# Hadamard bent sequences and spherical designs *

Minjia Shi, Danni Lu, Andrés Armario, Ronan Egan, Ferruh Özbudak, Patrick Solé

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## To cite this version:

Minjia Shi, Danni Lu, Andrés Armario, Ronan Egan, Ferruh Özbudak, et al. Hadamard bent sequences and spherical designs *. Journal of Algebraic Combinatorics, In press. hal-05243648

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# Hadamard bent sequences and spherical designs*

Minjia Shi, Danni Lu†, Andrés Armario‡, Ronan Egan§, Ferruh Özbudak¶, Patrick Solé||

## Abstract

In (Solé et al, ISIT 2021) a new notion of bent sequences appeared, where the Sylvester matrix of the Walsh-Hadamard transform is replaced by an arbitrary Hadamard matrix. We introduce an even more general notion of a bent sequence, defined in terms of the following data: a Hadamard matrix of order $n$ defined over the complex $q^{th}$ roots of unity (a so-called Butson matrix), an algebraic integer in the cyclotomic field of order $q$, and a Galois automorphism of that field. This new generalisation facilitates the existence of self-dual bent sequences in cases where the restricted definition fails. In particular, we construct self-dual bent sequences for various $q \le 60$ and lengths $n \le 21$. Computational construction methods comprise the resolution of polynomial systems by Gröbner bases and eigenspace computations. Infinite families of self-dual bent sequences can be constructed from regular Hadamard matrices, Bush-type Hadamard matrices, and generalized Boolean bent functions. As an application of Hadamard bent sequences, we derive a lower bound on the covering radius of the $\mathbb{Z}_q$-code attached to a Hadamard matrix, for the so-called Chinese Euclidean metric. To every Hadamard matrix we attach a spherical code, which is optimal as a packing code when $q = 4$. This code is a strength 2 spherical design under some mild conditions. Based on the covering properties of spherical designs, we bound above the covering radius of that spherical code for the standard Euclidean metric. This upper bound, in turn, yields an upper bound on the covering radius of the $\mathbb{Z}_q$-code for the Chinese Euclidean metric which is not far from the lower bound.

**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 with connections to difference sets, Boolean functions, strongly regular graphs, and symmetric cryptography [23]. They are classically defined in relation to the Sylvester matrix, the matrix of the Walsh-Hadamard transform. Recently a new notion of bent sequences appeared, motivated by a cryptographic problem (Physically Unclonable Functions), that is defined for any Hadamard matrix [31]. In that context, these sequences provide a lower bound on the covering radius of a Hadamard code for the Hamming metric, in the same way that Sylvester bent sequences provide a lower bound on the covering radius of the first order Reed-Muller code [21, Chap. 15]. Constructions and properties of these sequences in the self-dual case were investigated in [30]. The advantage of self-dual bent sequences over unrestricted bent sequences is that they enable the application of linear algebraic techniques to the problem. Note that every bent sequence is self-dual for

<footnote>*This research is supported by the National Natural Science Foundation of China (12071001).</footnote>

<footnote>†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. smjwcl.good@163.com, ludanni_in@163.com</footnote>

<footnote>‡Departamento de Matemática Aplicada I, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012, Seville, Spain. armario@us.es This research was supported by the Strategic R+D Project TED2021-130566B-100 from the Ministry of Science and Innovation of the Government of Spain.</footnote>

<footnote>§School of Mathematical Sciences, Dublin City University, Ireland. ronan.egan@dcu.ie</footnote>

<footnote>¶Ferruh Özbudak is with Faculty of Engineering and Natural Sciences, Sabancı University, 34956, Istanbul, E-mail: ferruh.ozbudak@sabanciuniv.edu</footnote>

<footnote>|| Patrick Solé is with I2M (Aix Marseille Univ, CNRS, Centrale Marseille), Marseilles, France. sole@enst.fr</footnote>
some matrix [29]. The extension of this notion to complex Hadamard matrices in the sense of Turyn, that is to say with entries in $\Omega_4 = \{\pm1, \pm i\}$, was done in [29]. A self-dual bent sequence $X$ attached to a matrix $H$ is an eigenvector of $H$ for an eigenvalue $\lambda$ of $H$ (some Gaussian integer) that have its entries in $\Omega_4$. Thus $HX = \lambda X$.

In [4] a more general definition involving **Butson** Hadamard matrices is introduced, that is to say Hadamard matrices with values in the complex roots of unity of a given order. If $H$ is an $n \times n$ Butson Hadamard matrix, they say that a vector $X$ with entries of norm 1 is $H$-bent if $HX = Y$ for some vector $Y$ with entries of norm $\sqrt{n}$. In this paper, the authors focus on solutions to the equation $HX = \lambda X$, where the bar denotes complex conjugation and $|\lambda| = \sqrt{n}$. The motivation for this twist on the definition of self-dual bent sequences was to obtain more bent sequences, in particular in relation with generalized bent functions.

In the present paper, we also extend the notion of self-dual Hadamard bent sequences to incorporate Butson Hadamard matrices. Secondly, we replace the notion of a conjugate by that of a 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. Such a multiplier induces an element of the Galois group of the cyclotomic field of order $q$. When $k=1$ this new definition reduces to that of [29, 30]. When $k=q-1$, we recover the definition of [4]. This change of definition affects the notion of a strong group, a matrix group that preserves the set of self-dual bent sequences. A tangible motivation for introducing this definition is that self-dual bent sequences with a multiplier can exist for some $k \in \{2, \dots, q-2\}$ in cases where existence fails for $k=1$ or $k=q-1$ or both. Hence this more general object can exist when the definitions of [29, 30] and [4] are too restrictive. Moreover, we apply these objects directly when bounding the covering radius of related Butson Hadamard codes and spherical codes.

We extend computational techniques utilising linear algebra and Gröbner bases from [29, 30] to this new setting. This allows us to obtain many examples of self-dual bent sequences from the matrices of the database [6]. General constructions of Butson Hadamard matrices with designed self-dual bent sequences are then derived from 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 [9, 25]. Strong necessary existence conditions for Butson matrices admitting self-dual bent sequences 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 in $\mathbb{Z}_q^n$ or on the Euclidean sphere. While the packing properties of the $\mathbb{Z}_q$-code with respect to the homogeneous and other metrics have been investigated by various authors [1, 14], we focus here on the Chinese Euclidean metric and the covering properties in connection with bent sequences. The Chinese Euclidean metric, which is first mentioned in [7], should not be confused with that in [20]. 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. Depending 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 [8]. In [28], spherical designs are established, providing a solid foundation for their theoretical exploration. Moreover, spherical designs motivate the definition of conformal $t$-designs [15]. Building spherical designs from Butson Hadamard matrices seems to be a new approach. Here we use upper bounds on the covering radius of spherical designs [12, 32] to control the covering radius of the 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 derives necessary conditions for the existence of Butson matrices admitting a self-dual bent sequence. Section 5 gives four general constructions of Butson Hadamard matrices with designed self-dual bent sequences. Section 6 describes our computational methods, providing computational results and examples demonstrating the existence of self-dual bent sequences with non-trivial multipliers. 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 attached spherical codes. Section 9 concludes the article, and proposes a conjecture that generalises a conjecture of Patterson and Wiedemann [24] which was since proved by K-U Schmidt in [26].

