#  Butson Hadamard matrices, bent sequences, and spherical codes

Minjia Shi, Danni Lu, Andrés Armario , Ronan Egan , Ferruh Ozbudak , Patrick
Solé Minjia Shi and Danni Lu are with the Key Laboratory of Intelligent
Computing Signal Processing, Ministry of Education, School of Mathematical
Sciences, Anhui University, Hefei 230601, China; State Key Laboratory of
integrated Service Networks, Xidian University, Xi’an, 710071, China.
<EMAIL_ADDRESS><EMAIL_ADDRESS>de Matemática Aplicada I,
Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012, Seville, Spain.
<EMAIL_ADDRESS>This research was supported by the Strategic R+D Project
TED2021-130566B-I00 from the Ministry of Science and Innovation of the
Government of Spain.School of Mathematical Sciences, Dublin City University,
Ireland<EMAIL_ADDRESS>Özbudak is with Faculty of Engineering and
Natural Sciences, Sabancı University, 34956, Istanbul, E-mail:
<EMAIL_ADDRESS>Patrick Solé is with I2M (Aix Marseille Univ,
CNRS, Centrale Marseille), Marseilles, France<EMAIL_ADDRESS>
###### Abstract

We explore a notion of bent sequence attached to the data consisting of an
Hadamard matrix of order $n$ defined over the complex $q^{th}$ roots of unity,
an eigenvalue of that matrix, and a Galois automorphism from the cyclotomic
field of order $q.$ In particular we construct self-dual bent sequences for
various $q\leq 60$ and lengths $n\leq 21.$ Computational construction methods
comprise the resolution of polynomial systems by Groebner bases and eigenspace
computations. Infinite families can be constructed from regular Hadamard
matrices, Bush-type Hadamard matrices, and generalized Boolean bent functions.
As an application, we estimate the covering radius of the code attached to
that matrix over $\mathbb{Z}_{q}.$ We derive a lower bound on that quantity
for the Chinese Euclidean metric when bent sequences exist. We give the
Euclidean distance spectrum, and bound above the covering radius of an
attached spherical code, depending on its strength as a spherical design.

Keywords: Hadamard matrices, Butson matrices, covering radius, bent sequences,
spherical codes
MSC (2020): Primary 05 B20, Secondary 05E99

##  1 Introduction

Bent sequences are important combinatorial objects that are relevant to
difference sets, Boolean functions, strongly regular graphs, and symmetric
cryptography [21]. They are classically defined in relation to the Sylvester
matrix, the matrix of the Walsh-Hadamard transform. Recently a new notion of
bent sequences, motivated by a cryptographic problem (PUFs=Physically
Unclonable Functions), appeared, that is defined for any Hadamard matrix [28].
Constructions and properties of these sequences in the self-dual case were
investigated in [27]. The advantage of self-dual bent sequences over
unrestricted bent sequences is that they allow linear algebra to bear on the
problem. Note that every bent sequence is self-dual for some matrix [26]. The
extension of this notion to complex Hadamard matrix in the sense of Turyn,
that is to say with entries in $\Omega_{4}=\\{\pm 1,\pm i\\}$, was done in
[26]. A self-dual bent sequence $X$ attached to a matrix $H$ is then an
eigenvector of $H$ for an eigenvalue $\lambda$ of $H$ (some Gaussian integer)
that have its entries in $\Omega_{4}.$ Thus $HX=\lambda X.$ A new twist in the
definition of self-dual bent sequences was given in [29] were it is suggested
to replace the above equation by $HX=\lambda\overline{X},$ where the bar
denotes complex conjugation. The motivation for that change was to obtain more
bent sequences, in particular in relation with generalized bent functions.

In the present paper, we extend the notion of self-dual bent sequences in two
directions. First, we consider Butson Hadamard matrices, that is to say
Hadamard matrices with values in the complex roots of unity of a given order.
Next, we replace the notion of conjugate by that of multiplier, that is a
bijective map of the form $z\mapsto z^{k}$ with $(k,q)=1$ which preserves
globally the complex $q^{th}$ roots of unity. When $k=1$ this new definition
reduces to that of [26, 27]. When $k=q-1$, we recover the definition of [29].
This change of definition affects the notion of a strong group, a matrix group
that preserves the set of self-dual bent sequences. We extend the
computational techniques (linear algebra and Groebner bases) from [26, 27] to
this new setting. This allows us to obtain many examples of self-dual bent
sequences from the matrices of the database [5]. General constructions of
Butson Hadamard matrices with designed self-dual bent sequences are then
derived from, respectively, Kronecker products of Butson matrices, regular
Butson matrices, Bush type Hadamard matrices, and generalized Walsh Hadamard
matrices. In the last case we use group invariant Butson matrices in the sense
of [8, 22]. Strong necessary existence conditions for Butson matrices
admitting self-dual bent sequences with trivial multiplier are given. In a
second part of the article, we turn our attention towards Butson-Hadamard
codes, and their metric properties with respect to covering and packing either
a finite set of strings or the Euclidean sphere. While the packing properties
of the code with respect to the homogeneous and other metrics have been
investigated by various authors [1, 13], we focus on the Chinese Euclidean
metric and the covering properties in connection with bent sequences. The
Chinese Euclidean metric, which is first mentioned in [6], should not be
confused with that in [18]. We also associate with every such code of length
$n$ a spherical code in dimension $2n.$ This code is shown to be optimal for
the minimum distance in the case of $q=4,$ as meeting the Levenshtein bound.
Pending on some properties of the Hadamard matrix, we can obtain spherical
designs of strength one or two. Spherical designs are finite sets of points on
the unit Euclidean sphere which occur naturally in relation to the “cubature
formulae” of numerical analysis, and can be thought of as analogues of
combinatorial designs [7]. In [25], the existence of spherical designs is
established, providing a solid foundation for their theoretical exploration.
Moreover, the utilization of spherical designs in motivating the definition of
conformal $t$-designs is demonstrated in [14]. Building spherical designs from
Butson Hadamard matrices seems to be a new approach. It enters here as a way
to use upper bounds on the covering radius of spherical designs [11, 30] to
control the covering radius of the said spherical code.

The material is arranged as follows. Section 2 collects the definitions needed
for the rest of the paper. Section 3 defines and constructs the strong
automorphism group of a Butson matrix with a given multiplier. Section 4
studies computational methods and results. Section 5 gives four general
constructions of Butson Hadamard matrices with designed self-dual bent
sequences. Section 6 derives a necessary condition for the existence of Butson
matrices admitting a self-dual bent sequence. Section 7 derives the Euclidean
distance spectrum of Butson Hadamard codes. Section 8 contains the lower and
upper bounds on the covering radius of Butson Hadamard codes and attending
spherical codes. Section 9 concludes the article.

##  2 Preliminaries

######  Definition 2.1.

A complex Hadamard matrix of order $n$ is an $n\times n$ matrix $H$ with
entries in the complex unit satisfying the equation $HH^{*}=nI,$ where the *
denotes the transpose conjugate. If all of its elements are in
$\Omega_{q}=\left\\{z\in C\mid z^{q}=1\right\\},$ for some integer $q,$ then
$H$ is said to be Butson type, and we write $H\in BH(n,q).$ A matrix $H\in
BH(n,q)$ is in dephased form when both first row and first columns only
contain ones.

######  Definition 2.2.

Two Hadamard matrices $H$ and $K$ of $BH(n,q)$ are called equivalent if there
are two permutation matrices $P_{1}$ and $P_{2}$ and two diagonal matrices
$D_{1}$ and $D_{2}$ with entries in $\Omega_{q}$ such that

$H=P_{1}D_{1}KP_{2}D_{2}.$

Every matrix of $BH(n,q)$ is equivalent to its dephased form.

######  Definition 2.3.

Given $H\in BH(n,q)$ the sequence $X\in\Omega_{q}^{n}$ is a bent sequence if
there is $Y\in\Omega_{q}^{n}$ such that $HX=\lambda Y$ for some
$\lambda\in\mathbb{Z}[\zeta_{q}].$

######  Definition 2.4.

For any $k$ coprime with $q$ we define the multiplier
$\mu_{k}:\Omega_{q}\longrightarrow\Omega_{q}$ by the rule $\mu_{k}(z)=z^{k}.$
Now a self-dual bent sequence attached to $H\in BH(n,q)$ is defined as
$X\in\Omega_{q}^{n}$ such that

$HX=\lambda\mu_{k}(X),$

where $\lambda$ is an element of $\mathbb{Z}[\zeta_{q}].$ (When $k=1$ it is
also an eigenvalue of $H$). When $k=1,$ this is the definition of self-dual
bent sequence in [26]. When $k=q-1,$ this is the definition of self-dual bent
sequence in [29].

######  Remark 2.1.

  1. (1)

Any multiplier $\mu_{k}$ can be extended into a Galois automorphism of the
cyclotomic field of index $q.$ Writing an element of that field as
$a=\sum\limits_{i=0}^{q-1}a_{i}\zeta^{i}$ for rationals $a_{i}$’s, we define
$\mu_{k}(a)=\sum\limits_{i=0}^{q-1}a_{i}\zeta^{ki}.$ In fact all Galois
automorphisms are of that form [16]. They thus satisfy the following three
properties.

