Generalized partially bent functions, generalized perfect arrays, and cocyclic Butson matrices

Abstract

In a recent survey, Schmidt compiled equivalences between generalized bent functions, group invariant Butson Hadamard matrices, and abelian splitting relative difference sets. We establish a broader network of equivalences by considering Butson matrices that are cocyclic rather than strictly group invariant. This result has several applications; for example, to the construction of Boolean functions whose expansions are generalized partially bent functions, including cases where no bent function can exist.

1 Introduction

Let $f:{\mathbb{Z}}_{{2}^{n}} \rightarrow {\mathbb {Z}}_{2}$ be a Boolean function with n a positive integer, and set $F(v) = (-1)^{f(v)}$ for $v \in {\mathbb {Z}}_{2}^{n}$ (throughout, we view ${\mathbb {Z}}_t$ for an integer $t>1$ as $\{0,1,\ldots , t-1\}$ under addition modulo t). The Walsh–Hadamard transform $\hat{F}$ of F is defined by

$$ \hat{F}(u) = \sum _{v \in {\mathbb {Z}}_{2}^n} (-1)^{u\cdot v} F(v), $$

where $u\cdot v$ denotes the inner product $uv^{\top}$ of u and v. The Walsh–Hadamard transform is used to analyze cryptographic properties of Boolean functions. A Boolean function f is bent if $|\hat{F}(u)|$ is constant for all $u \in {\mathbb {Z}}_{2}^{n}$. Parseval’s theorem (see, e.g., [5, (8.36), p. 322]) gives

$$ \sum _{v \in {\mathbb {Z}}_{2}^{n}}\hat{F}(v)^2 = 2^{2n}. $$

Hence, f can be bent only if n is even. If the Walsh–Hadamard transform takes no more than one non-zero absolute value, then it is plateaued.

A bent function is so-called because it is as far from being linear as possible. However, bent functions are not balanced (another desirable cryptographic property), while plateaued functions can be balanced and have large nonlinearity. These highly non-linear functions offer a robust defence against linear cryptanalysis [6, Chapter 3].

Bent functions are equivalent to certain Hadamard matrices and difference sets; see, e.g., [11, Lemma 14.3.2] and [6, Corollary 3.30]. The concept has been generalized, yielding equivalences between associated objects. Indeed, our paper is inspired by Schmidt’s survey [16], which describes equivalences between generalized bent functions, group invariant Butson Hadamard matrices, and splitting relative difference sets. There is also a connection to perfect arrays, not covered in [16].

We study how the aforementioned equivalences are affected when the property of being group invariant is broadened to cocyclic development. For example, a group-invariant Butson Hadamard matrix is a type of cocyclic matrix. As a consequence, we incorporate generalized partially bent functions [18], and construct a family of generalized partially bent functions with domain for which no generalized bent functions exist.

We now outline the paper. Preliminary definitions and results are given in Section 2. Section 3 is devoted to generalized perfect arrays and generalized partially bent functions. In Section 4, we prove the main theorem: a series of equivalences between cocyclic Butson Hadamard matrices, generalized perfect arrays, non-splitting relative difference sets, generalized plateaued functions, and generalized partially bent functions. (For certain parameters, the equivalences that we exhibit have those in [16] as special cases.) In Section 5 we give some examples illustrating the main theorem.

2 Background

We adopt the following definition from [16]. For integers $q, m, h>0$, and $\zeta _k$ the complex $k^{\textrm{th}}$ root of unity $\exp {(2\pi \sqrt{-1} /k)}$, a map $f:{\mathbb {Z}}_{q}^m \rightarrow {\mathbb {Z}}_{h}$ is a generalized bent function (GBF) if

$$ \Big|\sum _{x\in {\mathbb {Z}}_{q}^m} \zeta _h^{f(x)} \zeta _q^{-w\cdot x} \Big|^2=q^m\quad \forall \, w\in {\mathbb {Z}}_{q}^m, $$

|z| as usual denoting the modulus of $z\in \mathbb C$. Thus, a GBF for $q=h=2$ and even m is a bent function. For $h=q$, Kumar, Scholtz, and Welch [9] prove that GBFs exist if m is even or $q\not \equiv 2 \, \text{ mod } 4$. However, no GBF with $h=q$, m odd, and $q \equiv 2 \, \text{ mod } 4$ is known [10, p. 2].

A further generalization is relevant to our paper. If the values of

$$ \Big|\sum _{x\in {\mathbb {Z}}_{q}^m} \zeta _h^{f(x)} \zeta _q^{-w\cdot x} \Big|^2 $$

as w ranges over ${\mathbb {Z}}_{q}^{m}$ lie in $\{ 0,\alpha \}$ for a single non-zero $\alpha$, then f is a generalized plateaued function (cf. the definition of plateaued Walsh–Hadamard transform). Mesnager, Tang, and Qi [14] discuss such functions under the conditions that q is prime and h is a q-power. They call f an s-generalized plateaued function when $\alpha$ has the form $q^{m+s}$.

We examine the role of GBFs and generalized plateaued functions in cocyclic design theory [3, 6]. Some requisite definitions follow. Let G and U be finite groups, with U abelian. A map $\psi :G\times G\rightarrow U$ such that

$$ \psi (a,b)\psi (ab,c)=\psi (a,bc)\psi (b,c) \quad \forall \, a,b,c\in G $$

is a cocycle (over G, with coefficients in U). Cocycles $\psi$ are assumed to be normalized, meaning that $\psi (1,1)=1$. For any (normalized) map $\phi :G\rightarrow U$, the cocycle $\partial \phi$ defined by $\partial \phi (a,b)= \phi (a)^{-1}\phi (b)^{-1}\phi (ab)$ is a coboundary. The set of cocycles $\psi :G\times G \rightarrow U$ equipped with pointwise multiplication is an abelian group, $Z^2(G,U)$. Factoring out $Z^2(G,U)$ by the subgroup $B^2(G,U)$ of coboundaries gives the second cohomology group, $H^2(G,U)$. The elements of $H^2(G,U)$, namely cosets of $B^2(G,U)$, are cohomology classes. Each $\psi \in Z^2(G,U)$ is displayed as a cocyclic matrix $M_{\psi}$. That is, under an indexing of rows and columns by the elements of G, the $|G|\times |G|$ matrix $M_{\psi}$ has entry $\psi (a,b)$ in position (a, b). We focus on abelian G and cyclic U; say $G={\mathbb {Z}}_{s_1} \times \cdots \times {\mathbb {Z}}_{s_m}$ and $U=\langle \zeta _h\rangle \cong {\mathbb {Z}}_h$, where $\langle \zeta _{h}\rangle := \{\zeta _{h}^{i} \; | \; 0 \le i \le h-1\}$ is generated (multiplicatively) by $\zeta _h$.