## 2 Preliminaries

In this section we recall essential definitions and establish notation that will be used throughout the paper. For any complex number $z \in \mathbb{C}$, the complex conjugate is denoted by $\bar{z}$. The set of complex numbers of norm 1 is denoted by $\mathbb{T}$. Let $\zeta_q = e^{\frac{2\pi\sqrt{-1}}{q}}$ and let $\Omega_q = \{\zeta_q^j\}_{0 \le j \le q-1}$. We refer to an element $X \in \Omega_q^n$ as a **sequence** of length $n$, and typically treat it as a column vector. The all-ones vector of length $n$ will be denoted by $\mathbf{1}_n$. We denote by $M_n(\mathbb{C})$ the set of all $n \times n$ matrices with entries in $\mathbb{C}$, and denote the subset of matrices with entries in $\Omega_q$ by $M_n(q)$. The $n \times n$ identity and all-ones matrices are denoted by $I$ and $J$ respectively, where the dimensions and field of entries are gleaned from context. A **monomial** matrix is an $n \times n$ matrix with exactly one non-zero entry in every row and column. Any monomial matrix $M$ is uniquely written as a product $M = DP$ where $D$ is diagonal and $P$ is a permutation matrix. The set of all monomial matrices with complex entries, denoted by $\text{Mon}_n(\mathbb{C})$, form a group under matrix multiplication, and the subgroup of monomial matrices with non-zero entries in $\Omega_q$ is denoted by $\text{Mon}_n(q)$.

### 2.1 Butson Hadamard matrices

A complex **Hadamard** matrix of order $n$ is an $n \times n$ matrix $H$ with complex entries of norm 1 satisfying the equation $HH^* = nI$ where $H^*$ denotes the conjugate transpose of $H$. If all of its entries are in $\Omega_q$ for some integer $q$, then $H$ is said to be of **Butson** type, and we write $H \in BH(n, q)$. The group $\text{Mon}_n(q)^2$ acts on the set $M_n(q)$ via

$$H(M, N) = MHN^*.$$

It is clear that $H \in BH(n, q)$ if and only if $H(M, N) \in BH(n, q)$ for any pair $(M, N) \in \text{Mon}_n(q)$. Hence $\text{Mon}_n(q)^2$ partitions $BH(n, q)$ into orbits, called **equivalence classes**. Any two matrices in the same orbit are called **equivalent**. The stabilizer of a matrix $H$ under this action is the **automorphism group** of $H$, denoted by $\text{Aut}(H)$. The subgroup of $\text{Aut}(H)$ comprised of pairs of permutation matrices is called the **permutation automorphism group** of $H$, denoted $\text{PAut}(H)$. A matrix $H \in BH(n, q)$ is in **dephased** form when its first row and column each only contain ones. Every matrix of $BH(n, q)$ is equivalent to a dephased matrix.

### 2.2 Bent sequences and multipliers

The following definition of a bent sequence attached to a Butson Hadamard matrix $H$ appeared in [29].

**Definition 2.1.** Given $H \in BH(n, q)$ the sequence $X \in \Omega_q^n$ is a **bent sequence attached to H** if there is $Y \in \Omega_q^n$ such that $HX = \lambda Y$ for some $\lambda \in \mathbb{Z}[\zeta_q]$. A bent sequence $X$ is **self-dual** if $HX = \lambda X$ for some $\lambda \in \mathbb{Z}[\zeta_q]$.

In this paper we generalise the definition of a self-dual bent sequence. For any $k$ coprime to $q$ we define the **multiplier** $\mu_k : \Omega_q \to \Omega_q$ by the rule $\mu_k(z) = z^k$, and by abuse of notation we extend this componentwise to a map $\mu_k : \Omega_q^n \to \Omega_q^n$ of sequences, and a map $\mu_k : M_n(q) \to M_n(q)$ of matrices. 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 $a = \sum_{j=0}^{q-1} a_j \zeta_q^j$ for rational $a_j$'s, we define $\mu_k(a) = \sum_{j=0}^{q-1} a_j \zeta_q^{kj}$. In fact all Galois automorphisms are of that form [18]. They thus satisfy the following three properties.
(1) If $x \in \mathbb{Q}$, then $\mu_k(x) = x$.

(2) $\forall x, y \in \mathbb{Q}(\zeta_q)$, $\mu_k(x + y) = \mu_k(x) + \mu_k(y)$.

(3) $\forall x, y \in \mathbb{Q}(\zeta_q)$, $\mu_k(xy) = \mu_k(x)\mu_k(y)$.

We will adhere to the following definition throughout this paper.

**Definition 2.2.** Let $H \in BH(n, q)$. A self-dual bent sequence with multiplier $\mu_k$ attached to $H$ is defined to be a sequence $X \in \Omega_q^n$ such that

$$HX = \lambda\mu_k(X),$$

where $\lambda$ is an element of $\mathbb{Z}[\zeta_q]$. If $k = 1$, then $\lambda$ is also an eigenvalue of $H$.

When $k = 1$, this is the definition of a self-dual bent sequence. When $k = q-1$, this is the definition of self-dual bent sequence in [4]. 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 entries of $D$ lie in $\Omega_q$, which implies that $HD \in BH(n, q)$.

Let $k$ be a fixed integer coprime to $q$ and let $H \in BH(n, q)$. The **strong automorphism group** of $H$ with multiplier $\mu_k$, is the group

$$SAut(H, k) = \{M \in \mathrm{Mon}_n(q) \mid \mu_k(M)H = HM\}.$$

**Proposition 2.1.** 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 $\mathbb{Q}(\zeta_q)/\mathbb{Q}$. $\square$

**Remark 2.1.** The strong automorphism group with trivial multiplier of $H$ is the subgroup $SAut(H, 1) \le Aut(H)$ comprised of pairs of the form $(M, M)$. It follows from the definition of an automorphism that $(M, M) \in SAut(H, 1)$ if and only if $MH = HM$. Hence $SAut(H, 1)$ is in bijection with the group of matrices in $\mathrm{Mon}_n(q)$ that commute with $H$.

## 2.3 Butson Hadamard codes

A Butson matrix $H \in BH(n, q)$ is conveniently represented in logarithmic form, that is, the matrix $H = [\zeta_q^{\varphi_{i,j}}]_{i,j=1}^n$ is represented by the matrix $L(H) = [\varphi_{i,j} \mod q]_{i,j=1}^n$ 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 $C_H = \cup_{\alpha \in \mathbb{Z}_q} (F_H + \alpha \mathbf{1}_n)$. 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 [11]. Butson Hadamard codes are typically not linear, however under certain conditions they have been shown to have a propelinear structure, [3]. In the latter half of this paper we will require the following definitions.

