_{0}(x)+2a_{1}(x)+\ldots+2^{h-1}a_{h-1}(x),\ \ x\in\mathbb{F}_{2}^{n}.$

In paper Hodzic  it was proved that for the case $q=2^{k}$ and even $n$,
provided that $f$ is gbent its dual gbent $\widetilde{f}$ has the following
form

$\widetilde{f}(x)=b_{0}(x)+2b_{1}(x)+\ldots+2^{h-1}b_{h-1}(x),\ \
x\in\mathbb{F}_{2}^{n},$

where $b_{k-1}=\widetilde{b_{k-1}}$ and the dual of
$b_{j}=\widetilde{b_{k-1}}\oplus\big{(}\widetilde{b_{k-1}\oplus
b_{j}}\big{)}$. If $f$ is self-dual gbent then $b_{k-1}$ is self-dual Boolean
function and for $j=0,1,\ldots,k-1$ Boolean functions ${\big{(}b_{k-1}\oplus
b_{j}\big{)}}$ are self-dual. It follows the statement

###### Proposition 6

It holds
$\big{\lvert}\mathrm{SB}_{2^{k}}^{+}(n)\big{\rvert}\leqslant\big{\lvert}\mathrm{SB}_{2}^{+}(n)\big{\rvert}^{k}$.

Note that this bound is consistent with the results from the work
Decomposition .

###  6.2 Affinity of self-dual gbent function

In paper Singh  for the case when $q$ is divisible by $4$, necessary and
sufficient conditions for the bentness of generalized Boolean functions of the
form

$f(x)=\sum\limits_{i=1}^{n}\lambda_{i}x_{i}+\lambda_{0},$

where $\lambda_{0},\lambda_{1},\ldots,\lambda_{n}\in\mathbb{Z}_{q}$, were
obtained. Functions from this class are referred to as affine functions.

It is well known that Boolean bent function and, as a consequence, self-dual
Boolean bent function can not be affine. The next result shows the non-
existence of self-dual gbent functions within the class of affine functions.

###### Theorem 6.1

There are no self-dual generalized bent functions in $n$ variables of the form

$f(x)=\sum\limits_{i=1}^{n}\lambda_{i}x_{i}+\lambda_{0},$

where $\lambda_{0},\lambda_{1},\ldots,\lambda_{n}\in\mathbb{Z}_{q}$.

###### Proof

Let $f$ be an affine gbent function in $n$ variables (for the case $q$ not
divisible by $4$ if such exists, otherwise the result follows), namely

$f(x)=\sum\limits_{i=1}^{n}\lambda_{i}x_{i}+\lambda_{0},\ \
x\in\mathbb{F}_{2}^{n},$

where $\lambda_{0},\lambda_{1},\ldots,\lambda_{n}\in\mathbb{Z}_{q}$. It is
self-dual if and only if

$\displaystyle H_{f}(y)$
$\displaystyle=\sum\limits_{x\in\mathbb{F}_{2}^{n}}\omega^{f(x)}(-1)^{\langle
x,y\rangle}=\omega^{\lambda_{0}}\sum\limits_{x\in\mathbb{F}_{2}^{n}}\omega^{\sum\limits_{i=1}^{n}\lambda_{i}x_{i}+\frac{q}{2}\langle
x,y\rangle}$
$\displaystyle=\omega^{\lambda_{0}}\prod\limits_{i=1}^{n}\sum\limits_{x_{i}\in\mathbb{F}_{2}}\omega^{\lambda_{i}x_{i}+\frac{q}{2}y_{i}x_{i}}=\omega^{\lambda_{0}}\prod\limits_{i=1}^{n}\left(1+\omega^{\frac{q}{2}y_{i}+\lambda_{i}}\right),$

for any $y\in\mathbb{F}_{2}^{n}$.