Denote the set of $n \times n$ matrices with entries in a set S by $\mathcal {M}_{n}(S)$. A matrix $M\in \mathcal {M}_{n} (\langle \zeta _{k} \rangle )$ is a Butson (Hadamard) matrix if $M M^*=nI_n$, where $I_n$ is the $n\times n$ identity matrix and $M^*$ is the complex conjugate transpose of M. We write ${\text {BH}}(n,k)$ to denote the (possibly empty) set of all Butson matrices in $\mathcal {M}_{n} (\langle \zeta _{k} \rangle )$. For example, at every order n we have the Fourier matrix $\big[\zeta _n^{(i-1)(j-1)} \big]_{i,j=1}^n\in {\text {BH}}(n,n)$. Hadamard matrices of order n are the elements of ${\text {BH}}(n,2)$. We quote a number-theoretic constraint on the existence of elements of ${\text {BH}}(n,k)$.

Theorem 1

([3, Theorem 2.8.4]) If ${BH}(n,k)\ne \emptyset$ and $p_{1},\ldots ,p_{r}$ are the primes dividing k, then $n = a_{1}p_{1} + \cdots + a_{r}p_{r}$ for non-negative integers $a_1,\ldots , a_r$.

Two matrices $H, H'\in \mathcal {M}_{n} (\langle \zeta _{k} \rangle )$ are equivalent if $PHQ^{*} = H'$ for monomials $P, Q\in \mathcal {M}_n ( \langle \zeta _k\rangle \cup \{0\})$. This equivalence relation induces a partition of ${\text {BH}}(n,k)$.

Our interest is in cocyclic Butson matrices. Let G be a group of order n. A cocycle $\psi \in Z^2(G,\langle \zeta _k\rangle )$ such that $M_{\psi} \in {\text {BH}}(n,k)$ is orthogonal. In particular, group invariant Butson matrices are cocyclic. The orthogonal cocycles involved here are coboundaries, as we now explain. A matrix $X\in \mathcal {M}_{n}(\langle \zeta _k\rangle )$ is group invariant, over G, if $X=[x_{a,b}]_{a,b\in G}$ and $x_{ac,bc} = x_{a,b}$ for all $a,b,c\in G$. Such an X is equivalent to a group-developed matrix $[\chi (ab)]_{a,b\in G}$ for some map $\chi :G\rightarrow \langle \zeta _k\rangle$ (see, e.g., [3, 10.2.2]). In turn $[\chi (ab)]$ and $M_{\partial \chi }$ are equivalent: setting P to be the G-indexed diagonal matrix with $\chi (a)$ in row a, we have $P[\chi (ab)]P^*=M_{\partial \chi }$. A group-developed Butson matrix has constant row and column sum (in $\mathbb C$). Together with Theorem 1, there are strong restrictions on group-developed elements of ${\text {BH}}(n,k)$.

Lemma 1

([4, Lemma 5.2]) Set $r_{j} = \textrm{Re}(\zeta _k^j)$ and $s_{j} =\textrm{Im}(\zeta _k^j)$. A matrix in ${BH}(n,k)$ with constant row and column sums exists only if there are $x_{0},\ldots , x_{k-1} \in \{0,1,\ldots ,n\}$ such that $\big ( \sum _{j=0}^{k-1} r_{j}x_{j}\big )^{2} + \big (\sum _{j=0}^{k-1}s_{j}x_{j}\big )^{2} = n$ and $\sum _{j=0}^{k-1} x_{j} = n$.

It follows from Lemma 1 that if $k=2$ then n is an integer square, and if $k=4$ then n is the sum of two integer squares.

Cocyclic designs give rise to relative difference sets, and vice versa [3, Sections 10.4, 15.4]. Let E be a group with normal subgroup N, where $|N|=n$ and $|E:N|=v$. A (v,n,k,$\lambda$)-relative difference set in E relative to N (the forbidden subgroup) is a k-subset R of a transversal for N in E such that $|R\cap xR|=\lambda$ for all $x\in E\setminus N$. We call R abelian if E is abelian, and splitting if N is a direct factor of E.

The final piece of background concerns arrays. Let $\textbf{s}= (s_1,\ldots ,s_m)$ be an m-tuple of integers $s_i>1$, and let $G={\mathbb {Z}}_{s_1} \times \cdots \times {\mathbb {Z}}_{s_m}$. A h-ary s-array is just a set map $\phi :G\rightarrow {\mathbb {Z}}_h$ (normalized when necessary). If $h=2$, then the array is binary. For $w\in G$, the periodic autocorrelation of $\phi$ at shift w, denoted $AC_{\phi }(w)$, is defined by

$$ AC_{\phi }(w)= \sum _{g\in G} \zeta _h^{\phi (g)-\phi (g+w)}. $$

If $AC_{\phi }(w) = 0$ for all $w \ne 0$, then $\phi$ is perfect.

Lemma 2

Let $D_m$ be the $m^{th}$ Kronecker power of the $q\times q$ Fourier matrix, i.e., $(D_m)_{i,j}= \zeta _q^{\alpha _{i-1} \cdot \alpha _{j-1}}$, where $\alpha _0=(0,\ldots ,0), \alpha _1=(0,0,\ldots , 1),\ldots , \alpha _{q^m-1}=(q-1,\ldots ,q-1)$. Then, for any map $\phi :{\mathbb {Z}}_{q}^m \rightarrow {\mathbb {Z}}_{h}$,

$$ (AC_{\phi }(\alpha _0),\ldots , AC_{\phi }(\alpha _{q^m-1})) D_m = \Big( \Big| \sum _{x\in {\mathbb {Z}}_{q}^m} \zeta _h^{\phi (x)} \zeta _q^{-\alpha _0 \cdot x} \Big|^2, \ldots , \Big |\! \sum _{x\in {\mathbb {Z}}_{q}^m} \zeta _h^{\phi (x)} \zeta _q^{-\alpha _{q^m-1} \cdot x} \Big|^2\Big). $$