**Definition 2.3.** The Chinese Euclidean weight $w_{CE}(x)$ of a vector $x \in \mathbb{Z}_q^n$ is

$$\sum_{i=1}^{n} \left[ 2 - 2 \cos \left( \frac{2\pi x_i}{q} \right) \right],$$
and the **Chinese Euclidean distance** between the codewords $u$ and $v \in \mathbb{Z}_q^n$ is defined as

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

**Remark 2.2.** For $q = 2$, we have $w_{CE}(x) = 4w_H(x)$ where $w_H$ denotes the Hamming weight.

**Definition 2.4.** 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:

$$r_{CE}(C_H) = \max\{d_{CE}(x, C_H)\} \mid x \in \mathbb{Z}_q^n\} = \max_{x \in \mathbb{Z}_q^n} \min_{y \in C_H} d_{CE}(x, y).$$

## 2.4 Spherical codes

As is standard, the **Hermitian inner product** of $x$ and $y$ in $\mathbb{C}^n$ is defined as $\langle x, y \rangle = \sum_{j=1}^n x_j \overline{y_j}$. The **(squared) Euclidean distance** of $x$ and $y$ is then

$$d_E(x, y) = \langle x - y, x - y \rangle = \langle x, x \rangle + \langle y, y \rangle - 2\Re(\langle x, y \rangle).$$

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 $u, v \in \mathbb{Z}_q^n$.

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

$$\Upsilon_d = \{x = (x_1, x_2, \dots, x_d) \in \mathbb{R}^d : \|x\| = 1\}.$$

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 $(d, \rho, |X|)$. We denote by $A_d(\rho)$ the largest sized code $X \subseteq \Upsilon_d$ for a given $d$ with respect to $\rho$. That is, the function $A_d(\rho)$ can now be defined as

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

# 3 Associated digraphs and their automorphisms

Let $H \in BH(n, q)$. Our intention will be to construct the automorphism and strong automorphism groups of $H$. Inspired by [29, §3.1], we attach to $H$ a digraph $G(H)$ on $2qn$ vertices as follows.

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

The analogue of [29, Corollary 1] is stated below.

**Proposition 3.1.** Given $H \in BH(n, q)$, the automorphism group Aut($H$) is isomorphic to the automorphism group of $G(H)$.

*Proof.* Let $G(H) = (V, E)$ denote the digraph attached to $H$. Let $f: V \to V$ be 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

$$\begin{aligned} f(r(t, x)) &= r(t', y_t \cdot x) \\ f(c(s, x)) &= c(s', z_s \cdot x) \end{aligned}.$$
Thus, for every automorphism $f$ we can define two matrices $M, N \in \mathrm{Mon}_n(q)$, where $M = PD_M$ with $P$ the permutation matrix defined by $\pi_r(t) = t'$ and $D_M$ is the diagonal matrix with $t$-th entry equal to $\bar{y}_t$. Similarly, $N = QD_N$ with $Q$ the permutation matrix defined by $\pi_c(s) = s'$ and $D_N$ is the diagonal matrix with $s$-th entry equals to $\bar{z}_s$. Since $G(H) = G(MHN^*)$, it follows that $H = MHN^*$. $\square$

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 [29]. Define a modified digraph $G'(H)$ 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 of length two via an extra vertex $I_{s,x}$ say. The first claim of the following proposition is an analogue of [29, Corollary 2]. Moreover, we extend it to strong automorphism groups with any multiplier by means of the digraph $G^k(H) = (V^k, E^k)$ where $V^k = V \cup \{I(s, x) : 1 \le s \le n, x \in \Omega_q\}$ and $E^k = E \cup \{(r(s, y), I(s, x)), (I(s, x), c(s, x)) : 1 \le s \le n, x \in \Omega_q, y = x^k\}$.

**Proposition 3.2.** *The strong automorphism group of H with trivial multiplier $SAut(H, 1)$ is isomorphic to the automorphism group of $G'(H)$. In general, $SAut(H, k)$ is isomorphic to the automorphism group of $G^k(H)$ for $1 \le k \le q - 1$.*

*Proof.* For $SAut(H, 1)$ we consider the digraph $G'(H) = (V', E')$ defined above. If $f: V' \to V'$ an automorphism of $G'(H)$ then, by an argument on the out/in-degree of the vertices, $f$ also stabilizes the set of vertices $I(s, x)$. That is,

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

taking into account that $(f(r(s, x)), f(I(s, x)))$ and $(f(I(s, x)), f(c(s, x)))$ belong to $E'$, then we have $\pi_r(s) = \pi_c(s)$ and $y_s = w_{s,x} = z_s$. Therefore, $M = N$ where $(M, N)$ 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 $M = \mu_k(N)$ where $MHN^* = H$. As a consequence, $SAut(H, k)$ is isomorphic to the automorphism group of $G^k(H)$. $\square$

The automorphism group of $G'(H)$ can then be computed using standard graph theoretic programs like, e.g. Nauty [22].

# 4 Existence conditions

The existence conditions on a $BH(n, q)$ are well documented in the literature, and we refer the reader to the online catalogue of Complex Hadamard matrices at [6] and the numerous references contained within it for up to date research on this topic. In this section we derive some necessary conditions on the existence of an element of $BH(n, q)$ admitting a self-dual bent sequence with multiplier $\mu_k$. The following result is inspired by [10, Lemma 5.2].

**Theorem 4.1.** If $H$ in $BH(n, q)$ admits a self-dual bent sequence $X \in \Omega_q^n$ with multiplier $\mu_k$ then there are $q$ nonnegative integers $y_r \le n$ 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.* Expanding the first row of the matrix equation $HX = \lambda\mu_k(X)$ where $|\lambda| = \sqrt{n}$ yields, after dividing by $x_1^k$, the relation

$$\sum_{j=1}^{n} H_{1j} \frac{x_j}{x_1^k} = \lambda.$$
Let $y_r$ denote the number of occurrences of $\zeta_q^r$ among the summands in the above sum. By definition

$$ \sum_{r=0}^{q-1} y_r = n, \text{ where } 0 \le y_r \le n $$

and it follows immediately that

$$ \sum_{r=0}^{q-1} y_r \zeta_q^r = \lambda. $$

Taking the norm yields

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

where $a = \Re(\lambda)$ and $b = \Im(\lambda)$ are the real and imaginary parts of $\lambda$ respectively. $\square$

In the literature of combinatorics, the sequence $(y_r)_{r=0}^{q-1}$ is called a **composition** of $n$ into $q$ parts. By considering the different possible compositions of $n$ into $q$ parts we can apply Theorem 4.1 to eliminate the possibility of a $BH(n, q)$ admitting a self-dual bent sequence for certain $n$ and $q$.

**Example 4.1.** Let $n = 6$ and $q = 3$. For any $a + bi = y_0 + y_1\zeta_3 + y_2\zeta_3^2$ the quantity $a^2 + b^2$ is preserved by any permutation of the coefficients $y_0, y_1, y_2$. Up to permutation there are 7 compositions we need to check. The values of $a^2 + b^2$ for these compositions all come from the set $\{0, 3, 9, 12, 21, 36\}$, which does not contain 6.