For every $y\in\mathbb{F}_{2}^{n}$ denote

$\displaystyle\widehat{y}$
$\displaystyle=\left(y_{1},y_{2},\ldots,y_{n-1}\right)\in\mathbb{F}_{2}^{n-1},$
$\displaystyle P_{n-1}\left(\widehat{y}\right)$
$\displaystyle=\left(1+\omega^{\frac{q}{2}y_{1}+\lambda_{1}}\right)\left(1+\omega^{\frac{q}{2}y_{2}+\lambda_{2}}\right)\cdot\ldots\cdot\left(1+\omega^{\frac{q}{2}y_{n-1}+\lambda_{n-1}}\right),$
$\displaystyle a_{n-1}\left(\widehat{y}\right)$
$\displaystyle=\lambda_{1}y_{1}+\lambda_{2}y_{2}+\ldots+\lambda_{n-1}y_{n-1}.$

Then for any $y\in\mathbb{F}_{2}^{n}$ such that $y_{n}=0$ it holds

$P_{n-1}\left(\widehat{y}\right)\left(1+\omega^{\lambda_{n}}\right)=2^{n/2}\omega^{a_{n-1}\left(\widehat{y}\right)},$

and for any $y\in\mathbb{F}_{2}^{n}$ such that $y_{n}=1$:

$P_{n-1}\left(\widehat{y}\right)\left(1+\omega^{\frac{q}{2}+\lambda_{n}}\right)=2^{n/2}\omega^{a_{n-1}\left(\widehat{y}\right)+\lambda_{n}}.$

So, for any $\widehat{y}\in\mathbb{F}_{2}^{n-1}$ consider the system

$\begin{cases}P_{n-1}\left(\widehat{y}\right)\left(1+\omega^{\lambda_{n}}\right)=2^{n/2}\omega^{a_{n-1}\left(\widehat{y}\right)},\\\
P_{n-1}\left(\widehat{y}\right)\left(1-\omega^{\lambda_{n}}\right)=2^{n/2}\omega^{a_{n-1}\left(\widehat{y}\right)+\lambda_{n}}.\end{cases}$

It is equivalent to

$\begin{cases}P_{n-1}\left(\widehat{y}\right)\left(1+\omega^{\lambda_{n}}\right)=2^{n/2}\omega^{a_{n-1}\left(\widehat{y}\right)},\\\
P_{n-1}\left(\widehat{y}\right)\left(1-\omega^{\lambda_{n}}\right)=P_{n-1}\left(\widehat{y}\right)\left(1+\omega^{\lambda_{n}}\right)\cdot\omega^{\lambda_{n}}.\end{cases}$

Thus, we obtain the relation

$P_{n-1}\left(\widehat{y}\right)\left(1-\omega^{\lambda_{n}}\right)=P_{n-1}\left(\widehat{y}\right)\left(1+\omega^{\lambda_{n}}\right)\cdot\omega^{\lambda_{n}},$

and can note that $P_{n-1}\left(\widehat{y}\right)\neq 0$ since for any
$y\in\mathbb{F}_{2}^{n}$ we have

$H_{f}(y)=\omega^{\lambda_{0}}P_{n-1}\left(\widehat{y}\right)\left(1+\omega^{\frac{q}{2}y_{n}+\lambda_{n}}\right),$

and $f$ is gbent that is
$1-\omega^{\lambda_{n}}=\omega^{\lambda_{n}}+\left(\omega^{\lambda_{n}}\right)^{2}$.
The solutions of this equation are $\left(-1\pm\sqrt{2}\right)$. The norm of
every of these numbers is not $1$ therefore $\omega^{\lambda_{n}}$ can not be
a solution.

###  6.3 Self-dual gbent functions symmetric with respect to two variables

A generalized Boolean function $h\in\mathcal{GF}_{n+2}^{q}$ is said to be
symmetric with respect to two variables $y$ and $z$ if there exist functions
$f,g,s\in\mathcal{GF}_{n}^{q}$ such that

$h(z,y,x)=f(x)+(y\oplus z)g(x)+yzs(x),\ \
y,z\in\mathbb{F}_{2},x\in\mathbb{F}_{2}^{n}.$ (7)

In paper Generalized_bent  it was proved that a function of such form is gbent
if and only if the functions $f,f+g$ are gbent and $s(x)=q/2$,
$x\in\mathbb{F}_{2}^{n}$. We study the conditions for self-duality of
functions of such form.

###### Theorem 6.2

Let $h$ be a gbent function of the form $(\ref{equation_symmetric})$. Then $h$
is self-dual if and only if $f$ is regular gbent, $g=\widetilde{f}+(q-1)f$,
and $s(x)=q/2$, $x\in\mathbb{F}_{2}^{n}$.