    1. (i)

If $x\in\mathbb{Q},$ then $\mu_{k}(x)=x.$

    2. (ii)

$\forall x,y\in\mathbb{Q}(\zeta_{q}),\,\mu_{k}(x+y)=\mu_{k}(x)+\mu_{k}(y).$

    3. (iii)

$\forall x,y\in\mathbb{Q}(\zeta_{q}),\,\mu_{k}(xy)=\mu_{k}(x)\mu_{k}(y).$

  2. (2)

If $X$ is a bent sequence for $H\in BH(n,q)$ with $HX=\lambda Y,$ then $Y$ is
a self-dual bent sequence with trivial multiplier for the matrix $HD,$ where
$D$ is a diagonal matrix such that $X=DY.$ Note that the diagonal of $D$ lies
in $\Omega_{q},$ which implies that $HD\in BH(n,q).$

######  Definition 2.5.

The Chinese Euclidean weight $w_{CE}(x)$ of a vector $x\in\mathbb{Z}_{q}^{n}$
is

$\sum_{i=1}^{n}\left[2-2\cos(\frac{2\pi x_{i}}{q})\right],$

and the Chinese Euclidean distance between the codewords $u$ and
$v\in\mathbb{Z}_{q}^{n}$ is defined as

$d_{CE}(u,v)=w_{CE}(u-v).$

######  Definition 2.6.

A Butson matrix $H\in BH(n,q)$ is conveniently represented in logarithmic
form, that is, the matrix
$H=\left[\zeta_{q}^{\varphi_{i,j}}\right]_{i,j=1}^{n}$ is represented by the
matrix $L(H)=\left[\varphi_{i,j}\bmod q\right]_{i,j=1}^{n}$ with the
convention that $L_{i,j}\in\mathbb{Z}_{q}$ for all $i,j\in\\{1,2,\dots,n\\}.$
Given $H\in BH(n,q),$ we denote by $F_{H}$ the $\mathbb{Z}_{q}$-code of length
$n$ consisting of the rows of $L(H),$ and by
$C_{H}=\cup_{\alpha\in\mathbb{Z}_{q}}(F_{H}+\alpha\textbf{1})$ where 1 denotes
the all-one vector. The code $C_{H}\subseteq\mathbb{Z}_{q}^{n}$ is called a
Butson Hadamard code. Any subset of $\Omega_{q}^{n}$ for some integer $n$ is
called a polyphase code, following [10].

######  Definition 2.7.

Let $C_{H}$ be a Butson Hadamard code of length $n$ over $\mathbb{Z}_{q}$
coming from $H\in BH(n,q)$. The covering radius of $C_{H}$ with Chinese
Euclidean distance $d_{CE}$ is defined as:

$r_{CE}(C_{H})=\max\\{d_{CE}(x,C_{H})]\mid
x\in\mathbb{Z}_{q}^{n}\\}=\max\limits_{x\in\mathbb{Z}_{q}^{n}}\min\limits_{y\in
C_{H}}d_{CE}(x,y).$

##  3 Automorphism group

Denote the set of $n\times n$ monomial matrices with nonzero entries in
$\Omega_{q}$ by $M(n,q)$. The set $M(n,q)$ is a group generated by the set
$P_{n}$ of all permutation matrices of order $n$, and $\Delta(n,q)$, the group
of all diagonal matrices of order $n$ with entries in $\Omega_{q}$. The group
of all pairs of matrices from $M(n,q)$ acts on the set $\mathrm{BH}(n,q)$. The
action is given by

$(P,Q)H=PHQ^{\ast}$

for all $H\in\mathrm{BH}(n,q)$. The orbit of $H$ under this action is the
equivalence class of $H$. The stabilizer of $H$ under this action, i.e., the
set of all $(P,Q)\in M(n,q)^{2}$ such that

$PHQ^{\ast}=H$

is called the automorphism group of $H$, and is denoted $Aut(H)$. The subgroup
of $Aut(H)$ comprised of pairs of matrices in $P_{n}$ is called the
permutation automorphism group of $H$, denoted $PAut(H)$.

We define the strong automorphism group with trivial multiplier of $H$ to be
the subgroup $SAut(H,1)\leq Aut(H)$ comprised of pairs of the form $(P,P)$. It
follows from the definition of an automorphism that $(P,P)\in SAut(H,1)$ if
and only if $PH=HP$. Hence $SAut(H,1)$ is in bijection with the centralizer of
$H$ in $M(n,q)$.

Calculating the automorphism groups of $H\in\mathrm{BH}(n,q)$ is an onerous
task in general. The most efficient approaches all involve a similar idea -
constructing an appropriate larger $(0,1)$-matrix and using graph theoretic
tools such as nauty [20] which are highly efficient. In the following section
we condense the relevant details on computing the automorphism group of a
pairwise combinatorial design given in [17, Chapter 9] down for just the
purpose of handling a Butson Hadamard matrix.

###  3.1 The expanded and associated designs

######  Definition 3.1.

Let $H\in\mathrm{BH}(n,q)$. The expanded design $\mathcal{E}_{H}$ of $H$ is
the $nq\times nq$ matrix of the form

$\mathcal{E}_{H}=[\zeta_{q}^{i}H\zeta_{q}^{j}]_{0\leq i,j\leq q-1}.$

The associated design $\mathrm{A}_{H}$ of $H$ is obtained from the expanded
design by replacing each of its non-identity entries with $0$.

######  Example 3.1.

Let $\zeta=\zeta_{3}$. If

$H={\small\left[\begin{array}[]{rrr}1&1&1\\\ 1&\zeta&\zeta^{2}\\\
1&\zeta^{2}&\zeta\end{array}\right]}$

then $\mathcal{E}_{H}$ and $\mathrm{A}_{H}$ are

${\footnotesize\left[\begin{array}[]{ccccccccc}1&1&1&\zeta&\zeta&\zeta&\zeta^{2}&\zeta^{2}&\zeta^{2}\\\
1&\zeta&\zeta^{2}&\zeta&\zeta^{2}&1&\zeta^{2}&1&\zeta\\\
1&\zeta^{2}&\zeta&\zeta&1&\zeta^{2}&\zeta^{2}&\zeta&1\\\
\zeta&\zeta&\zeta&\zeta^{2}&\zeta^{2}&\zeta^{2}&1&1&1\\\
\zeta&\zeta^{2}&1&\zeta^{2}&1&\zeta&1&\zeta&\zeta^{2}\\\
\zeta&1&\zeta^{2}&\zeta^{2}&\zeta&1&1&\zeta^{2}&\zeta\\\
\zeta^{2}&\zeta^{2}&\zeta^{2}&1&1&1&\zeta&\zeta&\zeta\\\
\zeta^{2}&1&\zeta&1&\zeta&\zeta^{2}&\zeta&\zeta^{2}&1\\\
\zeta^{2}&\zeta&1&1&\zeta^{2}&\zeta&\zeta&1&\zeta^{2}\\\
\end{array}\right]}~{}\textit{and}~{}{\footnotesize\left[\begin{array}[]{ccccccccc}1&1&1&0&0&0&0&0&0\\\
1&0&0&0&0&1&0&1&0\\\ 1&0&0&0&1&0&0&0&1\\\ 0&0&0&0&0&0&1&1&1\\\
0&0&1&0&1&0&1&0&0\\\ 0&1&0&0&0&1&1&0&0\\\ 0&0&0&1&1&1&0&0&0\\\
0&1&0&1&0&0&0&0&1\\\ 0&0&1&1&0&0&0&1&0\\\ \end{array}\right]}$

respectively.

The following is a special case of [17, Theorem 9.6.12].

######  Theorem 3.1.

$Aut(H)\cong PAut(\mathcal{E}_{H})$.

The isomorphism of Theorem 3.1 is described in detail in [17, Section 9.6], we
summarize the main points here. For any $X\in M(n,q)$ there are unique
disjoint $(0,1)$-matrices $X_{\omega}$ such that
$X=\sum_{\omega\in\Omega_{q}}\omega{X_{\omega}}$. Let

$S_{\omega}=[\delta_{\omega\beta}^{\alpha}]_{\alpha,\beta\in\Omega_{q}}\quad\text{
and }\quad
T_{\omega}=[\delta_{\beta}^{\alpha\omega}]_{\alpha,\beta\in\Omega_{q}}.$ (1)

Let $\theta^{(1)}(X)=\sum_{\omega\in\Omega_{q}}T_{\omega}\otimes X_{\omega}$
and $\theta^{(2)}(X)=\sum_{\omega\in\Omega_{q}}S_{\omega}\otimes X_{\omega}$.
Define the map $\Theta:M(n,q)^{2}\rightarrow P_{nq}^{2}$ by
$\Theta:(X,Y)\mapsto(\theta^{(1)}(X),\theta^{(2)}(Y))$. Let
$Z_{q}=\\{(S_{\omega}\otimes I_{n},T_{\mu}\otimes
I_{n})\,\mid\,\omega,\mu\in\Omega_{q}\\}$. Then by [17, Theorem 9.6.7],
$\Theta(M(n,q)^{2})=C_{P_{nq}\times P_{nq}}(Z_{q})$, the centralizer of
$Z_{q}$ in $P_{nq}\times P_{nq}$. The restriction $\Theta:Aut(H)\rightarrow
PAut(\mathcal{E}_{H})$ is an isomorphism as in Theorem 3.1. This leads to the
following useful result.

######  Theorem 3.2 (Theorem 9.8.5, [17]).

$\Theta(Aut(H))=PAut(A_{H})\cap C_{P_{nq}\times P_{nq}}(Z_{q})$.