More generally, we may need to check more of the $\binom{n+q-1}{n}$ compositions of $n$ into $q$ parts, as not all permutations of the coefficients preserve the norm. The condition in Theorem 4.1 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 $BH(21, 3)$. We compile computational results of values $a^2 + b^2$ attainable for compositions of $n$ into $q$ parts for some small values of $n$ and $q$ in Table 1.

<table><caption>Table 1: Compositions of $n$ into $q$ parts</caption><thead><tr><th>n</th><th>q</th><th># of compositions</th><th>Values of a<sup>2</sup> + b<sup>2</sup></th></tr></thead><tbody><tr><td>2</td><td>2</td><td>3</td><td>0;4</td></tr><tr><td>6</td><td>3</td><td>28</td><td>0;3;9;12;21;36</td></tr><tr><td>6</td><td>4</td><td>84</td><td>0;2;4;8;10;16;18;20;26;36</td></tr><tr><td>6</td><td>6</td><td>462</td><td>0;1;3;4;7;9;12;13;16;19;21;25;27;28;31;36</td></tr><tr><td>8</td><td>2</td><td>9</td><td>0;4;16;36;64</td></tr><tr><td>8</td><td>4</td><td>165</td><td>0;2;4;8;10;16;18;20;26;32;34;36;40;50;64</td></tr><tr><td>10</td><td>6</td><td>3003</td><td>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</td></tr><tr><td>12</td><td>2</td><td>13</td><td>0;4;16;36;64;100;144</td></tr><tr><td>12</td><td>4</td><td>455</td><td>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</td></tr><tr><td>12</td><td>6</td><td>6188</td><td>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</td></tr><tr><td>13</td><td>6</td><td>8568</td><td>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</td></tr><tr><td>14</td><td>4</td><td>680</td><td>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</td></tr></tbody></table>
<table><tr><td>14</td><td>6</td><td>11628</td><td>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</td></tr><tr><td>21</td><td>3</td><td>253</td><td>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</td></tr></table>

# 5 General constructions

In this section we collect constructions of self-dual bent sequences and demonstrate the existence of families of self-dual bent sequences associated to different Butson Hadamard matrices. The simplest construction of self-dual bent sequences is recursive. Further constructions of self-dual bent sequences associated to matrices $BH(n, q)$ with multiplier $\mu_{q-1}$ are detailed in [4], including constructions of large quantities of self-dual bent sequences using Bush type Butson Hadamard matrices as a powerful tool. We note the relevant results from [4] here.

## 5.1 Kronecker product

If we preserve 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. $BH(m, q)$) admitting a self-dual bent sequence $X$ (resp. $Y$) with 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 $\mu_k(X) \otimes \mu_k(Y) = \mu_k(X \otimes Y)$. $\square$

<figure><img src="image_3.png" /></figure>

## 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.

**Proposition 5.2.** If $H$ is a regular Butson Hadamard matrix of order $n$, then for all $u \in \Omega_q$, we have that $u1_n$ 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 $H1_n = \sigma1_n$. The result follows by linearity. $\square$

<figure><img src="image_4.png" /></figure>

A strong existence condition for regular Butson matrices is Lemma 5.2 in [10]. A special type of regular Hadamard matrix, known as a Bush-type Hadamard matrix, can provide a much greater quantity of self-dual bent sequences due to its very specific block structure. A matrix $H \in BH(n^2, q)$ us said to be of **Bush type** if it can be 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 [16].

**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 $X^\top = (u_11_n, \dots, u_n1_n)$, where the $u_k$'s are arbitrary in $\Omega_q$, is a self-dual bent sequence. $\square$

<figure><img src="image_5.png" /></figure>
**Proposition 5.4** (cf. Proposition 4.5, [4]). If $H \in BH(4m^2, 4)$ is of Bush-type, then there are at least $2^{2m}$ self-dual bent sequences associated to $H$ with trivial multiplier, and at least $2^{2m}$ self-dual bent sequences associated to $-H$ with multiplier $\mu_3$.

**Proposition 5.5** (cf. Proposition 4.6, [4]). Let $q$ be odd and let $H$ be a Bush-type $BH(n^2, q)$, then there exist $q^n$ matrices in $BH(n^2, q)$, each admitting at least one self-dual bent sequence with trivial multiplier, and at least one self-dual bent sequence with multiplier $\mu_{q-1}$.

## 5.3 Fourier transforms

Let $q \ge 2$ and $r \ge 1$ be integers. We define the inner product $\cdot$ on $\mathbb{Z}_q^r$ as

$$ (x_1, \dots, x_r) \cdot (y_1, \dots, y_r) = x_1 y_1 + \dots + x_r y_r, \qquad (1) $$

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 $\Omega_q$ 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

$$ H(x, y) = \zeta_q^{x \cdot y}. \qquad (2) $$

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 $f: \mathbb{Z}_q^r \to \mathbb{Z}_q$ be a function. Let $X = X(f)$ be the $r \times 1$ matrix over $\Omega_q$ 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 all-ones column. Observe that when $X$ is a self-dual bent sequence attached to the conjugate matrix of (2), then $f: \mathbb{Z}_q^r \to \mathbb{Z}_q$ is a generalized bent function (in sense of [25]). 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 $q \ge 2$ and $m \ge 1$ be integers, and let $r = 2m$ and $n = q^r$. Let $H$ be the matrix of order $n$ given in (2). Let $\phi: \mathbb{Z}_q^m \to \mathbb{Z}_q^m$ be a permutation and let $f: \mathbb{Z}_q^r \to \mathbb{Z}_q$ be the map defined as

$$ f(x_1, x_2) = x_1 \cdot \phi(x_2). $$

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 multiplier $\mu_k$ if and only if all of the followings hold:

$$ (1) \quad x_1 \cdot x_2 + k\phi(x_1) \cdot \phi(x_2) = 0 \quad \text{for all } x_1, x_2 \in \mathbb{Z}_q^m. $$

$$ (2) \quad \lambda = q^m. $$

*Proof.* Let $(x_1, x_2) \in \mathbb{Z}_q^r$. The $(x_1, x_2)$-th entry of $HX$ is

$$ \begin{aligned} \sum_{y_1 \in \mathbb{Z}_q^m} \sum_{y_2 \in \mathbb{Z}_q^m} \zeta_q^{(x_1, x_2) \cdot (y_1, y_2)} \zeta_q^{f(y_1, y_2)} &= \sum_{y_1 \in \mathbb{Z}_q^m} \sum_{y_2 \in \mathbb{Z}_q^m} \zeta_q^{x_1 \cdot y_1 + x_2 \cdot y_2 + y_1 \cdot \phi(y_2)} \\ &= \sum_{y_2 \in \mathbb{Z}_q^m} \zeta_q^{x_2 \cdot y_2} \sum_{y_1 \in \mathbb{Z}_q^m} \zeta_q^{y_1 \cdot (x_1 + \phi(y_2))} = q^m \zeta_q^{\phi^{-1}(-x_1) \cdot x_2}. \end{aligned} $$

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

$$ \phi^{-1}(-x_1) \cdot x_2 = kx_1 \cdot \phi(x_2) \quad \text{for all } x_1, x_2 \in \mathbb{Z}_q^m. $$

Consider the change of variable $x'_1 = \phi^{-1}(-x_1) \iff \phi(x'_1) = -x_1$. Using this change of variable, the last condition is equivalent to the condition that $x_1 \cdot x_2 + k\phi(x_1) \cdot \phi(x_2) = 0$ for all $x_1, x_2 \in \mathbb{Z}_q^m$. $\square$
The following Corollary is immediate.