###### Proof

Let $F,FG$ be sign functions of regular gbent functions $f,f+g$. It is clear
that

$\omega^{h}=\begin{pmatrix}F\\\ FG\\\ FG\\\ -F\end{pmatrix}.$

Then the function $h$ is self-dual gbent if and only if

$\omega^{\widetilde{h}}=\mathcal{H}_{n+2}\omega^{h}=\frac{1}{2}\begin{pmatrix}\mathcal{H}_{n}&\mathcal{H}_{n}&\mathcal{H}_{n}&\mathcal{H}_{n}\\\
\mathcal{H}_{n}&-\mathcal{H}_{n}&\mathcal{H}_{n}&-\mathcal{H}_{n}\\\
\mathcal{H}_{n}&\mathcal{H}_{n}&-\mathcal{H}_{n}&-\mathcal{H}_{n}\\\
\mathcal{H}_{n}&-\mathcal{H}_{n}&-\mathcal{H}_{n}&\mathcal{H}_{n}\end{pmatrix}\begin{pmatrix}F\\\
FG\\\ FG\\\ -F\end{pmatrix}=\begin{pmatrix}F\\\ FG\\\ FG\\\
-F\end{pmatrix}=\omega^{h}.$

Consider the system

$\displaystyle\omega^{\widetilde{h}}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}\mathcal{H}_{n}&\mathcal{H}_{n}&\mathcal{H}_{n}&\mathcal{H}_{n}\\\
\mathcal{H}_{n}&-\mathcal{H}_{n}&\mathcal{H}_{n}&-\mathcal{H}_{n}\\\
\mathcal{H}_{n}&\mathcal{H}_{n}&-\mathcal{H}_{n}&-\mathcal{H}_{n}\\\
\mathcal{H}_{n}&-\mathcal{H}_{n}&-\mathcal{H}_{n}&\mathcal{H}_{n}\end{pmatrix}\begin{pmatrix}F\\\
FG\\\ FG\\\ -F\end{pmatrix}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}\mathcal{H}_{n}F+\mathcal{H}_{n}\left(FG\right)+\mathcal{H}_{n}\left(FG\right)+\mathcal{H}_{n}F\\\
\mathcal{H}_{n}F-\mathcal{H}_{n}\left(FG\right)+\mathcal{H}_{n}\left(FG\right)-\mathcal{H}_{n}F\\\
\mathcal{H}_{n}F+\mathcal{H}_{n}\left(FG\right)-\mathcal{H}_{n}\left(FG\right)-\mathcal{H}_{n}F\\\
\mathcal{H}_{n}F-\mathcal{H}_{n}\left(FG\right)-\mathcal{H}_{n}\left(FG\right)+\mathcal{H}_{n}F\end{pmatrix}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}\widetilde{F}-\widetilde{F}+2\widetilde{FG}\\\
2\widetilde{F}-\widetilde{FG}+\widetilde{FG}\\\
2\widetilde{F}+\widetilde{FG}-\widetilde{FG}\\\
-2\widetilde{FG}\end{pmatrix}=\begin{pmatrix}\widetilde{FG}\\\
\widetilde{F}\\\ \widetilde{F}\\\ -\widetilde{FG}\end{pmatrix}.$

Writing

$\begin{pmatrix}\widetilde{FG}\\\ \widetilde{F}\\\ \widetilde{F}\\\
-\widetilde{FG}\end{pmatrix}=\begin{pmatrix}F\\\ FG\\\ FG\\\ -F\end{pmatrix},$

we see that $\widetilde{f}=f+g$, or, equivalently, $g=\widetilde{f}+(q-1)f$.

Thus, we have

$h\left(z,y,x\right)=f(x)+\left(z\oplus
y\right)\left[\widetilde{f}(x)+(q-1)f(x)\right]+\frac{q}{2}zy.$

##  7 Symmetries

In paper Feulner  (see also Self-dual ) it was shown that the mapping

$f(x)\longrightarrow f\left(L\left(x\oplus c\right)\right)\oplus\left\langle
c,x\right\rangle\oplus d,$ (8)

where $L\in\mathcal{O}_{n}$, $c\in\mathbb{F}_{2}^{n}$, $\mathrm{wt}(c)$ is
even, $d\in\mathbb{F}_{2}$, preserves self-duality of a bent function. The
group which consists of mappings of such form is called an extended orthogonal
group and denoted by $\overline{\mathcal{O}}_{n}$ Feulner . It is known that
this group is a subgroup of $\mathrm{GL}\left(n+2,\mathbb{F}_{2}\right)$
Feulner .