Treating $A_{H}$ as the incidence structure of points and blocks, built in
Magma [4] functions construct the automorphism group $PAut(A_{H})$.

###  3.2 Automorphisms of digraphs

Inspired by [26, §3.1], we attach a digraph $G(H)$ on $2qn$ vertices.

  1. (1)

For each row $t$ of $H$, there are $q$ row vertices $r(t,x)$ where
$x\in\Omega_{q}$, and $q$ arcs $(r(t,x),r(t,\zeta_{q}x))$ for each
$x\in\Omega_{q}$.

  2. (2)

For each column $s$ of $H$, there are $q$ column vertices $c(s,x)$ where
$x\in\Omega_{q}$, and $q$ arcs $(c(s,x),c(s,\zeta_{q}x))$ for each
$x\in\Omega_{q}$.

  3. (3)

The entry $H_{ts}$ yields $q$ arcs $(r(t,x),c(s,H_{ts}x))$ for each
$x\in\Omega_{q}$.

The analogue of [26, Corollary 1] is stated below. The proof is given in the
proof of Proposition 3.3.

######  Proposition 3.1.

The automorphism group $Aut(H)$ is isomorphic to the automorphism group of
$G(H).$

We now consider the strong automorphism group of $C.$ Let $k$ be a fixed
integer coprime to $q$. Define the strong group with multiplier $\mu_{k}$, to
be

$SAut(H,k)=\\{M\in M(n,q)\mid\mu_{k}(M)H=HM\\}.$

######  Proposition 3.2.

The set of self-dual bent sequences for $H$ with multiplier $\mu_{k}$ is
preserved globally under the action of $SAut(H,k).$

###### Proof.

Let $X$ be such a sequence. Thus $HX=\lambda\mu_{k}(X)$ for some
$\lambda\in\mathbb{Z}[\zeta_{q}].$ Multiplying both sides by $\mu_{k}(M)$ we
see by using the relation $\mu_{k}(M)H=HM$ that $Y=MX$ is also a self-dual
bent sequence for $H$ with multiplier $\mu_{k}.$ Indeed

$\mu_{k}(M)HX=HMX=\lambda\mu_{k}(M)\mu_{k}(X)=\lambda\mu_{k}(MX).$

Observe that writing $\mu_{k}(M)\mu_{k}(X)=\mu_{k}(MX)$ uses the fact that
$\mu_{k}$ can be extended to an element of the Galois group of
$\mathbb{Q}(\zeta_{q})/\mathbb{Q}.$ ∎

A graph-theoretic algorithm to construct $SAut(H,k)$ in the case $k=1$ can be
immediately extended from the $BH(n,4)$ case in [26]. Define a modified
digraph $G^{\prime}(H)$ as follows. We connect every pair $r(s,x)$ and
$c(s,x)$ for $s=1,\dots,n$ and $x$ ranging over $\Omega_{q}$ by a path a
length two via an extra vertex $I_{s,x}$ say. The following proposition is an
analogue of [26, Corollary 2]. Moreover, the similar result hold for any $k$
with $1\leq k\leq q-1$ by mean of the digraph $G^{k}(H)=(V^{k},E^{k})$ where
$V^{k}=V\cup\\{I(s,x)\,\colon\,1\leq s\leq n,x\in\Omega_{q}\\}$ and
$E^{k}=E\cup\\{\big{(}r(s,y),I(s,x)\big{)},\,\big{(}I(s,x),c(s,x)\big{)}\colon\,1\leq
s\leq n,x\in\Omega_{q},y=x^{k}\\}$.

######  Proposition 3.3.

If $k=1,$ then the strong automorphism group of $H$ is isomorphic to the
automorphism group of $G^{\prime}(H).$ In general, $SAut(H,k)$ is isomorphic
to the automorphism group of $G^{k}(H)$ for $1\leq k\leq q-1$.

###### Proof.

Given $H\in BH(n,q)$, $G(H)=(V,E)$ denotes its associated digraph. If $f\colon
V\rightarrow V$ is an automorphism of $G(H)$, by definition $f$ is a
$1$-to-$1$ map such that $(f(v_{1}),f(v_{2}))\in E$ iff $(v_{1},v_{2})\in E$.
Looking at the out/in-degree of the vertices, then $f$ has to stabilize the
set of row/column vertices, respectively. Furthermore, $f$ maps cycles into
cycles. Hence, the general form of $f$ is

$\begin{array}[]{c}f(r(t,x))=r(t^{\prime},y_{t}\cdot x)\\\
f(c(s,x))=c(s^{\prime},z_{s}\cdot x)\end{array}$

with $y_{t},z_{s}\in\Omega_{q}$ satisfying that $(f(v_{1}),f(v_{2}))\in E$ iff
$(v_{1},v_{2})\in E$. Thus, for every automorphism $f$ we can define two
matrices $P,Q\in M(n,q)$, where $P=P_{1}D_{1}$ with $P_{1}$ the permutation
matrix defined by $\pi_{r}(t)=t^{\prime}$ and $D_{1}$ is the diagonal matrix
with $t$-th entry equal to $\overline{y}_{t}$ (where $\overline{y}$ denotes
the complex conjugation of $y$). Similarly, $Q=P_{2}D_{2}$ with $P_{2}$ the
permutation matrix defined by $\pi_{c}(s)=s^{\prime}$ and $D_{2}$ is the
diagonal matrix with $s$-th entry equals to $\bar{z}_{s}$. Since
$G(H)=G(PHQ^{*})$, it follows that $H=PHQ^{*}$. This proves Proposition 3.1.

For $SAut(H,1)$ we consider the digraph
$G^{\prime}(H)=(V^{\prime},E^{\prime})$ defined above. If $f\colon
V^{\prime}\rightarrow V^{\prime}$ an authomorphism of $G^{\prime}(H)$ then, by
an argument on the out/in-degree of the vertices, $f$ also stabilizes the set
of vertices $I(s,x)$. That is,

$f(I(s,x))=I(s^{\prime},w_{s,x}\cdot x)$

taking into account that $(f(r(s,x)),f(I(s,x)))$ and $(f(I(s,x)),f(c(s,x)))$
belong to $E^{\prime}$, then we have $\pi_{r}(s)=\pi_{c}(s)$ and
$y_{s}=w_{s,x}=z_{s}$. Therefore, $P=Q$ where $(P,Q)$ are the monomial
matrices associated to $f$. This completes the proof of the first part.

In general, for $SAut(H,k)$ we consider the digraph $G^{k}(H)=(V^{k},E^{k})$.
By a similar argument as for $k=1$, we conclude that $P=\mu_{k}(Q)$ where
$PHQ^{*}=H$. As a consequence, $SAut(H,k)$ is isomorphic to the automorphism
group of $G^{k}(H).$ ∎

The automorphism group of $G^{\prime}(H)$ can then be computed using standard
graph theoretic programs like, e.g. Nauty [20].

##  4 Computational techniques

In this section we extend the computational construction methods of self-dual
bent sequences of [26, 27] to Butson matrices.

###  4.1 Groebner bases

The system $HX=\lambda\mu_{k}(X)$ with $X\in\Omega_{q}^{n}$ can be solved
using Groebner bases, as it is equivalent to a polynomial system, which,
instead of being quadratic as in [26] is now of degree $q.$ More concretely,
we can consider the following steps.

  1. (1)

Construct the ring $P$ of polynomial functions in $n$ variables
$X_{i},i=1,\dots,n.$

  2. (2)

Construct the linear constraints $HX=\lambda\mu_{k}(X).$

  3. (3)

Construct the constraints $\forall i\in\\{1,2,\dots,n\\},X_{i}^{q}=1.$

  4. (4)

Compute a Groebner basis for the ideal $I$ of $P$ determined by constraints
(ii) and (iii).

  5. (5)

Compute the solutions as the zeros determined by $I$.

Complexity: As is well-known, the complexity of computing Groebner bases can
be doubly exponential in the number of variables [2]. However, for the system
at hand, it might at most be simply exponential [12].

###  4.2 Linear algebra

Consider the system $HX=\lambda\mu_{k}(X).$ Denote by $t$ the multiplicative
order of $k$ modulo $q,$ implying $\mu_{k}^{t}=1.$ We can then obtain, by
repeated application of $\mu_{k}$ the equations

$\mu_{k}^{i}(H)\mu_{k}^{i}(X)=\mu_{k}^{i}(\lambda)\mu_{k}^{i+1}(X),i=1,2,\dots,t-1.$

By successive substitution, we get

$\prod_{i=0}^{t-1}\mu_{k}^{t-1-i}(H)X=\prod_{i=0}^{t-1}\mu_{k}^{t-1-i}(\lambda)X.$

From this equation, we can give an algorithm for the construction of self-dual
bent sequences.

  1. (1)

Construct $H\in BH(n,q).$

  2. (2)

Compute $\prod_{i=0}^{t-1}\mu_{k}^{t-1-i}(H)$ and
$\prod_{i=0}^{t-1}\mu_{k}^{t-1-i}(\lambda).$

  3. (3)

Compute a basis of the eigenspace associated to the eigenvalue
$\prod_{i=0}^{t-1}\mu_{k}^{t-1-i}(\lambda)$ of
$\prod_{i=0}^{t-1}\mu_{k}^{t-1-i}(H).$

  4. (4)

Let $B$ denote a matrix with rows such a basis of size $l\leq n.$ Pick $B_{l}$
a $l$-by-$l$ submatrix of $B$ that is invertible, by the algorithm given
below.