Proof

We adapt the proof of the lemma (for Boolean functions) in [2, Section 2]. First,

$$ \sum _{i\ge 0} AC_{\phi} (\alpha _i) \zeta _q^{\alpha _i\cdot \alpha _j}= \sum _{i\ge 0} \sum _{k\ge 0}\zeta _h^{\phi (\alpha _k)- \phi (\alpha _k+\alpha _i) } \zeta _q^{\alpha _i\cdot \alpha _j}. $$

After replacing $\alpha _i$ by $\alpha _i-\alpha _k$, the double summation becomes

$$\begin{aligned} \sum _{i\ge 0} \sum _{k\ge 0} \zeta _h^{\phi (\alpha _k)-\phi (\alpha _i) } \zeta _q^{\alpha _i\cdot \alpha _j- \alpha _k\cdot \alpha _j}&=\sum _{k\ge 0}\zeta _h^{\phi (\alpha _k) } \zeta _q^{- \alpha _k\cdot \alpha _j} \sum _{i\ge 0}\zeta _h^{-\phi (\alpha _i) } \zeta _q^{\alpha _i\cdot \alpha _j}\\ &= \Big |\sum _{x\in {\mathbb {Z}}_{q}^m} \zeta _h^{\phi (x)} \zeta _q^{-\alpha _{j} \cdot x} \Big |^2, \end{aligned}$$

as required.$\square$

Our fundamental motivating result is extracted mostly from [16].

Theorem 2

Let $f:{\mathbb {Z}}_{q}^m \rightarrow {\mathbb {Z}}_{h}$ be a map. The following are equivalent : 

• 1. f is a GBF; 

• 2. $M_{\partial f}\in {B\!H}(q^m,h);$

• 3. f is a perfect h-ary $(q,\ldots , q)$-array.

Additionally, if h is prime and divides $q^m$, then (1)–(3) are equivalent to

• 4. $\{(f(x),x) \; | \; x\in {\mathbb {Z}}_{q}^m\}$ is a splitting $(q^m,h,q^m,q^m/h)$-relative difference set in ${\mathbb {Z}}_{h}\times {\mathbb {Z}}_{q}^m$.

Proof

The equivalences $(1)\Leftrightarrow (2) \Leftrightarrow (4)$ come from Propositions 2.3 and 2.7 of [16] (h prime is a sufficient condition to ensure $(2)\Rightarrow (4)$). Lemma 2 implies $(1)\Leftrightarrow (3)$. $\square$

We investigate the effect on Theorem 2 when non-coboundary cocyclic Butson matrices, generalized perfect arrays, and non-splitting abelian relative difference sets are considered in (2), (3), (4), respectively. To this end, we need some material of a more specialized nature, which is presented over the next two sections.

3 More on arrays and bent functions

There is an equivalence between binary arrays and non-splitting abelian relative difference sets, as set out in [8]. Subsequently, a bridge to the theory of cocyclic Hadamard matrices was identified [7]. The main tool here is the notion of a generalized perfect binary array (GPBA). Guided by [1, Section 3], we extend the notion of GPBA from binary to h-ary arrays, $h\ge 2$, and show how this conforms with a variant of bent functions.

Definition 1

Let $\phi :G \rightarrow \mathbb {Z}_{h}$ be an s-array, where $\textbf{s}= (s_1,\ldots ,s_m)$ and $G={\mathbb {Z}}_{s_1} \times \cdots \times {\mathbb {Z}}_{s_m}$. Let $\textbf{z}=(z_1,\ldots ,z_m)\in \{ 0,1\}^m$. The expansion of $\phi$ of type $\textbf{z}$ is the map $\phi '$ from $E:= \mathbb {Z}_{(z_1(h-1)+1)s_1} \times \cdots \times \mathbb {Z}_{(z_m(h-1)+1)s_m}$ to $\mathbb {Z}_{h}$ defined by

$$ \phi ':(g_1, \ldots , g_m) \, \mapsto \, \phi (a) +b \, \text{ mod } h , $$

where $b=\sum _{i=1}^m \lfloor g_i/s_i\rfloor$ and $a\equiv (g_1, \ldots , g_m) \, \text{ mod } \mathbf{{s}}$, i.e., $a = (g_1 \, \text{ mod } s_1, \ldots , g_m \, \text{ mod } s_m)$.

We distinguish two subgroups of the extension group E in Definition 1:

$$\begin{array}{l} L=\{(g_1,\ldots ,g_m)\in E \, \, | \,\, g_i=y_is_i \, \text {with } 0\le y_i< h\text { if }z_i=1, \text { and }y_i=0\text { if } z_i=0\},\\ K=\{ (g_1,\ldots ,g_m)\in L\,\, |\,\, {\sum _i (g_i/s_i) \equiv 0 \, \text{ mod } h}\}. \end{array} $$

Note that

• $L\cong {\mathbb {Z}}_h^n$ where $n=\textrm{wt}(\textbf{z})=\sum _i z_i$;

• $E/L\cong G$;

• if $\textbf{z}\ne \textbf{0}$ then $L/K= \langle (0,\ldots , 0, s_i, 0 , \ldots , 0) +K\rangle \cong {\mathbb {Z}}_h$, for any i such that $z_i=1$.

With these subgroups of E now defined, we will be able to see how the expansion of an $\textbf{s}$-array is natural, and how it allows us to generalize the notion of perfect array.

Lemma 3

Let $\phi$ be a h-ary $(s_1,\ldots , s_m)$-array with expansion $\phi ':E\rightarrow {\mathbb {Z}}_h$. If $e\in E$ and $g= (g_1,\ldots , g_m)\in L$, then $\phi '(e+g)\equiv \phi '(e)+b \, \text{ mod } h$ where $b=\sum _i g_i/s_i$.

Proof

This is routine, from the definitions. $\square$

Corollary 1

Under the hypotheses of Lemma 3, $AC_{\phi '}(g)= \zeta _h^{-b} |E|$ for any $g\in L$.

Definition 2

A h-ary $\textbf{s}$-array $\phi$ with expansion $\phi ':E\rightarrow {\mathbb {Z}}_h$ of type $\textbf{z}$ is generalized perfect if $AC_{\phi '}(g)= 0$ for all $g \in E\setminus L$; in short, $\phi$ is a $\textrm{GPhA}(\textbf{s})$ of type $\textbf{z}$. We write $\textrm{GPhA}(c^m)$ when $\textbf{s}$ is the vector $(c,\ldots , c)$ of length m for a constant c.