In paper Kutsenko_CC  known results were generalized within isometric mappings
from the set of all mappings of all Boolean functions in $n\geqslant 4$
variables into itself, which preserve the Hamming distance. Namely it was
proved the group of automorphisms of self-dual Boolean bent functions
coincides with the extended orthogonal group.

In paper Quaternary  it was proved that the mappings of the form

$f(x)\longrightarrow f\left(Lx\right)+d,$

where $L\in\mathcal{O}_{n}$, $d\in\mathbb{Z}_{4}$, preserve self-duality of a
quaternary self-dual gbent function.

In current work we set the form (8) for generalized case. The following result
provides the construction of mappings of such form preserving the (anti-)self-
duality of a Boolean function.

###### Theorem 7.1

The mapping of the set of all generalized Boolean functions in $n$ variables
to itself of the form

$f(x)\longrightarrow f\left(L\left(x\oplus c\right)\right)+\frac{q}{2}\langle
c,x\rangle+d,$

where $L\in\mathcal{O}_{n}$, $c\in\mathbb{F}_{2}^{n}$, $\mathrm{wt}(c)$ is
even, $d\in\mathbb{Z}_{q}$, preserves (anti-)self-duality of a gbent function.

###### Proof

Let $f\in\mathrm{SB}_{q}^{+}(n)\cup\mathrm{SB}_{q}^{-}(n)$ that is
$\widetilde{f}=f+\frac{q}{2}\varepsilon$ for some
$\varepsilon\in\mathbb{F}_{2}$. Consider a function
$g(x)=f\left(L\left(x\oplus c\right)\right)+\frac{q}{2}\langle c,x\rangle+d$,
where $L\in\mathcal{O}_{n}$, $c\in\mathbb{F}_{2}^{n}$, $\mathrm{wt}(c)$ is
even, $d\in\mathbb{Z}_{q}$. Its generalized Walsh–Hadamard transform is

$\displaystyle H_{g}(y)$
$\displaystyle=\sum\limits_{x\in\mathbb{F}_{2}^{n}}{\omega^{g(x)}(-1)^{\left\langle
x,y\right\rangle}}=\sum\limits_{x\in\mathbb{F}_{2}^{n}}{\omega^{f\left(L\left(x\oplus
c\right)\right)+\frac{q}{2}\left\langle
c,x\right\rangle+d+\frac{q}{2}\left\langle x,y\right\rangle}}$
$\displaystyle=\omega^{d}\sum\limits_{x\in\mathbb{F}_{2}^{n}}{\omega^{\frac{q}{2}\left\langle
x,y\oplus c\right\rangle+f\left(L\left(x\oplus
c\right)\right)}}=\omega^{d}\sum\limits_{z\in\mathbb{F}_{2}^{n}}{\omega^{\frac{q}{2}\left\langle
L^{-1}z\oplus c,y\oplus c\right\rangle+f\left(z\right)}}$
$\displaystyle=\omega^{d+\frac{q}{2}\left\langle
c,y\right\rangle+\frac{q}{2}\left\langle
c,c\right\rangle}\sum\limits_{z\in\mathbb{F}_{2}^{n}}{\omega^{\frac{q}{2}\left\langle
z,L\left(y\oplus c\right)\right\rangle+f\left(z\right)}}$
$\displaystyle=\omega^{d+\frac{q}{2}\left\langle
c,y\right\rangle}2^{n/2}\omega^{\widetilde{f}\left(L\left(y\oplus
c\right)\right)}=2^{n/2}\omega^{f\left(L\left(y\oplus
c\right)\right)+\frac{q}{2}\langle c,y\rangle+d+\frac{q}{2}\varepsilon}$
$\displaystyle=2^{n/2}\omega^{g(y)+\frac{q}{2}\varepsilon}=2^{n/2}\omega^{\widetilde{g}(y)},$

hence $\widetilde{g}(y)=g(y)+\frac{q}{2}\varepsilon$ for any
$y\in\mathbb{F}_{2}^{n}$.