  5. (5)

For all $Z\in\Omega_{q}^{l}$ solve the system in $C$ given by $Z=CB_{l}.$

  6. (6)

Compute the remaining $n-l$ entries of $CB.$

  7. (7)

If these entries are in $\Omega_{q}$ declare $CB$ a self-dual bent sequence
attached to $\prod_{i=0}^{t-1}\mu_{k}^{t-1-i}(H).$

  8. (8)

Sieve out the self-dual bent sequence that satisfies $HX=\lambda\mu_{k}(X).$

To construct $B_{l}$ we apply a greedy algorithm. We construct the list $J$ of
the indices of the columns of $B_{l}$ as follows.

  1. (i)

Initialize $J$ at $J=[1].$

  2. (ii)

Given a column of index ${l}$ we compute the ranks $r$ and $r^{\prime}$ of the
submatrices of $B$ with $l$ rows and columns defined by the respective lists
$J$ and $J^{\prime}=Append(J,{l}).$

  3. (iii)

If $r<r^{\prime},$ then update $J:=J^{\prime}.$

  4. (iv)

Repeat until $\left|J\right|=rank(B).$

######  Remark 4.1.

If the first column of $B$ is zero, step (i) does not make sense, but then
there is no self-dual bent sequence in that situation, as all eigenvectors
have first coordinate zero.

Complexity: If we estimate at $O(n^{3})$ the complexity of computation of an
eigenspace basis, then the total complexity is $O(n^{3}+nkq^{k})$ where $k$ is
the dimension of the relevant eigenspace.

###  4.3 Numerical results

In Table 1, we apply the two previous construction techniques to the matrices
of database [5]. Each row corresponding to two values of $(n,q).$ Blanks mean
that we cannot conclude, due to the computational burden. Explicit self-dual
bent sequences can be seen at the bottom line of
http://math.ahu.edu.cn/smj/list.htm. In Table 2 we recall the distribution of
matrices per classes of $BH(n,q)$ in the said database.

Table 1: Statistics of self-dual bent sequences of Butson matrices $n$ | $q$ | #$\lambda$ | #$X$ of per $\lambda$ | $n$ | $q$ | #$\lambda$ | #$X$ of per $\lambda$
---|---|---|---|---|---|---|---
2 | 2 | 0 | 0 | 3 | 3 | 1 | 1
4 | 2 | 2 | 2; 2 | 4 | 4 | 3 | 8; 8; 4
5 | 5 | 2 | 6; 10 | 6 | 3 | 0 | 0
6 | 4 | 0 | 0 | 6 | 6 | 0 | 0
7 | 6 | 5 | 6; 4; 6; 0; 6 | 7 | 7 | 2 | 9; 3
8 | 2 | 0 | 0 | 8 | 4 | 0 | 0
8 | 4 | 0 | 0 | 8 | 8 | 4 | 16; 16; 16; 16
8 | 20 | 0 | 0 | 8 | 20 | 0 | 0
9 | 3 | 2 | 22; 12 | 9 | 6 | 2 | 12; 4
9 | 9 | 2 | 48; 0 | 9 | 10 | 2 | 2; 0
10 | 4 | 2 | 12; 12 | 10 | 5 | 0 | 0
10 | 5 | 0 | 0 | 10 | 6 | 0 | 0
10 | 6 | 0 | 0 | 10 | 10 | 0 | 0
11 | 11 | 2 | 15; 5 | 12 | 2 | 0 | 0
12 | 3 | 2 | 15; 1 | 12 | 4 | 0 | 0
12 | 4 | 0 | 0 | 12 | 6 | 2 | 96; 96
12 | 6 | 0 | 0 | 12 | 12 | 4 | 72; 96; 72;108
12 | 36 | 3 |  | 13 | 6 | 0 | 0
13 | 6 | 0 | 0 | 13 | 6 | 0 | 0
13 | 13 | 2 |  | 13 | 60 | 5 |
14 | 4 | 0 | 0 | 14 | 4 | 0 | 0
14 | 6 | 0 | 0 | 14 | 7 | 0 | 0
14 | 10 | 0 | 0 | 14 | 14 | 0 | 0
15 | 15 | 2 |  | 16 | 2 | 2 | 20; 20
16 | 2 | 2 | 8; 4 | 16 | 2 | 2 | 0
16 | 2 | 2 | 0 | 16 | 2 | 2 | 0
16 | 4 | 4 | 240; 240; 268; 180 | 16 | 4 | 2 | 0
16 | 8 | 4 |  | 16 | 8 | 2 |
16 | 16 | 4 |  | 21 | 3 | 0 | 0

Table 2: Number of matrices in $BH(n,q)$ listed in the database [5] # | $BH(n,q)$
---|---
1 |  $BH(2,2),BH(3,3),BH(4,2),BH(4,4),BH(5,5),BH(6,3),BH(6,4),BH(6,6),BH(7,6),$ $BH(7,7),BH(8,2),BH(8,8),BH(9,3),BH(9,6),BH(9,9),BH(9,10),BH(10,4),BH(10,10),$ $BH(11,11),BH(12,2),BH(12,3),BH(12,12),BH(12,36),BH(13,13),BH(13,60)$
2 |  $BH(8,4),BH(8,20),BH(10,5),BH(10,6),BH(12,4),BH(12,6),BH(14,4),BH(16,4),$ $BH(16,8)$
3 | $BH(13,6)$
5 | $BH(16,2)$

##  5 General constructions

###  5.1 Kronecker product

If we keep the field of definition and the multiplier then we can take the
Kronecker product of Butson matrices with self-dual bent sequences and obtain
another Butson matrix with a self-dual bent sequence, as the next result
shows.

######  Proposition 5.1.

Let $H$ (resp. $K$) be a Butson matrix in $BH(n,q)$ (resp. $BH(m,q)$)
affording a self dual bent sequence $X$ (resp. $Y$) for the multiplier
$\mu_{k}.$ Then $X\otimes Y$ is a self-dual bent sequence with multiplier
$\mu_{k},$ for $H\otimes K\in BH(mn,q).$

###### Proof.

From $HX=\lambda\mu_{k}(X)$ and $KY=\nu\mu_{k}(Y)$ we obtain, taking Kronecker
products

$H\otimes K(X\otimes Y)=\lambda\nu\mu_{k}(X)\otimes\mu_{k}(Y).$

The result follows by $\mu_{k}(X)\otimes\mu_{k}(Y)=\mu_{k}(X\otimes Y).$ ∎

Example: Taking $q=3,$ we write $j$ for a complex cubic root of unity. We find
an example of application of this proposition for $n=3,\,m=9,\ k=2$ when
$X=(1,j,j),$ and $Y=(1,1,1,1,j^{2},j,1,j,j^{2}).$

###  5.2 Regular matrices

A Butson Hadamard matrix $H$ of order $n$ is $regular$ if it has constant row
and column sum. A direct connection between self-dual bent sequences and
regular Butson Hadamard matrices is as follows. Denote by ${\bf 1}_{n}$ the
all-one vector of length $n.$

######  Proposition 5.2.

If $H$ is a regular Butson Hadamard matrix of order $n,$ then for all
$u\in\Omega_{q},$ we have that $u{\bf 1}_{n}$ is a self-dual bent sequence for
$H$ with trivial multiplier.

###### Proof.

Denote by $\sigma$ the sum of elements of any row. By definition of regular
Butson Hadamard matrices $H{\bf 1}_{n}=\sigma{\bf 1}_{n}.$ The result follows
by linearity. ∎

A strong existence condition for regular Butson matrices is Lemma 5.2 in [9].

###  5.3 Bush type

A Bush-type Hadamard matrix, say $H,$ is a Butson Hadamard matrix of order
$n^{2}$ over the $q^{th}$ roots of unity which is subdivided into $n^{2}$
blocks $H_{11},H_{12},\dots,H_{nn}$ of order $n$ such that
$JH_{ij}=H_{ij}J=\delta_{ij}nJ,$ where $J$ denotes the all ones matrices. Bush
type Butson matrices were constructed recently in [15]. They have many self-
dual bent sequences attached to $H$ as the next result shows.

######  Proposition 5.3.

If $H$ is a Bush-type Hadamard matrix of order $n^{2}$ over the $q^{th}$ roots
of unity, then there are at least $q^{n}$ self-dual bent sequences with
trivial multiplier attached to $H.$

###### Proof.

From the definition, we see that the sequence $X$ defined by $X^{T}=(u_{1}{\bf
1}_{n},\dots,u_{n}{\bf 1}_{n}),$ where the $u_{k}$’s are arbitrary in
$\Omega_{q},$ is a self-dual bent sequence. ∎

###  5.4 Fourier transforms

Let $q\geq 2$ be an integer, not necessarily a prime power. Let $r\geq 1$ be
an integer. We define the inner product $\cdot$ on $\mathbb{Z}_{q}^{r}$ as

$\displaystyle(x_{1},\ldots,x_{r})\cdot(y_{1},\ldots,y_{r})=x_{1}y_{1}+\cdots+x_{r}y_{r},$
(2)

where we use the ring structure of $\mathbb{Z}_{q}$ on the right-hand side.
Note that $x\cdot y=y\cdot x$ for all $x,y\in\mathbb{Z}_{q}^{r}$.