So a GPhA$(\textbf{s})$ of type $\textbf{0}$ is exactly a perfect h-ary $\textbf{s}$-array.

Definition 3

(cf. [18, Definition 2.2]) A map $f:{\mathbb {Z}}_{q}^m \rightarrow {\mathbb {Z}}_{h}$ such that $|AC_f(x)|\in \{ 0, q^m\}$ for all $x\in \mathbb {Z}_{q}^m$ is a generalized partially bent function (GPBF).

Let $\phi$ be a h-ary $(q,\ldots , q)$-array of type $\textbf{1}$. By Corollary 1, $|\phi '(g)|=(hq)^m \, \forall g\in L$. If $\phi$ is generalized perfect, then by definition $|\phi '(g)|=0 \, \forall x\in \mathbb {Z}_{hq}^m\setminus L$, so $\phi '$ is generalized partially bent. However, the converse does not hold, as evidenced by the following simple example. Define $\phi :{\mathbb {Z}}_{2}^2 \rightarrow {\mathbb {Z}}_{2}$ by $\phi (0,1)=1$ and $\phi (0,0)=\phi (1,0)=\phi (1,1)=0$. The expansion of $\phi$ of type $\textbf{1}$ is a GPBF, but $\phi$ is not a $\textrm{GP2A}(2^2)$ of type $\textbf{1}$ (writing $\textbf{1}$ for the all 1s vector). We obtain the converse by imposing more conditions.

Proposition 1

Let $\phi$ be an array ${\mathbb {Z}}_{h}^m \rightarrow {\mathbb {Z}}_{h}$ such that for each $y=(y_1,\ldots , y_m)\in {\mathbb {Z}}_{h}^m \setminus \{\textbf{0}\}$ with $\sum _i y_i\equiv 0 \, \text{ mod } h$, there exists $x=(x_1,\ldots ,x_m)\in {\mathbb {Z}}_{h}^m$ satisfying

$$\begin{aligned} \phi (x+y)+ \sum _i \left\lfloor (x_i+y_i)/h\right\rfloor \not \equiv \phi (x)+\phi (y) \, \text{ mod } h. \end{aligned}$$

(1)

Then the expansion $\phi '$ of $\phi$ of type $\textbf{1}$ is a GPBF if and only if $\phi$ is a GPhA$(h^m)$ of type $\textbf{1}$.

Proof

In this proposition, $E = {\mathbb {Z}}_{h^2}^m$ and $L = \{ 0, h, \ldots , (h-1)h\}^m \cong {\mathbb {Z}}_h^m$. Suppose that $\phi '$ is a GPBF. Then $\phi$ is a $\textrm{GPhA}(h^m)$ if $|AC_{\phi '}(g)|<h^{2m}$ for all $g\in E\setminus L$. So we prove that $\phi^{\prime}(w)-\phi^{\prime}(w+g) \not \equiv \phi^{\prime}(x)- \phi^{\prime}(x+g)\, \text{ mod } h$ for some $w, x\in E$. Taking $w=0$, and assuming that $\phi$ is normalized, this non-congruence becomes $\phi '(x+g)\not \equiv \phi '(x)+ \phi '(g)$.

Suppose that

$$ \phi^{\prime}(0)-\phi^{\prime}(g), \, \phi^{\prime}(g)-\phi ^{\prime}(2g) , \, \ldots , \, \phi^{\prime}((h-1)g) -\phi ^{\prime}(h g) $$

are all congruent modulo h (otherwise, the required x may be found as a multiple of g). Adding these h terms gives

$$ \phi^{\prime}(0)-\phi^{\prime}(hg) \equiv 0 \, \text{ mod } h \, \Rightarrow \, \phi ^{\prime}(hg) \equiv 0 \, \text{ mod } h. $$

Consequently $\sum _i g_i\equiv 0\, \text{ mod } h$.

If $g=(g_1,\ldots , g_m)$ with $0\le g_i< h$, then the right-hand side of (1) for $y=g$ is $\phi '(x)+ \phi '(g)$, and the left-hand side is $\phi^{\prime}(x+g)$; so we are done.

Now let $g=a+l$ with $a = (g_1 \, \text{ mod } h, \ldots , g_m \, \text{ mod } h)$ and $l\in L$. Then $\sum _i a_i\equiv 0$, because $hg = ha$ in E. Using Lemma 3, and adding $b=\sum _i l_i/h$ to both sides of (1) for $y=a$, we see that $\phi^{\prime}(x+g)\not \equiv \phi^{\prime}(x) + \phi^{\prime}(g)$. This completes the proof. $\square$

4 Equivalences between arrays, bent functions, and associated combinatorial objects

Let $\textbf{s}$, $\textbf{z}$, G, K, L, E be as in Section 3, with $\textbf{z}\ne \textbf{0}$. We have a short exact sequence

$$\begin{aligned} 1 \longrightarrow \langle \zeta _h\rangle \overset{\iota }{\longrightarrow} E/{K} \overset{\beta}{\longrightarrow} G \longrightarrow 0 , \end{aligned}$$

(2)

where $\beta (g+K)\equiv g \, \text{ mod } \mathbf{{s}}$ and $\iota$ sends $\zeta _h$ to a generator of $L/K\cong \mathbb {Z}_h$. In the standard way we extract a cocycle $\mu _{\textbf{z}}\in Z^2(G,\langle \zeta _h\rangle )$ from (2), depending on the choice of a transversal map $\tau :G\rightarrow E/K$ (see, e.g., [3, $\S \,12.1.3$]). Set $\tau (x) = x+K$ (a mild abuse of notation), so that $\beta \circ \tau = \textrm{id}_G$; then $\mu _{\textbf{z}}(x, y)= \iota ^{-1}(\tau (x)+ \tau (y) - \tau (x+y))$.

Proposition 2

(cf. [7, Lemma 3.1]) Define $\gamma _t\in Z^2(\mathbb {Z}_t,\langle \zeta _h\rangle )$ by $\gamma _t(j,k)= \zeta _h^{\lfloor (j+k)/t\rfloor }$. Then

• (i) $\mu _{\textbf{z}}(x,y)=\prod _{i \, { \text {with}} \, z_i=1} \gamma _{s_i}(x_i,y_i);$

• (ii) $\mu _{\textbf{z}}\in B^2(G,\langle \zeta _h\rangle )$ if and only if $s_{i}$ is coprime to h whenever $z_{i} =1$.