By using the mappings of this form we can clarify, for instance, the
classification of quaternary self-dual bent functions in $4$ variables given
in Quaternary  and formed by $8$ classes. Namely, the representatives with
vectors of values $(0330302132010110)$ and $(3123231322030300)$ from the
classes $4$ and $5$ respectively are related by the transformation

$f(x)\longrightarrow f(L(x\oplus c))+\frac{q}{2}\langle c,x\rangle+d,$

where

$L=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{pmatrix},\ \
c=(1001),\ \ d=3.$

The representatives with vectors of values $(2022220222020200)$ and
$(2123230332121210)$ from the classes $2$ and $7$ respectively are related by
the transformation

$f(x)\longrightarrow f(L(x\oplus c))+\frac{q}{2}\langle c,x\rangle+d,$

where

$L=\begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{pmatrix},\ \
c=(0101),\ \ d=1.$

Thus, the classification of quaternary self-dual bent functions in $4$
variables is given in the Table 1.

By a slight change of the parameters mentioned in Theorem 7.1, it is possible
to obtain the class of mapping that define a bijection between self-dual and
anti-self-dual gbent functions in $n$ variables.

Table 1: Classification of quaternary self-dual bent functions in $4$ variables Representative from equivalence class | Size
---|---
$0220202022000000$ | $24$
$2022220222020200$ | $64$
$0330313133110110$ | $48$
$0330302132010110$ | $120$
$1321213122010100$ | $96$
$0220213023100000$ | $48$
Number of quaternary self-dual bent functions in four variables | $400$

###### Proposition 7

The mapping of the set of all generalized Boolean functions in $n$ variables
to itself of the form

$f(x)\longrightarrow f(\pi(x))+g(x),$

with

$\pi(x)=L\left(x\oplus c\right),\ \ g(x)=\frac{q}{2}\langle c,x\rangle+d,\ \
x\in\mathbb{F}_{2}^{n},$

where $L\in\mathcal{O}_{n}$, $c\in\mathbb{F}_{2}^{n}$, $\mathrm{wt}(c)$ is
odd, $d\in\mathbb{Z}_{q}$, is a bijection between the sets
$\mathrm{SB}_{q}^{+}(n)$ and $\mathrm{SB}_{q}^{-}(n)$.

###### Proof

Let $f\in\mathrm{SB}_{q}^{+}(n)\cup\mathrm{SB}_{q}^{-}(n)$ and
$\widetilde{f}=f+\frac{q}{2}\varepsilon$ for some
$\varepsilon\in\mathbb{F}_{2}$. One can show that for
$g(x)=f\left(L\left(x\oplus c\right)\right)+\frac{q}{2}\langle c,x\rangle+d$,
where $L\in\mathcal{O}_{n}$, $c\in\mathbb{F}_{2}^{n}$, $\mathrm{wt}(c)$ is
odd, $d\in\mathbb{Z}_{q}$, it holds
$H_{g}(y)=2^{n/2}\omega^{\widetilde{g}(y)+q/2}$, $y\in\mathbb{F}_{2}^{n}$.

From the existence of such bijections it follows that the cardinalities of the
sets of self-dual and anti-self-dual gbent functions coincide.

###### Corollary 3

It holds
$\left\lvert\mathrm{SB}_{q}^{+}(n)\right\rvert=\left\lvert\mathrm{SB}_{q}^{-}(n)\right\rvert$.

##  8 Conclusion

In current paper self-dual generalized bent functions were explored. A group
of primary and secondary constructions was presented. The general form of
self-dual Maiorana–McFarland gbent functions and their metrical properties
were studied. The non-existence of affine self-dual gbent functions was shown.
We also gave the description of self-dual gbent functions symmetric with
respect to two variables. The properties of sign functions of self-dual gbent
functions were considered.

It is interesting to find other symmetries, if any, distinct from the ones
that were found in this work. It involves the study of the automorphisms
gropus of the considered gbent functions with respect to Hamming or Lee
metrics. The study of connection with self-dual Boolean functions also seems
to be a promising task.

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