**Corollary 5.1.** Assume the notation of Theorem 5.1. Consider the set

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

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 $1 \le d \le q-1$ 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

$$x_1 \cdot x_2 + k\phi(x_1) \cdot \phi(x_2) = (1 + kd^2)x_1 \cdot x_2$$

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 $HX = q^m \bar{X}$.

## 5.4 Group invariant matrices

Next, we consider another special type of regular Butson Hadamard matrices known as group invariant Butson Hadamard matrices, admitting self-dual bent sequences. A matrix $H$ with rows and columns indexed by the elements of an (additive) group $G$ is **group invariant** if, up to permutations of rows and columns, $H_{g,k} = H_{g+\ell,k+\ell}$ for all $g, \ell, k \in G$, see e.g., [9]. Note that other definitions are used by various authors in the literature. It is known that if $H \in BH(n, q)$ is group invariant, then $G$ is isomorphic to a subgroup of $PAut(H)$ acting regularly on the rows and columns of $H$, see e.g., [19, Chapter 10].

Let $q \ge 2$ and $m \ge 1$ be integers and let $r = 2m$ and $n = q^r$. Let $H$ 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

$$H((x_1, x_2), (y_1, y_2)) = \zeta_q^{(x_1 - y_1) \cdot (x_2 - y_2)} = \zeta_q^{g((x_1, x_2) - (y_1, y_2))} \quad \text{for } g \text{ from Remark 5.1.} \quad (3)$$

This implies that

$$H((x_1 + a_1, x_2 + a_2), (y_1 + a_1, y_2 + a_2)) = H((x_1, x_2), (y_1, y_2))$$

for all $(a_1, a_2) \in \mathbb{Z}_q^r$ and hence the matrix $H$ defined in (3) is $\mathbb{Z}_q^r$-group invariant in the sense of [9].

**Proposition 5.6.** The matrix $H$ defined by Equation (3) is Butson Hadamard.

*Proof.* 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

$$\sum_{y_1 \in \mathbb{Z}_q^m} \sum_{y_2 \in \mathbb{Z}_q^m} \zeta_q^{(x_1 - y_1) \cdot (x_2 - y_2)} \zeta_q^{-(y_1 - z_1) \cdot (y_2 - z_2)} = \sum_{y_1 \in \mathbb{Z}_q^m} \sum_{y_2 \in \mathbb{Z}_q^m} \zeta_q^{x_1 \cdot x_2 - z_1 \cdot z_2 + y_1 \cdot (z_2 - x_2) + y_2 \cdot (z_1 - x_1)} \\ = \zeta_q^{x_1 \cdot x_2 - z_1 \cdot z_2} \sum_{y_1 \in \mathbb{Z}_q^m} \zeta_q^{y_1 \cdot (z_2 - x_2)} \sum_{y_2 \in \mathbb{Z}_q^m} \zeta_q^{y_2 \cdot (z_1 - x_1)}.$$

These arguments and the usual orthogonality relations imply that $HH^* = q^r I_n$. □

Note that the matrix defined in (2) 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 $g: \mathbb{Z}_q^n \to \mathbb{Z}_q$ is a generalized bent function (see [25, Proposition 2.3]).
Now we are ready to present the analogous result of Theorem 5.1 for the group invariant matrix H defined in (3). We also show that H is a Butson type Hadamard matrix in the next theorem.

**Theorem 5.2.** Let $q \ge 2$ and $m \ge 1$ be integers, and let $r = 2m$ and $n = q^r$. Let H be the matrix of order $n$ given in (3). Let $\phi: \mathbb{Z}_q^m \to \mathbb{Z}_q^m$ be a permutation and let $f: \mathbb{Z}_q^r \to \mathbb{Z}_q$ be the map, different from Theorem 5.1, defined as

$$f(x_1, x_2) = x_1 \cdot \phi(x_2) - x_1 \cdot x_2,$$

where $\phi$ is non-trivial so $f$ is not the zero map. 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 multiplier $\mu_k$ if and only if all of the followings hold:

$$(1) \quad kx_1 \cdot \phi^2(x_2) - (k+1)x_1 \cdot \phi(x_2) + x_1 \cdot x_2 = 0 \quad \text{for all } x_1, x_2 \in \mathbb{Z}_q^m.$$

$$(2) \quad \lambda = q^m.$$

*Proof.* Let $(x_1, x_2) \in \mathbb{Z}_q^r$. The $(x_1, x_2)$-th entry of $HX$ is

$$\sum_{y_1 \in \mathbb{Z}_q^m} \sum_{y_2 \in \mathbb{Z}_q^m} \zeta_q^{(x_1-y_1) \cdot (x_2-y_2)} \zeta_q^{f(y_1, y_2)} = \sum_{y_1 \in \mathbb{Z}_q^m} \sum_{y_2 \in \mathbb{Z}_q^m} \zeta_q^{x_1 \cdot x_2 + y_1 \cdot y_2 - x_1 \cdot y_2 - x_2 \cdot y_1 - y_1 \cdot y_2 + y_1 \cdot \phi(y_2)} \\ = \zeta_q^{x_1 \cdot x_2} \sum_{y_2 \in \mathbb{Z}_q^m} \zeta_q^{-x_1 \cdot y_2} \sum_{y_1 \in \mathbb{Z}_q^m} \zeta_q^{y_1 \cdot (\phi(y_2) - x_2)} = q^m \zeta_q^{x_1 \cdot x_2 - x_1 \cdot \phi^{-1}(x_2)}.$$

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

$$x_1 \cdot x_2 - x_1 \cdot \phi^{-1}(x_2) = -kx_1 \cdot x_2 + kx_1 \cdot \phi(x_2) \quad \text{for all } x_1, x_2 \in \mathbb{Z}_q^m.$$

By a suitable change of variables as in the proof of Theorem 5.1, the last condition is equivalent to the condition that $kx_1 \cdot \phi^2(x_2) - (k+1)x_1 \cdot \phi(x_2) + x_1 \cdot x_2 = 0$ for all $x_1, x_2 \in \mathbb{Z}_q^m$. $\square$

The following corollary is immediate.