Let $n=q^{r}$ and $H$ be the $n\times n$ matrix over $\mathbb{C}$ defined as
follows: For $x,y\in\mathbb{Z}_{q}^{r}$, the entry $H(x,y)$ of $H$
corresponding to row $x$ and column $y$ is given by

$\displaystyle H(x,y)=\zeta_{q}^{x\cdot y}.$ (3)

It follows from the orthogonality of group characters that $H$ is a Butson
type Hadamard matrix of order $n$ belonging to $BH(n,q).$

Let $f:\mathbb{Z}_{q}^{r}\rightarrow\mathbb{Z}_{q}$ be a function. Let
$X=X(f)$ be the $r\times 1$ matrix over $\mathbb{C}$ such that the $x$-th
entry of $X$ is $\zeta_{q}^{f(x)}$. We assume that $f$ is not the zero
function so that $X$ is not the trivial column matrix, i.e. not the full one
column. Recall that if there exists a multiplier
$\mu_{k}:\Omega_{q}\rightarrow\Omega_{q}$ and
$\lambda\in\mathbb{Z}[\zeta_{q}]$ such that $HX=\lambda\mu_{k}(X)$, then $X$
is called a self-dual bent sequence attached to $H$ with multiplier $\mu_{k}$
and the coefficient $\lambda$ (see Definition 2.4 above). Let us point out
that when $X$ is a self-dual bent sequence attached to the conjugate matrix of
(3), then $f:\mathbb{Z}_{q}^{r}\rightarrow\mathbb{Z}_{q}$ is a generalized
bent function (in sense of [22]).

In the following theorem, when $r$ is even and $f$ is a generalized Maiorana-
McFarland (MM) function, we characterize all $f$ such that $X(f)$ is self-
dual.

######  Theorem 5.1.

Let $q\geq 2$ be an integer, not necessarily a prime power. Let $m\geq 1$ be
an integer. Let $r=2m$. Let
$\phi:\mathbb{Z}_{q}^{m}\rightarrow\mathbb{Z}_{q}^{m}$ be a permutation. Let
$f:\mathbb{Z}_{q}^{r}\rightarrow\mathbb{Z}_{q}$ be the map defined as

$\displaystyle f(x_{1},x_{2})=x_{1}\cdot\phi(x_{2}),{}$

where $\cdot$ is the Euclidean inner product on $\mathbb{Z}_{q}^{m}$ defined
as in (2).

Let $1\leq k\leq q-1$ and $\mu_{k}(z)=z^{k}$ be as in Definition 2.4. Put
$n=q^{r}$ and let $H$ be the matrix of order $n$ given in (3). Let $X$ be the
$n\times 1$ column such that $(x_{1},x_{2})$-th entry of $X$ is
$\zeta_{q}^{f(x_{1},x_{2})}$. Then $X$ is a self-dual bent sequence attached
to $H$ with

$\displaystyle HX=\lambda\mu_{k}(X){}$

if and only if all of the followings hold:

  * (1)

$x_{1}\cdot x_{2}+k\phi(x_{1})\cdot\phi(x_{2})=0$ for all
$x_{1},x_{2}\in\mathbb{Z}_{q}^{m}$.

  * (2)

$\lambda=q^{m}$.

###### Proof.

Let $(x_{1},x_{2})\in\mathbb{Z}_{q}^{r}$. The $(x_{1},x_{2})$-th entry of $HX$
is

$\displaystyle\begin{array}[]{l}\displaystyle\sum_{y_{1}\in\mathbb{Z}_{q}^{m}}\sum_{y_{2}\in\mathbb{Z}_{q}^{m}}\zeta_{q}^{(x_{1},x_{2})\cdot(y_{1},y_{2})}\zeta_{q}^{f(y_{1},y_{2})}=\sum_{y_{1}\in\mathbb{Z}_{q}^{m}}\sum_{y_{2}\in\mathbb{Z}_{q}^{m}}\zeta_{q}^{x_{1}\cdot
y_{1}+x_{2}\cdot y_{2}+y_{1}\cdot\phi(y_{2})}\\\ \\\
\displaystyle=\sum_{y_{2}\in\mathbb{Z}_{q}^{m}}\zeta_{q}^{x_{2}\cdot
y_{2}}\sum_{y_{1}\in\mathbb{Z}_{q}^{m}}\zeta_{q}^{y_{1}\cdot\left(x_{1}+\phi(y_{2})\right)}=q^{m}\zeta_{q}^{\phi^{-1}(-x_{1})\cdot
x_{2}}.\end{array}{}$ (7)

Hence $X$ is a self-dual bent sequence if and only if $\lambda=q^{m}$ and

$\displaystyle\phi^{-1}(-x_{1})\cdot x_{2}=kx_{1}\cdot\phi(x_{2})\;\;\mbox{for
all $x_{1},x_{2}\in\mathbb{Z}_{q}^{m}$}.{}$

Consider the change of variable
$x_{1}^{\prime}=\phi^{-1}(-x_{1})\iff\phi(x_{1}^{\prime})=-x_{1}$. Using this
change of variable, the last condition is equivalent to the condition that
$x_{1}\cdot x_{2}+k\phi(x_{1})\cdot\phi(x_{2})=0\;\;\mbox{for all
$x_{1},x_{2}\in\mathbb{Z}_{q}^{m}$}.$ ∎

The following corollary is immediate.

######  Corollary 5.1.

Let $q\geq 2$ be an integer, not necessarily a prime power. Let $m\geq 1$ be
an integer. Let $r=2m$. Let $H$ be the matrix of order $n$ given in (3).
Consider the set

$\displaystyle S=\left\\{\frac{-1}{d^{2}}:1\leq d\leq
q-1,\;\;\gcd(d,q)=1\right\\}.{}$

If $k\in S$, then there exists a self-dual bent sequence $X$ attached to $H$
with $HX=q^{m}\mu_{k}(X)$.

###### Proof.

Let $1\leq d\leq q-1$ be an integer with $\gcd(d,q)=1$. Note that the map
$\phi(x)=dx$ is a permutation on $\mathbb{Z}_{q}^{m}$. Moreover we have

$\displaystyle x_{1}\cdot
x_{2}+k\phi(x_{1})\cdot\phi(x_{2})=(1+kd^{2})x_{1}\cdot x_{2}{}$

for all $x_{1},x_{2}\in\mathbb{Z}_{q}^{m}$. If $k\in S$, then $1+kd^{2}=0$,
which completes the proof. ∎

######  Remark 5.1.

Taking $d=1$ in Corollary 5.1, then for $k=q-1$ and $\phi(x)=x$, we have that
$g(x_{1},x_{2})=x_{1}\cdot x_{2}$ is a (self-dual) generalized bent function
with $HX=q^{m}\;\overline{X}$.

Next, we consider some group invariant Butson type Hadamard matrices having
self-dual bent sequences.

Let $q\geq 2$ be an integer, not necessarily a prime power. Let $m\geq 1$ be
an integer and $r=2m$. Let $n=q^{r}$ and $H$ be the $n\times n$ matrix over
$\mathbb{C}$ defined as follows: For
$(x_{1},x_{2}),(y_{1},y_{2})\in\mathbb{Z}_{q}^{r}$, the entry
$H((x_{1},x_{2}),(y_{1},y_{2}))$ of $H$ corresponding to the row
$(x_{1},x_{2})$ and the column $(y_{1},y_{2})$ is given by

$\displaystyle
H((x_{1},x_{2}),(y_{1},y_{2}))=\zeta_{q}^{(x_{1}-y_{1})\cdot(x_{2}-y_{2})}{=\zeta_{q}^{g((x_{1},x_{2})-(y_{1},y_{2}))}\quad\mbox{for
$g$ from Remark \ref{rm1}}}.$ (8)

This implies that

$\displaystyle
H((x_{1}+a_{1},x_{2}+a_{2}),(y_{1}+a_{1},y_{2}+a_{2}))=H((x_{1},x_{2}),(y_{1},y_{2})){}$

for all $(a_{1},a_{2})\in Z_{q}^{r}$ and hence the matrix $H$ defined in (8)
is $\mathbb{Z}_{q}^{r}$-group invariant in the sense of [8]. Note that the
matrix defined in (3) is not $\mathbb{Z}_{q}^{r}$-group invariant. Let us
point out that $H(x,y)=\zeta_{q}^{g(x-y)}$ is a group invariant Butson type
Hadamard matrix when $g\colon\mathbb{Z}_{q}^{n}\rightarrow\mathbb{Z}_{q}$ is a
generalized bent function (see [22, Proposition 2.3]).

Now we are ready to present the analogous result of Theorem 5.1 for the group
invariant matrix $H$ defined in (8). We also show that $H$ is a Butson type
Hadamard matrix in the next theorem.

######  Theorem 5.2.

Let $q\geq 2$ be an integer, not necessarily a prime power. Let $m\geq 1$ be
an integer. Let $r=2m$. Let
$\phi:\mathbb{Z}_{q}^{m}\rightarrow\mathbb{Z}_{q}^{m}$ be a permutation. Let
$f:\mathbb{Z}_{q}^{r}\rightarrow\mathbb{Z}_{q}$ be the map, different from
Theorem 5.1, defined as

$\displaystyle f(x_{1},x_{2})=x_{1}\cdot\phi(x_{2})-x_{1}\cdot x_{2},{}$

where $\cdot$ is the Euclidean inner product on $\mathbb{Z}_{q}^{m}$ defined
as in (2). Assume that $f$ is not the zero map.