In the opposite direction, each cocycle $\psi \in Z^2(G, \langle \zeta _h\rangle )$ determines a central extension $E_{\psi }$ of $\langle \zeta _h\rangle$ by G: namely, the group with elements $\{ (\zeta _h^j,g) \mid 0\le j< h,\,g\in G\}$ and multiplication defined by $(u,g)(v,h)=(uv\,\, \psi (g,h),gh)$. More properly, the central extension is the short exact sequence

$$\begin{aligned} 1\longrightarrow \langle \zeta _h\rangle \overset{\iota '}{\longrightarrow} E_{\psi } \overset{\beta '}{\longrightarrow} G \longrightarrow 0, \end{aligned}$$

(3)

where $\iota '(u)=(u,0)$ and $\beta '(u,x)=x$.

The next two results mimic Proposition 4 and Lemma 3 of [1], respectively.

Proposition 3

If $\mu _{\textbf{z}}$ and $\psi \in Z^2(G,\langle \zeta _h\rangle )$ are in the same cohomology class, say $\psi =\mu _{\textbf{z}}\partial \phi$, then (2) and (3) are equivalent as short exact sequences. Specifically, for the transversal map $\tau$ as defined before Proposition 2, the map $\Gamma$ sending $(u,x)\in E_{\psi}$ to $\iota (u\phi (x)^{-1})+\tau (x) \in E/K$ is an isomorphism that makes the diagram

$$ \begin{array}{ccccccccc} 1 &{} \longrightarrow &{} \langle \zeta _h\rangle &{} \overset{\iota^{\prime}}{\longrightarrow} &{} E_{\psi } &{} \overset{\beta^{\prime}}{\longrightarrow}&{} G &{} \longrightarrow &{} 0\\ &{} &{} \Vert &{} &{} {{\Gamma }} \downarrow &{} &{} \Vert &{} &{} \\ 1 &{} \longrightarrow &{} \langle \zeta _h\rangle &{} \overset{\iota }{\longrightarrow}&{} E/K &{} \overset{\beta }{\longrightarrow } &{} G &{} \longrightarrow &{} 0 \end{array} $$

commute.

Remark 1

In Proposition 3, the $\phi$ has multiplicative target group $\langle \zeta _h\rangle$. When considering $\phi$ as an array, we may replace the multiplicative group $\langle \zeta _h\rangle$ by the additive group ${\mathbb {Z}}_h$, without bothering to change notation. Likewise, note that $E_{\psi}$ is treated multiplicatively, whereas E and its subgroups and quotients are treated additively.

Lemma 4

Assuming the set-up of Proposition 3, $\Gamma$ maps the subset $\{(1,x)\, | \, x\in G\}$ of $E_{\psi }$ onto $\{g+K\in E/K \mid \phi^{\prime}(g) \equiv 0 \, \text{ mod } h \}$.

Proof

As $\phi^{\prime}$ is constant on each coset of K in E by Lemma 3, the stated subset of E/K is well-defined. If $\phi (x) = \zeta _h^j$ then $\Gamma ((1,x))= -jy + x+K$ where $\iota (\zeta _h) = y+K$ generates L/K. Remember that y may be chosen as $(0,\ldots ,0,s_{i},0,\ldots ,0)$ for some i. Again by Lemma 3, $\phi^{\prime}(-j y + x)= j - j \sum _i(y_i/s_i) \equiv 0 \, \text{ mod } h$. Conversely, suppose that $\phi '(g)=0$. Put $a\equiv g \, \text{ mod } \textbf{s}$ and $b\equiv \sum _i\lfloor g_i/s_i\rfloor \, \text{ mod } h$; so $\phi (a)=\phi^{\prime}(g)-b\equiv -b \, \text{ mod } h$. Therefore, because $g- a-(0,\ldots , 0, bs_i , 0, \ldots , 0) \in K$, we get that $g+K= \iota (\phi (a)^{-1})+a+K = \Gamma ((1,a))$. $\square$

Remark 2

$\{(1,x)\, | \, x\in G\}$ is a full transversal for the cosets of $\langle \zeta _h\rangle$ in $E_{\psi }$.

Next we present two lemmas about special subsets of E, to be used in the proof of the impending theorem. For $0\le i\le h-1$, define $N_{\phi^{\prime}}^i=\{g\in E \, |\, \phi^{\prime}(g)\equiv i \, \text{ mod } h\}$ and $L_i=\{g\in L \, | \sum _k (g_k/s_k) \equiv i \, \text{ mod } h\}$.

Lemma 5

$N_{\phi^{prime}}^{i}+L_{j}= N_{\phi^{prime}}^{i+j}$ (elementwise sum in E), reading indices modulo h.

Proof

If $x\in N_{\phi^{prime}}^i$ and $g\in L_j$, then $\phi^{prime}(x+g)\equiv \phi^{prime}(x) +\sum _k (g_k/s_k) \equiv i+j$ by Lemma 3. Hence $N_{\phi ^{prime}}^{i}+L_j\subseteq N_{\phi^{prime}}^{i+j}$. Since $-L_j = L_{h-j}$, this containment implies that $N_{\phi^{prime}}^{i+j}-L_j\subseteq N_{\phi^{prime}}^{i}$, and so $N_{\phi^{prime}}^{i+j} = N_{\phi^{prime}}^{i}+L_j$. $\square$

Lemma 6

For all i, j and $e\in E$, $|N_{\phi^{prime}}^i\cap (e+N_{\phi^{prime}}^{i})|= |N_{\phi^{prime}}^j\cap (e+N_{\phi ^{prime}}^{j})|$.

Proof

The equation $x-y= e$ has precisely $|N_{\phi^{prime}}^i\cap (e+N_{\phi^{prime}}^{i})|$ solutions $(x,y)\in N_{\phi^{prime}}^i\times N_{\phi^{prime}}^i$. By Lemma 5, for $g\in L_{j-i}$ each such (x, y) gives a solution $(\tilde{x}, \tilde{y}) = (x+g,y+g)\in N_{\phi '}^j\times N_{\phi '}^j$ of the equation $\tilde{x}-\tilde{y}= e$. Thus $|N_{\phi^{prime}}^i\cap (e+N_{\phi^{prime}}^{i})| \le |N_{\phi^{prime}}^j\cap (e+N_{\phi^{prime}}^{j})|$. The equality follows after swapping i and j. $\square$

We also need a fact about vanishing sums of roots of unity (see, e.g., [3, Lemma 2.8.5]).