**Corollary 5.2.** Assume the notation of Theorem 5.2, with the condition that $q \ge 3$. Consider the set

$$S = \left\{ \frac{1}{d} : 1 \le d \le q-1, \text{gcd}(d, q) = 1, d \neq 1 \right\}.$$

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 $1 \le d \le q-1$ 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

$$kx_1 \cdot \phi^2(x_2) - (k+1)x_1 \cdot \phi(x_2) + x_1 \cdot x_2 = (kd^2 - (k+1)d + 1) x_1 \cdot x_2 = (kd-1)(d-1)x_1 \cdot x_2$$

for all $x_1, x_2 \in \mathbb{Z}_q^m$. If $k \in S$, then $kd-1=0$, which completes the proof. $\square$

**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 $m \ge 1$ is any integer, there are other maps $\phi$ providing self-dual bent sequences satisfying the conditions of Theorem 5.2.

# 6 Computational techniques

In this section we extend the computational construction methods of self-dual bent sequences of [29, 30] to Butson matrices. These algorithms can be applied given some matrix $H \in BH(n, q)$. The
first uses Gröbner basis techniques. We refer to [2] for background reading on the topic. The second approach uses linear algebra techniques. These techniques are applied to the Butson matrices that are available in the online database at [6].

## 6.1 Gröbner bases

The system $HX = \lambda\mu_k(X)$ with $X \in \Omega_q^n$ can be solved using Gröbner bases, as it is equivalent to a polynomial system, which, instead of being quadratic as in [29] is now of degree $q$. More concretely, we can consider the following steps.

(1) Construct the ring $P$ of polynomial functions in $n$ variables $X_j, j = 1, \dots, n$.

(2) Construct the linear constraints $HX = \lambda\mu_k(X)$.

(3) Construct the constraints $X_j^q = 1$ for all $j \in \{1, 2, \dots, n\}$.

(4) Compute a Gröbner basis for the ideal $I$ of $P$ determined by constraints of (2) and (3).

(5) Compute the solutions as the zeros determined by $I$.

**Remark 6.1.** As is well-known, the complexity of computing Gröbner bases can be doubly exponential in the number of variables [2]. However, for the system at hand, it might at most be simply exponential [13]. Moreover, this algorithm can be modified to search for self-dual bent sequences under the assumption that they are preserved under some automorphism group, which would reduce the number of variables so searches in higher dimensions may be feasible.

## 6.2 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

$$ \mu_k^j(H)\mu_k^j(X) = \mu_k^j(\lambda)\mu_k^{j+1}(X), j = 1, 2, \dots, t-1. $$

By successive substitution, we get

$$ \prod_{j=0}^{t-1} \mu_k^{t-1-j}(H)X = \prod_{j=0}^{t-1} \mu_k^{t-1-j}(\lambda)X. $$

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

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

(2) Compute $\prod_{j=0}^{t-1} \mu_k^{t-1-j}(H)$ and $\prod_{j=0}^{t-1} \mu_k^{t-1-j}(\lambda)$.

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

(4) For some $m \le n$, let $B$ denote an $m \times n$ matrix with rows such a basis. Pick $B_m$, a $m \times m$ submatrix of $B$ that is invertible, by the algorithm given below.

(5) For all $Z \in \Omega_q^l$ solve the system in $C$ given by $Z = CB_l$.

(6) Compute the remaining $n-m$ entries of $CB$.

(7) If these entries are in $\Omega_q$ declare $CB$ a self-dual bent sequence attached to $\prod_{j=0}^{t-1} \mu_k^{t-1-j}(H)$.

(8) Sieve out the self-dual bent sequence that satisfies $HX = \lambda\mu_k(X)$.
**Algorithm to construct $B_m$**: We apply a greedy algorithm. We construct the list $L$ of the indices of the columns of $B_m$ as follows.

(i) Initialize $J$ at $J = [1]$.

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

(iii) If $r < r'$, then update $J := J'$.

(iv) Repeat until $|J| = m$.

**Remark 6.2.** If the first column of $B$ is zero, step (i) is redundant, but there is no self-dual bent sequence in that situation, as all eigenvectors have first coordinate zero.

**Remark 6.3.** 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.

## 6.3 Numerical results

In this section we apply the two previous constructions to the matrices of database [6], to construct and in many cases classify the self-dual bent sequences with all different possible multipliers. In Table 2 we provide the total number of self-dual bent sequences found attached to matrices in $BH(n, q)$ for distinct pairs $(n, q)$ and constants $\lambda$. Blanks in the table mean that we could not conclude our algorithm, due to the computational burden. For the sake of brevity we did not further categorise them here according to different possible multipliers. For more detailed results with this categorisation complete, we refer the reader to the weblink found at the bottom line of http://math.ahu.edu.cn/smj/list.htm. Nevertheless, we highlight here some interesting examples where no self-dual bent sequences with trivial multiplier exist, but we do find self-dual bent sequences with multiplier $\mu_k$ when $k \neq 1$. These examples motivate the introduction of this more general definition.

**Example 6.1.** The following matrices listed in [6] admit self-dual bent sequences with multiplier $\mu_k$. This list is not exhaustive, only examples where $n \le 10$ are included.

* The $BH(7, 7)$ admits sequences when $k \in \{3, 5, 6\}$.

* The $BH(8, 8)$ admits numerous sequences, only when $k = 7$.

* The $BH(9, 9)$ admits sequences when $k \in \{2, 5, 8\}$.

* The $BH(9, 10)$ admits numerous sequences, only when $k = 3$.

* The $BH(10, 4)$ admits numerous sequences, only when $k = 3$.

For completeness, in Table 3 we recall the distribution of matrices per classes of $BH(n, q)$ in the database [6].