Let $1\leq k\leq q-1$ and $\mu_{k}(z)=z^{k}$ be as in Definition 2.4. Put
$n=q^{r}$ and let $H$ be the matrix of order $n$ given in (8). We have that
$H$ is a Butson type Hadamard matrix.

Moreover, let $X$ be the $n\times 1$ column such that $(x_{1},x_{2})$-th entry
of $X$ is $\zeta_{q}^{f(x_{1},x_{2})}$. Then $X$ is a self-dual bent sequence
attached to $H$ with

$\displaystyle HX=\lambda\mu_{k}(X){}$

if and only if all of the followings hold:

  * (1)

$kx_{1}\cdot\phi^{2}(x_{2})-(k+1)x_{1}\cdot\phi(x_{2})+x_{1}\cdot x_{2}=0$ for
all $x_{1},x_{2}\in\mathbb{Z}_{q}^{m}$.

  * (2)

$\lambda=q^{m}$.

###### Proof.

We first prove that $H$ defined in (8) is a Butson type Hadamard matrix. Let
$(x_{1},x_{2}),(z_{1},z_{2})\in\mathbb{Z}_{q}^{r}$. The entry of $HH^{*}$
corresponding to the row $(x_{1},x_{2})$ and the column $(z_{1},z_{2})$ is

$\displaystyle\begin{array}[]{l}\displaystyle\sum_{y_{1}\in\mathbb{Z}_{q}^{m}}\sum_{y_{2}\in\mathbb{Z}_{q}^{m}}\zeta_{q}^{(x_{1}-y_{1})\cdot(x_{2}-y_{2})}\zeta_{q}^{-(y_{1}-z_{1})\cdot(y_{2}-z_{2})}=\sum_{y_{1}\in\mathbb{Z}_{q}^{m}}\sum_{y_{2}\in\mathbb{Z}_{q}^{m}}\zeta_{q}^{x_{1}\cdot
x_{2}-z_{1}\cdot z_{2}+y_{1}\cdot(z_{2}-x_{2})+y_{2}\cdot(z_{1}-x_{1})}\\\ \\\
\displaystyle=\zeta_{q}^{x_{1}\cdot x_{2}-z_{1}\cdot
z_{2}}\sum_{y_{1}\in\mathbb{Z}_{q}^{m}}\zeta_{q}^{y_{1}\cdot(z_{2}-x_{2})}\sum_{y_{2}\in\mathbb{Z}_{q}^{m}}\zeta_{q}^{y_{2}\cdot(z_{1}-x_{1})}.\end{array}{}$
(12)

These arguments and the usual orthogonality relations imply that
$HH^{*}=q^{r}I_{n}$. This completes the proof of the statement that $H$
defined in (8) is a Butson type Hadamard matrix.

Let $(x_{1},x_{2})\in\mathbb{Z}_{q}^{r}$. The $(x_{1},x_{2})$-th entry of $HX$
is

$\displaystyle\begin{array}[]{l}\displaystyle\sum_{y_{1}\in\mathbb{Z}_{q}^{m}}\sum_{y_{2}\in\mathbb{Z}_{q}^{m}}\zeta_{q}^{(x_{1}-y_{1})\cdot(x_{2}-y_{2})}\zeta_{q}^{f(y_{1},y_{2})}=\sum_{y_{1}\in\mathbb{Z}_{q}^{m}}\sum_{y_{2}\in\mathbb{Z}_{q}^{m}}\zeta_{q}^{x_{1}\cdot
x_{2}+y_{1}\cdot y_{2}-x_{1}\cdot y_{2}-x_{2}\cdot y_{1}-y_{1}\cdot
y_{2}+y_{1}\cdot\phi(y_{2})}\\\ \\\ \displaystyle=\zeta_{q}^{x_{1}\cdot
x_{2}}\sum_{y_{2}\in\mathbb{Z}_{q}^{m}}\zeta_{q}^{-x_{1}\cdot
y_{2}}\sum_{y_{1}\in\mathbb{Z}_{q}^{m}}\zeta_{q}^{y_{1}\cdot\left(\phi(y_{2})-x_{2}\right)}=q^{m}\zeta_{q}^{x_{1}\cdot
x_{2}-x_{1}\cdot\phi^{-1}(x_{2})}.\end{array}{}$ (16)

Hence $X$ is a self-dual bent sequence if and only if $\lambda=q^{m}$ and

$\displaystyle x_{1}\cdot x_{2}-x_{1}\cdot\phi^{-1}(x_{2})=-kx_{1}\cdot
x_{2}+kx_{1}\cdot\phi(x_{2})\;\;\mbox{for all
$x_{1},x_{2}\in\mathbb{Z}_{q}^{m}$}.{}$

By a suitable change of variables as in the proof of Theorem 5.1, the last
condition is equivalent to the condition that
$kx_{1}\cdot\phi^{2}(x_{2})-(k+1)x_{1}\cdot\phi(x_{2})+x_{1}\cdot
x_{2}=0\;\;\mbox{for all $x_{1},x_{2}\in\mathbb{Z}_{q}^{m}$}.$ ∎

The following corollary is immediate.

######  Corollary 5.2.

Let $q\geq 3$ be an integer, not necessarily a prime power. Let $m\geq 1$ be
an integer. Let $r=2m$. Let $H$ be the matrix of order $n$ given in (8).
Consider the set

$\displaystyle S=\left\\{\frac{1}{d}:1\leq d\leq q-1,\;\;\gcd(d,q)=1,\;\;d\neq
1\right\\}{}$

If $k\in S$, then there exists a self-dual bent sequence $X$ attached to $H$
with $HX=q^{m}\mu_{k}(X)$.

###### Proof.

Let $1\leq d\leq q-1$ be an integer with $d\neq 1$. Note that the map
$\phi(x)=dx$ is a permutation on $\mathbb{Z}_{q}^{m}$. Also $f$ is not the
zero map as $d\neq 1$. Moreover, we have

$\displaystyle
kx_{1}\cdot\phi^{2}(x_{2})-(k+1)x_{1}\cdot\phi(x_{2})+x_{1}\cdot
x_{2}=\left(kd^{2}-(k+1)d+1\right)x_{1}\cdot x_{2}=(kd-1)(d-1)x_{1}\cdot
x_{2}{}$

for all $x_{1},x_{2}\in\mathbb{Z}_{q}^{m}$. If $k\in S$, then $kd-1=0$, which
completes the proof. ∎

######  Remark 5.2.

Corollaries 5.1 and 5.2 only present very simple examples satisfying
conditions of Theorems 5.1 and 5.2, respectively. For example, if $q=2$ and
$m\geq 1$ is any integer, there are also such maps $\phi$ proving self-dual
bent sequences satisfying the conditions of Theorem 5.2.

##  6 Existence conditions

The following result is inspired by [9, Lemma 5.2]. The hypothesis holds true
when $H$ admits a self-dual bent sequence with $k=1.$

######  Theorem 6.1.

If $H$ in $BH(n,q)$ admits an eigenvalue $\lambda$ in $\mathbb{Z}[\zeta_{q}],$
with an eigenvector in $\Omega_{q}^{n},$ then there are $q$ nonnegative
integers $y_{r}\leq n$ summing up to $n,$ and such that

$n=\left(\sum_{r=0}^{q-1}y_{r}c_{r}\right)^{2}+\left(\sum_{r=0}^{q-1}y_{r}s_{r}\right)^{2}$

where $\zeta_{q}^{r}=c_{r}+is_{r}$ and $i=\zeta_{4}.$

###### Proof.

Projecting the eigenvalue equation $Hx=\lambda x$ on the first coordinate
yields after dividing out by $x_{1}$ the relation

$\sum_{j=1}^{n}H_{1j}\frac{x_{j}}{x_{1}}=\lambda.$

Let $y_{r}$ denote the number of occurrences of $\zeta_{q}^{r}$ among the
summands in the above sum. By definition

$\sum_{r=0}^{q-1}y_{r}=n,\;\mbox{where $0\leq y_{r}\leq n$}.$

A well-known consequence of the Hadamard equation is that

$n=|\lambda|^{2}=a^{2}+b^{2},$

where $a=\Re(\lambda),\,b=\Im(\lambda).$ The result follows upon computing $a$
and $b$ from the above expression for $\lambda.$ ∎

######  Remark 6.1.

In the literature of combinatorics, the sequence $(y_{r})_{r=0}^{q-1}$ is
called a composition of $n$ into $q$ parts.