Lemma 7

For prime h, if $\sum _{i=0}^{h-1} \alpha _i\zeta _h^i =0$ with $\alpha _i\in {\mathbb {Z}}$, then $\alpha _0=\alpha _1=\cdots =\alpha _{h-1}$.

Theorem 3

Let $\phi$ be a h-ary $\textbf{s}$-array of type $\textbf{z}\ne \textbf{0}$, where h is a prime dividing $v:=|G|=\prod _i s_i$ (Definition 1), and let

$$ R=\{g+K\in E/K \mid \phi '(g)\equiv 0 \, \text{ mod } h\}. $$

Then $\phi$ is a GPhA$(\textbf{s})$ of type $\textbf{z}$ if and only if R is a (v, h, v, v/h)-relative difference set in E/K with forbidden subgroup L/K.

Proof

For $e\in E$ and $0\le k<h$, define $B_k^{(e)}= \sum _{i=0}^{h-1} |N_{\phi '}^i\cap (N_{\phi '}^{i-k} - e)|$. We readily see that

$$ AC_{\phi '}(e)= \sum _{g\in E} \zeta _h^{\phi '(g)-\phi '(g+e)} =\sum _{k=0}^{h-1} B_k^{(e)}\zeta _h^k. $$

If $e\not \in L$ then we use Lemma 7 and $\sum _{k=0}^{h-1} B_k^{(e)}=|E|$ to infer

$$\begin{aligned} AC_{\phi '}(e)=0 \, \Leftrightarrow \, B_k^{(e)}= |E|/h\quad \forall \, k. \end{aligned}$$

(4)

Suppose that $\phi$ is generalized perfect. If $e\not \in L$ then $|N_{\phi '}^i\cap (e+N_{\phi '}^{i}) | =|E|/h^2$ by (4) and Lemma 6. On the other hand, $|N_{\phi '}^i\cap (e+N_{\phi '}^{i})|=0$ if $e\in L\setminus K$, by Lemma 3. Hence the number of solutions $(x+K, y+K)\in R\times R$ of $x+K-(y+K)= e+K$ is 0 if $e\in L\setminus K$ and $|E|/\! \left( |K|h^2\right)$ if $e\not \in L$. Accordingly R is an $\left( \left| E:L\right| ,\, \left| L:K\right| ,\,\left| R\right| , \,|E|/(|K|h^2)\right)$-relative difference set in E/K, with forbidden subgroup L/K. Also $|E:L| = |G|$, $|L:K|=h$, and $|R|=|G|$ by Lemma 4. Thus R has the claimed parameters.

Now suppose that R is a (v, h, v, v/h)-relative difference set in E/K with forbidden subgroup L/K. Then $|N_{\phi '}^i\cap (N_{\phi '}^{i}-e)| =|E|/h^2$ for any $e \in E \setminus L$; thus $B_0^{(e)}= |E|/h$. Further, if $z\in L_k$ then $N_{\phi '}^{i-k}-e+z = N_{\phi '}^i-e$ by Lemma 5, giving $B_0^{(e)}= B_k^{(e-z)}$. Since $B_0^{(e)}$ is constant as e ranges over $E\setminus L$, this means that

$$ B_0^{(e)}=B_{i}^{(e)}=|E|/h \quad \forall \, i \, \, \text {and} \, \, \forall \, e \not \in L. $$

By (4), $\phi$ is a $\textrm{GPhA}(\textbf{s})$. $\square$

Remark 3

For an equivalence between difference sets and almost perfect arrays, see [15].

Proposition 4

([4, Theorem 4.1]) Let H be a finite group whose order is divisible by a prime h. Then $\psi \in Z^2(H,\langle \zeta _h\rangle )$ is orthogonal if and only if $\{(1,x)\,\, |\,\,x\in H\}$ is a (|H|, h, |H|, |H|/h)-relative difference set in $E_{\psi}$ with forbidden subgroup $\langle (\zeta _h,1)\rangle$.

Theorem 4

For prime h, a (normalized) h-ary $\textbf{s}$-array $\phi$ is a GPhA$(\textbf{s})$ of type $\textbf{z}\not =\textbf{0}$ if and only if $\mu _{\textbf{z}}\partial \phi$ is orthogonal.

Proof

This is a consequence of Theorem 3, Proposition 4, and Lemma 4. $\square$

The next theorem connects generalized plateaued functions to GPhAs.

Theorem 5

Let $\phi :{\mathbb {Z}}_{q}^m \rightarrow {\mathbb {Z}}_{h}$ be a map, where h is a prime dividing q. The following are equivalent : 

• 1. $\phi$ is a GPhA$(q^m)$ of type $\textbf{1};$

• 2. The expansion $\phi ':{\mathbb {Z}}_{hq}^m \rightarrow {\mathbb {Z}}_{h}$ of $\phi$ of type $\textbf{1}$ is a generalized plateaued function, i.e.,

  $$ \Big | \sum _{x\in {\mathbb {Z}}_{hq}^m} \zeta _h^{\phi '(x)} \zeta _{hq}^{-v\cdot x} \Big |^2=\left\{ \begin{array}{cl} (h^2q)^{m} &{} \quad v\in \mathcal{F}\\ 0 &{} \quad v\in {\mathbb {Z}}_{hq}^m\setminus \mathcal {F}, \end{array}\right. $$

  where $\mathcal{F}=\{v \in {\mathbb {Z}}_{hq}^m \mid v\equiv \textbf{1} \, \text{ mod } h\}$.