<table><caption>Table 2: Statistics of self-dual bent sequences of Butson matrices</caption><thead><tr><th>n</th><th>q</th><th>#λ</th><th>#X of per λ</th><th>n</th><th>q</th><th>#λ</th><th>#X of per λ</th></tr></thead><tbody><tr><td>2</td><td>2</td><td>0</td><td>0</td><td>3</td><td>3</td><td>1</td><td>1</td></tr><tr><td>4</td><td>2</td><td>2</td><td>2; 2</td><td>4</td><td>4</td><td>3</td><td>8; 8; 4</td></tr><tr><td>5</td><td>5</td><td>2</td><td>6; 10</td><td>6</td><td>3</td><td>0</td><td>0</td></tr><tr><td>6</td><td>4</td><td>0</td><td>0</td><td>6</td><td>6</td><td>0</td><td>0</td></tr><tr><td>7</td><td>6</td><td>5</td><td>6; 4; 6; 0; 6</td><td>7</td><td>7</td><td>2</td><td>9; 3</td></tr><tr><td>8</td><td>2</td><td>0</td><td>0</td><td>8</td><td>4</td><td>0</td><td>0</td></tr></tbody></table>
<table><caption>Table 3: Number of matrices in $BH(n, q)$ listed in the database [6]</caption><tr><td>8</td><td>4</td><td>0</td><td>0</td><td>8</td><td>8</td><td>4</td><td>16; 16; 16; 16</td></tr><tr><td>8</td><td>20</td><td>0</td><td>0</td><td>8</td><td>20</td><td>0</td><td>0</td></tr><tr><td>9</td><td>3</td><td>2</td><td>22; 12</td><td>9</td><td>6</td><td>2</td><td>12; 4</td></tr><tr><td>9</td><td>9</td><td>2</td><td>48; 0</td><td>9</td><td>10</td><td>2</td><td>2; 0</td></tr><tr><td>10</td><td>4</td><td>2</td><td>12; 12</td><td>10</td><td>5</td><td>0</td><td>0</td></tr><tr><td>10</td><td>5</td><td>0</td><td>0</td><td>10</td><td>6</td><td>0</td><td>0</td></tr><tr><td>10</td><td>6</td><td>0</td><td>0</td><td>10</td><td>10</td><td>0</td><td>0</td></tr><tr><td>11</td><td>11</td><td>2</td><td>15; 5</td><td>12</td><td>2</td><td>0</td><td>0</td></tr><tr><td>12</td><td>3</td><td>2</td><td>15; 1</td><td>12</td><td>4</td><td>0</td><td>0</td></tr><tr><td>12</td><td>4</td><td>0</td><td>0</td><td>12</td><td>6</td><td>2</td><td>96; 96</td></tr><tr><td>12</td><td>6</td><td>0</td><td>0</td><td>12</td><td>12</td><td>4</td><td>72; 96; 72;108</td></tr><tr><td>12</td><td>36</td><td>3</td><td>0</td><td>13</td><td>6</td><td>0</td><td>0</td></tr><tr><td>13</td><td>6</td><td>0</td><td>0</td><td>13</td><td>6</td><td>0</td><td>0</td></tr><tr><td>13</td><td>13</td><td>2</td><td>0</td><td>13</td><td>60</td><td>5</td><td>0</td></tr><tr><td>14</td><td>4</td><td>0</td><td>0</td><td>14</td><td>4</td><td>0</td><td>0</td></tr><tr><td>14</td><td>6</td><td>0</td><td>0</td><td>14</td><td>7</td><td>0</td><td>0</td></tr><tr><td>14</td><td>10</td><td>0</td><td>0</td><td>14</td><td>14</td><td>0</td><td>0</td></tr><tr><td>15</td><td>15</td><td>2</td><td>0</td><td>16</td><td>2</td><td>2</td><td>20; 20</td></tr><tr><td>16</td><td>2</td><td>2</td><td>8; 4</td><td>16</td><td>2</td><td>2</td><td>0</td></tr><tr><td>16</td><td>2</td><td>2</td><td>0</td><td>16</td><td>2</td><td>2</td><td>0</td></tr><tr><td>16</td><td>4</td><td>4</td><td>240; 240; 268; 180</td><td>16</td><td>4</td><td>2</td><td>0</td></tr><tr><td>16</td><td>8</td><td>4</td><td>0</td><td>16</td><td>8</td><td>2</td><td>0</td></tr><tr><td>16</td><td>16</td><td>4</td><td>0</td><td>21</td><td>3</td><td>0</td><td>0</td></tr></table>

<table><thead><tr><th>#</th><th>BH(n, q)</th></tr></thead><tbody><tr><td>1</td><td>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)</td></tr><tr><td>2</td><td>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)</td></tr><tr><td>3</td><td>BH(13, 6)</td></tr><tr><td>5</td><td>BH(16, 2)</td></tr></tbody></table>

## 7 Distance spectrum of Butson Hadamard codes

For the remainder of the paper we turn our attention to Butson Hadamard codes and related spherical codes. We refer the reader back to Section 2.3 and Section 2.4 for notation and definitions used in this section.

Let $H \in BH(n, q)$. Recall that $F_H$ is the $\mathbb{Z}_q$-code of length $n$ consisting of the rows of $H$ in logarithmic form and we denote by $C_H = \cup_{\alpha \in \mathbb{Z}_q} (F_H + \alpha \mathbf{1})$. In this section we let $w = \zeta_q$ to ease notation, and denote by $w^{CH}$ the set of all vectors $\{[w^{x_j}]_{1 \le j \le n} | x \in C_H\}$.

In the next proposition we compute the possible Chinese Euclidean distances of the Butson Hadamard code $C_H$, which are also the squared Euclidean distances of the spherical code $\phi(w^{CH})$ to be defined below.
**Proposition 7.1.** The Chinese Euclidean distances of $C_H$ are

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

*Proof.* Observe that $w^{F_H+i}$ for $w^i w^{F_H}$, and that by definition of $C_H$ we have $w^{C_H} = w^{F_H} \cup w^{F_H+1} \cup \dots \cup w^{F_H+q-1}$. The Chinese Euclidean distances of $C_H$ can then be computed as follows. Let $x = w^i h, y = w^j h'$, where $h, h' \in w^{F_H}$ and $i, j \in \{0, 1, \dots, q-1\}$. We have

$$ \langle x, y \rangle = \begin{cases} 0, & h \neq h', \\ nw^t, & h = h', \end{cases} $$

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

The map $\psi: \mathbb{T} \to \mathbb{R}^2, x + iy \mapsto (x, y)$ is an isometry from $(\mathbb{T}^n, d_E)$ to $(\mathcal{Y}_{2n}, \delta)$, where $\delta(U, V) = \|U - V\|^2$, for all $U, V \in \mathcal{Y}_{2n}$. Note that $(\psi(a), \psi(b)) = \Re(\langle a, b \rangle)$, where $a$ and $b \in \mathbb{C}^n$, and $(\cdot, \cdot)$ 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 $\phi(w^{C_H}) \subseteq \mathcal{Y}_{2n} \subseteq \mathbb{R}^{2n}$ has size $nq$ and a distance of

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

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

**Proposition 7.2.** If $H \in BH(n, q)$ exists, then the following inequality holds:

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

We will require a special case of Levenshtein bound [11, 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 $|X| \le L_3(s) = \frac{d(2+(d+1)s)(1-s)}{1-d s^2}$, 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, i.e., matrices in $BH(n, 4)$.

**Corollary 7.2.** If $H \in BH(n, 4)$, then the spherical code $\phi(w^{C_H})$ 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

Finally, we consider lower and upper bounds on the covering radius of Butson Hadamard codes and the related spherical codes developed in the previous section. As it will be clear from context we continue to write $w = \zeta_q$ to ease notation.

## 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

$$ \theta(C, x) = \max_{y \in C} \{|\langle x, y \rangle|\}. $$
**Proposition 8.1.** If $X$ is a bent sequence with multiplier $\mu_k$ attached to $H \in BH(n, q)$, then its corresponding polyphase code $w^{CH}$ 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

$$|\langle h_i, X \rangle| = |\lambda| |\mu_k(X_i)| = \sqrt{n}.$$

Since $|\langle w^t h_i, X \rangle| = |\langle h_i, X \rangle|$ for all $i = 1, \dots, n$ and all $t = 1, \dots, q$ the result follows. $\square$

We consider the polyphase codes attached to some $H \in BH(n, q)$ when $q$ equals 4, 6 and 8. As established earlier, we have $d_{CE}(u, v) = d_E(x, y)$, where $x = w^u$ and $y = w^v$ with $u, v \in \mathbb{Z}_q^n$. Using this identity and expanding $\langle x - y, x - y \rangle$, we obtain

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

for all $x, y \in \Omega_q^n$.