######  Example 6.1.

$H\in BH(6,3)$, $n=6,q=3.$ There are precisely
$\binom{6+3-1}{6}=\binom{8}{6}=28$ compositions of $6$ into $3$ parts, namely,

$\displaystyle 6=6+0+0=0+6+0=0+0+6=1+2+3=5+1+0=5+0+1=1+5+0$
$\displaystyle=2+1+3=1+0+5=0+1+5=0+5+1=2+2+2=4+2+0=4+0+2$
$\displaystyle=0+4+2=2+4+0=2+0+4=0+2+4=4+1+1=1+4+1=1+1+4$
$\displaystyle=3+3+0=3+0+3=0+3+3=3+2+1=3+1+2=1+3+2=2+3+1$

$6=6+0+0=0+6+0=0+0+6,$ when $a^{2}+b^{2}=36.$
$6=5+1+0=5+0+1=1+5+0=1+0+5=0+1+5=0+5+1,$ when $a^{2}+b^{2}=21.$
$6=4+2+0=4+0+2=0+4+2=2+0+4=0+2+4=2+4+0,$ when $a^{2}+b^{2}=12.$
$6=4+1+1=1+1+4=1+4+1=3+3+0=3+0+3=0+3+3,$ when $a^{2}+b^{2}=9.$
$6=3+2+1=3+1+2=1+3+2=2+3+1=1+2+3=2+1+3,$ when $a^{2}+b^{2}=3.$
$6=2+2+2,$ when $a^{2}+b^{2}=0.$
The above $28$ cases satisfy $\sum_{r=0}^{q-1}y_{r}=6,$ but $a^{2}+b^{2}\neq
6,$ so $BH(6,3)$ does not contain matrices having eigenvalues of the type of
Theorem 6.1. This is confirmed by computation on matrices of $BH(6,3).$

######  Remark 6.2.

The condition in Theorem 6.1 is necessary but not sufficient as $BH(8,4)$
contains two matrices neither leading to self-dual bent sequences. Still,
$8=2^{2}+2^{2},$ with $y_{0}=4,\,y_{1}=y_{2}=2,\,y_{3}=0.$ A similar
phenomenon occurs for $BH(21,3).$

Table 3: Parameters excluded by Theorem 6.1 n | q | # of compositions | values of $a^{2}+b^{2}$
---|---|---|---
2 | 2 | 3 | 0;4
6 | 3 | 28 | 0;3;9;12;21;36
6 | 4 | 84 | 0;2;4;8;10;16;18;20;26;36
6 | 6 | 462 | 0;1;3;4;7;9;12;13;16;19;21;25;27;28;31;36
8 | 2 | 9 | 0;4;16;36;64
8 | 4 | 165 | 0;2;4;8;10;16;18;20;26;32;34;36;40;50;64
10 | 6 | 3003 |  0;1;3;4;7;9;12;13;16;19;21;25;27;28;31;36;37;39;43;48;49;52; 57;61;63;64; 67;73;75;76;79;81;84;91;100
12 | 2 | 13 | 0;4;16;36;64;100;144
12 | 4 | 455 |  0;2;4;8;10;16;18;20;26;32;34;36;40;50;52;58;64;68;72;74;80;82;90;100;104; 122;144
12 | 6 | 6188 |  0;1;3;4;7;9;12;13;16;19;21;25;27;28;31;36;37;39;43;48;49;52;57;61;63;64; 67;73;75;76;79;81;84;91;93;97;100;103;108;109;111;112;117;121;124;133;144
13 | 6 | 8568 |  0;1;3;4;7;9;12;13;16;19;21;25;27;28;31;36;37;39;43;48;49;52;57;61;63;64; 67;73;75;76;79;81;84;91;93;97;100;103;108;109;111;112;117;121;124;127;129;133;139; 144;147;157;169
14 | 4 | 680 |  0;2;4;8;10;16;18;20;26;32;34;36;40;50;52;58;64;68;72;74;80;82;90;98;100;104; 106;116;122;130;144;148;170;196
14 | 6 | 11628 |  0;1;3;4;7;9;12;13;16;19;21;25;27;28;31;36;37;39;43;48;49;52;57;61;63;64; 67;73;75;76;79;81;84;91;93;97;100;103;108;109;111;112;117;121;124;127;129;133;139; 144;147;148;151;156;157;163;169;172;183;196
21 | 3 | 253 |  0;3;9;12;21;27;36;39;48;57;63;75;81;84;93;108;111;117;129;144;147;156;171;183; 189;201;225;228;237;273;279;324;327;381;441

##  7 Distance spectrum of Butson Hadamard codes

######  Definition 7.1.

The unit sphere $\Upsilon_{d}$ in Euclidean $d$-space $\mathbb{R}^{d}$ is the
set of all unit norm vectors:

$\Upsilon_{d}\triangleq\left\\{x=(x_{1},x_{2},\dots,x_{d})\in\mathbb{R}^{d}:\left\|x\right\|=1\right\\}$

######  Definition 7.2.

The hermitian inner product of $x$ and $y$ in $\mathbb{C}^{n}$ is defined as

$\left\langle x,y\right\rangle=\sum_{i=1}^{n}x_{i}\overline{y_{i}},$

with $\overline{z}$ denoting the complex conjugate of $z.$ The (squared)
Euclidean distance of $x$ and $y$ is then

$d_{E}(x,y)=\left\langle x-y,x-y\right\rangle=\left\langle
x,x\right\rangle+\left\langle y,y\right\rangle-2\Re(\left\langle
x,y\right\rangle).$

######  Definition 7.3.

A spherical code in dimension $d$ is a finite set $X\subseteq\Upsilon_{d}.$
Its minimum distance for the squared Euclidean distance is denoted by $\rho.$
Its parameters are denoted compactly by $(d,\rho,|X|).$ The function
$A_{d}(\rho)$ can then be defined as

$A_{d}(\rho)=\max\\{|X|\mid X\text{ spherical code of parameters
}(d,\rho,|X|)\\}.$

In the next proposition we compute the possible Chinese Euclidean distances of
${C_{H}}$, which are also the squared Euclidean distances of the spherical
code $\phi(\zeta_{q}^{C_{H}})$ to be defined below.

######  Proposition 7.1.

The Chinese Euclidean distances of ${C_{H}}$ are

$\left\\{d_{E}(x,y)\mid x\neq y,x,y\in
w^{C_{H}}\right\\}=\left\\{2n\right\\}\cup\left\\{2n(1-\cos\frac{2\pi
t}{q})\mid t=1,2,\dots,\left\lfloor\frac{q}{2}\right\rfloor\right\\}.$

###### Proof.

For simplicity put $w=\zeta_{q}$ and write $w^{F_{H}+i}$ for $w^{i}w^{F_{H}}.$
Note that, by definition of $C_{H}$ we have $w^{C_{H}}=w^{F_{H}}\cup
w^{F_{H}+1}\cup\cdots\cup w^{F_{H}+q-1}.$ The Chinese Euclidean distances of
${C_{H}}$ can then be computed as follows. Let $x=w^{i}h,y=w^{j}h^{\prime},$
where $h,h^{{}^{\prime}}\in w^{F_{H}}$ and
$i,j\in\left\\{0,1,\dots,q-1\right\\}.$ We have

$\left\langle x,y\right\rangle=\left\\{\begin{array}[]{ll}0,&h\neq
h^{{}^{\prime}},\\\ nw^{t},&h=h^{{}^{\prime}},\end{array}\right.$

where $t\in\left\\{1,2,\dots,q-1\right\\}.$ From the equation
$d_{E}(x,y)=2n-2\Re(\left\langle x,y\right\rangle),$ the result follows. ∎

The map $\psi:\mathbb{C}\longrightarrow\mathbb{R}^{2},x+iy\longmapsto(x,y)$ is
an isometry from $\left(\mathbb{C}^{n},d_{E}\right)$ to
$\left(\mathbb{R}^{2n},\delta\right),$ where
$\delta(U,V)=\left\|U-V\right\|^{2},$ for all $U,V\in\mathbb{R}^{2n}.$ Note
that $(\psi(a),\psi(b))=\Re\big{(}\left\langle a,b\right\rangle\big{)},$ where
$a$ and $b\in\mathbb{C}^{n},$ and $(,)$ denotes the standard inner product in
$\mathbb{R}^{2n}.$ For normalization purposes, we will let
$\phi(z)=\frac{\psi(z)}{\sqrt{n}},$ for all $z\in\mathbb{C}^{n}.$

######  Corollary 7.1.

The spherical code
$\phi(\zeta_{q}^{C_{H}})\subseteq\Upsilon_{2n}\subseteq\mathbb{R}^{2n}$ has a
size of $nq$ and a distance of

$\rho=\frac{d_{CE}}{n}=2(1-\cos\frac{2\pi}{q}).$

We note an unexpected consequence for the parameters of a Butson Hadamard
matrix.

######  Proposition 7.2.

If $H\in BH(n,q)$ exits, then the following inequality holds.

$nq\leq A_{2n}(2(1-\cos{\frac{2\pi}{q}})).$

We will require a special case of Levenshtein bound [10, Theorem 2.5.1].

######  Theorem 7.1.

Let $X$ be a spherical code with parameters $(d,\rho,|X|)$ and let
$s=1-\frac{\rho}{2}.$ Then the size $|X|$ is necessarily bounded according to
$|X|\leq L_{3}(s)=\frac{d(2+(d+1)s)(1-s)}{1-ds^{2}},$ for
$s\in[0,\frac{1}{(\sqrt{d+3}+1)}).$

We deduce from Corollary 7.1 the following optimality result in relation to
complex Hadamard matrices in the Turyn sense.