Proof

Let $u=(y_1 q, \ldots , y_m q ) \in L$ and $v=(y_1' q+a_1, \ldots , y_m' q+a_m) \in E= {\mathbb {Z}}_{hq}^m$ where $0\le y_j, y_j'\le h-1$ and $0\le a_j\le q-1$. Then $u \cdot v \equiv (a_1y_1+\cdots +a_my_m) q \, \text{ mod } hq$. Hence, if $\phi$ is a GPhA$(q^m)$ of type $\textbf{1}$, then by Lemma 2 and Corollary 1,

$$ \Big |\!\sum _{x\in {\mathbb {Z}}_{hq}^m} \zeta _h^{\phi '(x)} \zeta _{hq}^{-v \cdot x} \Big |^2=\sum _{u\in L} AC_{\phi '}(u) \zeta _{hq}^{u \cdot v}= (hq)^m\! \sum _{0\le y_1,\ldots ,y_m\le h-1} \zeta _{hq}^{-(y_1\, + \, \cdots \, +\, y_m)q \, +\, u \cdot v} $$

The rightmost displayed summation is equal to

$$ (hq)^m\sum _{0\le y_1,\dots ,y_m\le h-1} \zeta _{h}^{(a_1-1)y_1\, +\, \cdots \, +\, (a_m-1)y_m}= \left\{ \begin{array}{cl} (h^2q)^m &{} \quad a_k \equiv 1 \, \text{ mod } h\, \, \forall \,k \\ 0 &{} \quad \text {otherwise} . \end{array}\right. $$

This proves (1) $\Rightarrow$ (2). We get $(2)\Rightarrow (1)$ similarly, appealing once more to Lemma 2 and taking into account that $D_m D_m^*=(hq)^m I_m$. $\square$

Now we can fulfil our intention as stated just after Theorem 2.

Theorem 6

Let h be a prime divisor of q, and let $\phi :{\mathbb {Z}}_{q}^m \rightarrow {\mathbb {Z}}_{h}$ be an array with expansion $\phi '$ of type $\textbf{z}\ne \textbf{0}$.

• (a) The following are equivalent : 

  • (i) $\mu _{\textbf{z}}\partial \phi$ is symmetric and orthogonal, i.e., $M_{\mu _{\textbf{z}}\partial \phi }$ is a symmetric Butson Hadamard matrix; 

  • (ii) $\phi$ is a $G\!P\!h\!A(q^m)$ of type $\textbf{z};$

  • (iii) $\{g+K\in E/K \mid \phi '(g)=0\}$ is a non-splitting $(q^m,h,q^m,q^{m}/h)$-relative difference set in E/K with forbidden subgroup L/K.

• (b) If $\textbf{z}=\textbf{1}$ then (i)–(iii) are equivalent to

  • (iv) $\phi '$ is a generalized plateaued function, i.e.,

    $$ \Big | \sum _{x\in {\mathbb {Z}}_{hq}^m} \zeta _h^{\phi '(x)} \zeta _{hq}^{-v\cdot x}\Big |^2= \left\{ \begin{array}{cl} (h^2q)^{m} &{} \quad v\in \mathcal{F}\\ 0 &{} \quad \text {otherwise}, \end{array}\right. $$

    where $\mathcal{F}= \{ v\in \mathbb {Z}_{hq}^m \mid v \equiv \textbf{1} \, \text{ mod } h \}$.

• (c) Let $h=q$ and $\textbf{z}=\textbf{1}$. Suppose that, for all $y\in {\mathbb {Z}}_{h}^m\setminus \{\mathbf{{0}}\}$ with $\sum y_i\equiv 0\, \text{ mod } h$, there exists $x\in {\mathbb {Z}}_{h}^m$ satisfying (1). Then (i)–(iv) are equivalent to

  • (v) $\phi '$ is a GPBF.

Proof

The equivalences $\mathrm{(i)}\Leftrightarrow \mathrm{(ii)}$, $\mathrm{(ii)}\Leftrightarrow \mathrm{(iii)}$, $\mathrm{(ii)}\Leftrightarrow \mathrm{(iv)}$, and $\mathrm{(ii)}\Leftrightarrow \mathrm{(v)}$ follow from Theorems 4, 3, 5, and Proposition 1. (Proposition 2 (ii) justifies non-splitting in (iii).) $\square$

Remark 4

In [18, Definition 2.2], a map $\phi :{\mathbb {Z}}_{q}^m \rightarrow {\mathbb {Z}}_{q}$ is called a generalized partially bent function if $(q^m-N_F)(q^m-N_C)=q^m$ where

$$ N_F= |\{v\in {\mathbb {Z}}_{q}^m \mid \sum _{x\in {\mathbb {Z}}_{q}^m} \zeta _q^{\phi (x)-v\cdot x}=0 \}| \quad \text {and} \quad N_C=|\{v\in {\mathbb {Z}}_{q}^m \mid AC_{-\phi }(v)=0 \}|. $$

Previously, Carlet [2, Definition 1] introduced partially bent functions for $q=2$. The coincidence with our Definition 3 is shown in [17, Theorem 2] and [13, Proposition 8]. Observe that, for $\textbf{z}=\textbf{1}$, we have $|L| \cdot |\mathcal{F}|= (hq)^m$, $|L|=(hq)^m-N_C$, and $|\mathcal{F}|=(hq)^m-N_F$.

Remark 5

For q prime, if $\phi :{\mathbb {Z}}_q^m\rightarrow {\mathbb {Z}}_q$ is a $\textrm{GPqA}(q^m)$ of type $\textbf{1}$, then $\phi '$ is a 2m-generalized plateaued function.

5 Examples

Example 1

Let $\phi$ be the map on ${\mathbb Z}_2^3$ with layers

$$ A_0= \left[ \begin{array}{cc} 0 &{} 1 \\ 1 &{} 1 \end{array} \right] \quad \text {and} \quad A_1= \left[ \begin{array}{cc} 0 &{} 1 \\ 0 &{} 0 \end{array} \right] . $$

Here $A_i$ is the layer on $\{i\} \times {\mathbb Z}_2 \times {\mathbb Z}_2$, and $\phi (i,j,k)=A_i(j,k)$. Then $\phi$ is a GPBA($2^3$), i.e., a GP2A($2^3$) of type $\textbf{1}$. It has orthogonal cocycle $\mu _{\textbf{1}}\partial _2\partial _3 \partial _4\partial _6$, where $\partial _i=\partial \phi _i$ for the multiplicative Kronecker delta $\phi _i$ of $\alpha _i$, with $\alpha _0=(0,0,0)$, $\alpha _1=(0,0,1)$, etc. We label rows and columns with the elements of ${\mathbb {Z}}_2^3=\{\alpha _0,\ldots ,\alpha _7\}$ in this ordering, and display the cocyclic Hadamard matrix $M_{\mu _{\textbf{1}}\partial \phi }$ as a Hadamard (entrywise) product $M_{\mu _{\textbf{1}}}\circ M_{\partial \phi }$ in logarithmic form:

$$ \begin{aligned} \left[ \begin{array}{cccccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 \\ 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 \end{array}\right] \circ \left[ \begin{array}{cccccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 \end{array}\right] = \left[ \begin{array}{cccccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 \end{array} \right] .\end{aligned} $$