The simple inequality $\Re(\langle x, y \rangle) \le |\langle x, y \rangle|$ in the complex field shows that

$$r_{CE}(C_H) \ge 2n - 2\theta(C, x).$$

Combining this fact with 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

$$r_{CE}(C_H) \ge 2n - 2\sqrt{n}.$$

**Remark 8.1.** If $q = 2$, $n = 2^m$ and $H$ is the Sylvester Hadamard matrix, then $r_{CE}(C_H) \ge 4(2^{m-1} - 2^{m/2-1})$. It is well known that the covering radius of the first order Reed-Muller code, $\rho(R(1, m))$, is equal to $2^{m-1} - 2^{m/2-1}$ when $m$ is even. For $m$ odd, the situation is much more difficult, but its asymptotic behavior was conjectured by Patterson and Widemann in [24], and solved by K-U. Schmidt in [26]. Concretely, he proved that $\rho(R(1, m))$ is asymptotically equivalent to $2^{m-1} - 2^{m/2-1}(1 + o(1))$. By Remark 2.2, we conclude that $r_{CE}(C_H)$ is asymptotically equivalent to $2n - 2\sqrt{n}(1 + o(1))$ for the mentioned values of $q, n$ and $H$.

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><caption>Table 4: The covering radius of $C_H$ for $H \in BH(n, 4)$</caption><thead><tr><th>BH(n, 4)</th><th>k</th><th>Lower bound</th><th>True value</th></tr></thead><tbody><tr><td>BH(4, 4)</td><td>1;3</td><td>4</td><td>4</td></tr><tr><td>BH(10, 4)</td><td>3</td><td>14</td><td>14</td></tr><tr><td>BH(16, 4)(F<sub>4</sub> ⊗ F<sub>4</sub>)</td><td>1;3</td><td>24</td><td>?</td></tr></tbody></table>

<table><caption>Table 5: The covering radius of $C_H$ for $H \in BH(n, 6)$</caption><thead><tr><th>BH(n, 6)</th><th>k</th><th>Lower bound</th><th>True value</th></tr></thead><tbody><tr><td>BH(7, 6)</td><td>5</td><td>9</td><td>9</td></tr><tr><td>BH(9, 6)</td><td>1;5</td><td>12</td><td>12</td></tr></tbody></table>
<table><caption>Table 6: The covering radius of $C_H$ for $H \in BH(n, 8)$</caption><tr><td>BH(12, 6)(F<sub>6</sub> &otimes; F<sub>2</sub>)</td><td>5</td><td>2(12 &minus; &radic;12) &asymp; 17.072</td><td>?</td></tr></table>

<table><thead><tr><td>BH(n, 8)</td><td>k</td><td>Lower bound</td><td>True value</td></tr></thead><tbody><tr><td>BH(8, 8)</td><td>7</td><td>2(8 &minus; &radic;8) &asymp; 10.343</td><td>2(8 &minus; &radic;8)</td></tr></tbody></table>

## 8.2 Upper bounds

In this subsection, we derive upper bounds on the covering radius of the spherical code $\phi(w^{C_H})$ (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.** ([8]) A spherical design of strength $t$, is a spherical code $X$ of size $N$ on the $d$-dimensional unit $d$-sphere $\Upsilon_d$ such that for every polynomial of degree at most $t$ we have

$$ \frac{1}{N} \sum_{x \in X} f(x) = \int_{s \in \Upsilon_d} f(s). $$

It is antipodal if $X = -X$.

We will require the following characterizations for $t = 1, 2$. The proof is omitted.

**Proposition 8.2.** ([32])

(1) A spherical code $X$ is a **1-design** if its center of mass is the origin or, more concretely, if for all coordinate indices $i$ the equation $\sum_{x \in X} x_i = 0$ is satisfied.

(2) 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.

$$ \sum_{x \in X} (x_i^2 - x_j^2) = 0, \quad \sum_{x \in X} x_i x_j = 0. $$

The impact on the covering radius is as described in the following Propositions and Theorems.

**Proposition 8.3.** If $H \in BH(n, q)$ with $q$ is even, then the covering radius of $\phi(w^{C_H})$ is at most $\sqrt{2}$.

*Proof.* The spherical code $\phi(w^{C_H})$ is antipodal, because $-1$ is a power of $w$. The result follows then by [32, Theorem 1]. $\square$

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

**Proposition 8.4.** If $H$ is dephased, then $\phi(w^{C_H})$ is a **1-design** and its covering radius is at most $\sqrt{2}$.

*Proof.* Note that $H^*$ 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 $\sum_{k=0}^{q-1} w^k = 0$. Hence the said spherical code is a 1-design. The result follows by Table 1 of [12]. $\square$

**Theorem 8.1.** If $H \in BH(n, q)$ is dephased, then $\phi(w^{C_H})$ is a 2-design and its covering radius is at most $\sqrt{2(1 - \frac{1}{2n})}$. $\square$
Proof. We have seen in Proposition 8.3 that $\phi(w^{C_H})$ is a 1-design. Denote by $h_j$ the column number $j$ of $H$. 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 $H^*$. 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 $\phi(w^{C_H})$ is a 2-design. The upper bound follows then by [12, 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 $\phi(w^{C_H})$ 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 $\phi(w^{C_H})$ is a 2-design. It is antipodal by the evenness of $q$. The result follows then by Theorem 3 of [32]. □

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.

$$ r_{CE}(C_H) \le 2n - \sqrt{2n} $$

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 and multipliers. 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 with trivial multipier. 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 [26, 27], 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, and we conjecture this to be the case. While this conjecture is only based on five values, it is one more than the number of values known by Patterson and Widemann in [24] to propose the conjecture that K-U. Schmidt eventually solved in [26]. Furthermore, the conjecture of Patterson and Widemann is a particular case of our conjecture, as was shown in Remark 8.1. Hence, K-U Schmidt's work in [26] could be considered as evidence underpinning this conjecture.
On the other hand, some metric properties of the self-dual Boolean bent functions (sequences) in $n$ variables as well as of the anti-self-dual Boolean bent functions (sequences) have been studied by Kutsenko in [17]. He denotes these sets by $SB^+(n)$ and $SB^-(n)$, respectively. The sets of analogs to $SB^+(n)$ and $SB^-(n)$ in our setting can be defined as follows. Given $H \in BH(n,q)$ and a multiplier $\mu_k$, $SB_{H,\mu_k}^\lambda$ denotes the set of self-dual sequences $X$ with $HX = \lambda\mu_k(X)$ for every admissible $\lambda \in \mathbb{Z}[\zeta_q]$. Consider the Chinese Euclidean distance. What can be said about the minimal distance and the metrical regularity of the sets $SB_{H,\mu_k}^\lambda$? We have left this question as an open problem to be treated in a near future.

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

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