######  Corollary 7.2.

If $H\in BH(n,4),$ then the spherical code $\phi(\zeta_{q}^{C_{H}})$ is
optimal in dimension $2n.$

###### Proof.

By Corollary 7.1 we know that, in the notation of Theorem 7.1, we have
$s=\cos(\frac{2\pi}{4})=0,$ and $d=2n.$ The result follows by Theorem 7.1
since the size of the code is $4n=2d.$ ∎

##  8 Covering radius of Butson Hadamard codes

###  8.1 Lower bound

######  Definition 8.1.

The deviation of an arbitrary vector $x\in\Omega_{q}^{n}$ from a polyphase
code $C$ is defined as

$\theta(C,x)=\max_{y\in C}\\{\left|\left\langle x,y\right\rangle\right|\\},$

where $\left\langle x,y\right\rangle$ denotes the hermitian inner product of
$x$ and $y.$

######  Proposition 8.1.

If $X$ is a bent sequence for $H\in BH(n,q),$ then its corresponding polyphase
code $\zeta_{q}^{C_{H}}$ has deviation $\theta(C,X)=\sqrt{n},$ where
$X\in\Omega_{q}^{n}.$

###### Proof.

$HX=\lambda\mu_{k}(X),$ where $k$ and $q$ are coprime. Let $h_{i}$ be the
$i$-th row of $H,$ then

$\left|\left\langle
h_{i},X\right\rangle\right|=\left|\lambda\right|\left|\mu_{k}(X_{i})\right|=\sqrt{n}.$

Since
$\left|\left\langle\zeta_{q}^{t}h_{i},X\right\rangle\right|=\left|\left\langle\
h_{i},X\right\rangle\right|$ for all $i=1,\cdots,n$ and all $t=1,\cdots,q$ the
result follows. ∎

We consider the polyphase codes attached to some $H\in BH(n,q)$ when $q$
equals $4,6$ and $8.$ By induction on $n,$ we can get
$d_{CE}(u,v)=d_{E}(x,y),$ where $x=\zeta_{q}^{u}$ and $y=\zeta_{q}^{v}$ with
$u,v\in\mathbb{Z}_{q}^{n}.$
It can be seen by expanding $\left\langle x-y,x-y\right\rangle$ that

$Re(\left\langle
x,y\right\rangle)=\frac{2n-d_{E}(x,y)}{2}=\frac{2n-d_{CE}(u,v)}{2},$

for all $x,y\in\Omega_{q}^{n}.$
The simple inequality $Re(\left\langle x,y\right\rangle)\leq\left|\left\langle
x,y\right\rangle\right|$ in the complex field shows that

$r_{CE}(C_{H})\geq 2n-2\theta(C,x).$

Combining this fact with the above proposition 8.1 yields the following bound.

######  Corollary 8.1.

If there is a bent sequence $X$ for $H\in BH(n,q),$ then the covering radius
of its attached $\mathbb{Z}_{q}$-code $C_{H}$ is bounded below as

$r_{CE}(C_{H})\geq 2n-2\sqrt{n}.$

The following three tables indicate the lower bounds and the true values of
the covering radius when $q$ equals $4,6$ and $8,$ respectively, under the
existence of the Butson Hadamard matrix with self-dual bent sequences. The
second column displays the index $k$ of the multiplier $\mu_{k}.$ A question
mark indicates that the computation could not be completed.

Table 4: The covering radius of $C_{H}$ for $H\in BH(n,4)$ $BH(n,4)$ | $k$ | Lower bound | True value
---|---|---|---
$BH(4,4)$ | 1;3 | 4 | 4
$BH(10,4)$ | 3 | 14 | 14
$BH(16,4)(F_{4}\otimes F_{4})$ | 1;3 | $24$ | ?
Table 5: The covering radius of $C_{H}$ for $H\in BH(n,6)$ $BH(n,6)$ | $k$ | Lower bound | True value
---|---|---|---
$BH(7,6)$ | 5 | 9 | 9
$BH(9,6)$ | 1;5 | 12 | 12
$BH(12,6)(F_{6}\otimes F_{2})$ | 5 | $2(12-\sqrt{12})\approx 17.072$ | ?
Table 6: The covering radius of $C_{H}$ for $H\in BH(n,8)$ $BH(n,8)$ | $k$ | Lower bound | True value
---|---|---|---
$BH(8,8)$ | 7 | $2(8-\sqrt{8})\approx 10.343$ | $2(8-\sqrt{8})$

###  8.2 Upper bounds

In this subsection, we derive upper bounds on the covering radius of the
spherical code $\phi(\zeta_{q}^{C_{H}})$ (after normalization to belong to the
unit sphere) defined in the preceding section, as a function of its strength
as a spherical design. This bound is then directly an upper bound on the
covering radius of the code $C_{H}$ for the Chinese Euclidean distance.
However, the two quantities may or may not coincide.

######  Definition 8.2.

A spherical code $X$ is a 1-design if its center of mass is the origin or,
more concretely, for all coordinate indices $i$ satisfy $\sum_{x\in
X}x_{i}=0.$ It is antipodal if $X=-X.$

######  Proposition 8.2.

If $H\in BH(n,q)$ with $q$ is even, then the covering radius of
$\phi(\zeta_{q}^{C_{H}})$ is at most $\sqrt{2}.$

###### Proof.

The spherical code $\phi(\zeta_{q}^{C_{H}})$ is antipodal, because $-1$ is a
power of $\zeta_{q}.$ The result follows then by [30, Theorem 1]. ∎

The same bound can be obtained from different hypotheses on $H.$

######  Proposition 8.3.

If $H$ is dephased, then $\phi(\zeta_{q}^{C_{H}})$ is a 1-design and its
covering radius is at most $\sqrt{2}.$

###### Proof.

Note that $H^{t}$ is also a Hadamard matrix. By taking scalar products between
columns of $H$ and its first column, which is all-one, we see that $\sum_{x\in
X}x_{i}=0$ for $i>1.$ That $\sum_{x\in X}x_{1}=0,$ follows by the well-known
property of the roots of unity that $\sum_{k=0}^{q-1}\zeta_{q}^{k}=0.$ Hence
the said spherical code is a $1$-design.

The result follows by Table 1 of [11]. ∎

######  Definition 8.3.

A spherical code $X$ is a 2-design if it is a 1-design, and if, furthermore,
for all pairs $i\neq j$ of coordinate indices the following two relations
hold.

$\sum_{x\in X}(x_{i}^{2}-x_{j}^{2})=0,\;\sum_{x\in X}x_{i}x_{j}=0.$

######  Theorem 8.1.

If $H\in BH(n,q)$ is dephased, then $\phi(\zeta_{q}^{C_{H}})$ is a 2-design
and its covering radius is at most $\sqrt{2(1-\frac{1}{2n})}.$

###### Proof.

We have seen in Proposition 8.3 that $\phi(\zeta_{q}^{C_{H}})$ is a 1-design.
Denote by $h_{j}$ the column number $j$ of $H.$ Write $w=\zeta_{q}.$ That

$\sum_{x\in X}x_{i}^{2}=\sum_{x\in X}x_{j}^{2}=q$

is immediate by the fact that the columns of $H$ have the same norm, by the
Hadamard property of $H^{t}.$ Further, still using that property we get

$\sum_{x\in
X}x_{i}x_{j}=\sum_{t=0}^{q-1}(\psi(w^{t}h_{i}),\psi(w^{t}h_{j}))=\frac{1}{{n}}\sum_{t=0}^{q-1}\Re(\langle
h_{i},h_{j}\rangle)=0.$

Thus $\phi(\zeta_{q}^{C_{H}})$ is a 2-design. The upper bound follows then by
[11, Table 1]. ∎

This bound can be sharpened under some extra conditions.

######  Theorem 8.2.

If $H\in BH(n,q),$ with $q$ even, is dephased, then $\phi(\zeta_{q}^{C_{H}})$
is an antipodal 2-design and its covering radius is at most
$\sqrt{2(1-\frac{1}{\sqrt{2n}})}.$

###### Proof.

We see by Theorem 8.1 that $\phi(\zeta_{q}^{C_{H}})$ is a 2-design. It is
antipodal by the evenness of $q.$ The result follows then by Theorem 3 of
[30]. ∎

To compare with the results on the previous subsection we can give the
following corollary.

######  Corollary 8.2.

If $H\in BH(n,q),$ with $q$ even, is dephased, then the covering radius of
$C_{H}$ for the Chinese Euclidean distance is bounded above as follows.

$r_{CE}(C_{H})\leq 2n-\sqrt{2n}$

###### Proof.

Immediate upon considering the normalization of the spherical code. ∎

##  9 Conclusion and open problems

In this paper we have explored a new definition of self-dual bent sequences in
relation with Butson Hadamard matrices. The strong group of automorphisms
which preserves the set of bent sequences for a given multiplier has been
introduced, and given a generation algorithm. We have given computational
methods to construct, for a given matrix, such a sequence. We have also given
infinite families of Butson Hadamard matrices with attending self-dual bent
sequences. We have also given an arithmetic criterion that allows to rule out
all the matrices of $BH(n,q)$ from admitting eigenvalues allowing to define
such sequences. In a second part, we have explored the code attached to the
Butson Hadamard matrix over a finite ring and its concomitant spherical code
from the viewpoint of the Chinese Euclidean distance, and of the standard
Euclidean distance. The existence of bent sequences implies then a lower bound
on the covering radius of these codes. Upper bounds on the covering radius of
the spherical code can be provided by showing its strength as a spherical
design. The main open problem left is then to know for $n\to\infty$ which
bound is closer to the true value of the covering radius of $C_{H},$ the lower
bound of Corollary 8.1 or the upper bound of Corollary 8.2? A similar question
was solved for the Sylvester matrix and the generalized Sylvester matrix in
[23, 24], using difficult techniques from number theory, probability, and
discrepancy theory. In our case, Tables 4, 5, 6 seem to indicate that the true
value equals the lower bound.

Acknowledgement: The authors are indebted to Denis Krotov for helpful
discussions.

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