The expansion $\phi ' :{\mathbb {Z}}_{4}^3 \rightarrow {\mathbb {Z}}_{2}$ is defined by the layers $B_i$ on $\{i\}\times {\mathbb {Z}}_4\times {\mathbb {Z}}_4$, $0\le i\le 3$, where

$$\begin{aligned} B_i=\left\{ \begin{array}{cc} \left[ \begin{array}{cc} A_0 &{} \ A_0 \oplus J \\ A_0 \oplus J &{} \, A_0 \end{array} \right] &{} \quad i=0,2 \\ \left[ \begin{array}{cc} A_1 \oplus J &{} A_1 \\ A_1 &{} A_1 \oplus J \end{array} \right]&\quad i=1,3, \end{array} \right. \end{aligned}$$

J denoting the all 1s matrix. We have $L\!=\!\langle (2,0,0), (0,2,0), (0,0,2)\rangle \!=\! \{(0,0,0),(0,0,2),$ $(0,2,0),(0,2,2), (2,0,0), (2,0,2), (2,2,0),(2,2,2)\}$,

$$ AC_{\phi^{\prime}}(v)=\left\{ \begin{array}{cl} (-1)^{\textrm{wt}(v)} \, 64 &{} \quad v\in L\\ 0 &{} \quad v\notin L, \end{array}\right. $$

$\mathcal{F}=\{(1,1,1),(1,1,3),(1,3,1), (1,3,3), (3,1,1),(3,1,3),(3,3,1), (3,3,3)\}$, and

$$ \begin{aligned}\Big |\sum _{x\in {\mathbb {Z}}_{4}^3} \zeta _2^{\phi^{\prime}(x)} \zeta _4^{-v\cdot x}\Big |^2= \left\{ \begin{array}{cl} 512 &{} \quad v\in \mathcal{F}\\ 0 &{} \quad v\notin \mathcal{F}. \end{array}\right. \end{aligned}$$

Therefore $\phi '$ is a GPBF.

Example 2

The map $\begin{aligned}\phi = \left[ \begin{array}{cccc} 0 &{} 1 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \end{array} \right] \end{aligned}$ on ${\mathbb {Z}}_4^2$ is a GPBA$(4^2)$ of type $\textbf{1}$. It has orthogonal cocycle $\mu _{\textbf{1}}\partial \phi$. If we label rows and columns with the elements of ${\mathbb {Z}}_4^2=\{\alpha _0=(0,0),\alpha _1=(0,1), \alpha _2=(0,2),\ldots ,\alpha _{15}=(3,3)\}$, then the cocyclic Hadamard matrix $M_{\mu _{\textbf{1}}}\circ M_{\partial \phi }$ in logarithmic form is the Hadamard product of

$$ \begin{aligned}\left[ \begin{array}{cccccccccccccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 \\ 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \end{array}\right] \end{aligned} $$

with

$$ \begin{aligned}\left[ \begin{array}{cccccccccccccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 \end{array}\right] \end{aligned}$$

which is equal to

$$\begin{aligned} \left[ \begin{array}{cccccccccccccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 \end{array}\right] .\end{aligned} $$

The expansion $\phi ' :{\mathbb {Z}}_{8}^2 \rightarrow {\mathbb {Z}}_{2}$ is defined by

$$\begin{aligned} \left[ \begin{array}{cccccccc} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 \\ 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \end{array} \right] , \end{aligned}$$

with $L=\{(0,0),(0,4),(4,0),(4,4)\}$,

$$ AC_{\phi^{\prime}}(v)= \left\{ \begin{array}{cl} (-1)^{\textrm{wt}(v)}\, 64 &{} \quad v\in L\\ 0 &{} \quad v\notin L, \end{array}\right. $$

$$\begin{aligned} \mathcal{F}&=\{(1,1),(1,3),(1,5),(1,7), (3,1),(3,3),(3,5),(3,7), (5,1),(5,3), (5,5),(5,7),\\ &\quad \,\, (7,1),(7,3), (7,5),(7,7) \}, \end{aligned}$$

and

$$ \Big |\sum _{x\in {\mathbb {Z}}_{8}^2} \zeta _2^{\phi^{\prime}(x)} \zeta _8^{-v\cdot x} \Big |^2= \left\{ \begin{array}{cl} 256 &{} \quad v\in \mathcal{F}\\ 0 &{} \quad v\notin \mathcal{F}. \end{array}\right. $$

Therefore $\phi '$ is a GPBF.

Example 3

The map $\begin{aligned}\phi = \left[ \begin{array}{ccc} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 2 &{} 2 &{} 1 \end{array}\right] \end{aligned}$ on ${\mathbb {Z}}_3^2$ is a GP3A$(3^2)$ of type $\textbf{1}$. It has orthogonal cocycle $\mu _{\textbf{1}}\partial \phi$. Labeling the rows and columns with the elements of ${\mathbb {Z}}_3^2= \{\alpha _0=(0,0),\alpha _1=(0,1),\alpha _2=(0,2), \ldots ,\alpha _8=(2,2)\}$, we display the cocyclic Butson matrix $M_{\mu _{\textbf{1}}}\circ M_{\partial \phi }$:

$$\begin{aligned} \left[ \begin{array}{ccccccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 2 \\ 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} 2 &{} 2 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 2 &{} 1 &{} 1 &{} 2 \\ 0 &{} 1 &{} 1 &{} 1 &{} 2 &{} 2 &{} 1 &{} 2 &{} 2 \end{array} \right] \circ \left[ \begin{array}{ccccccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 2 &{} 0 &{} 0 &{} 2 &{} 1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 2 &{} 1 &{} 2 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 &{} 2 &{} 1 &{} 1 &{} 1 &{} 1 &{} 2 \\ 0 &{} 2 &{} 2 &{} 1 &{} 2 &{} 1 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 2 &{} 1 &{} 1 &{} 2 \\ 0 &{} 0 &{} 2 &{} 1 &{} 0 &{} 1 &{} 2 &{} 0 &{} 0 \\ 0 &{} 2 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 2 &{} 0 \\ 0 &{} 1 &{} 1 &{} 2 &{} 1 &{} 2 &{} 0 &{} 0 &{} 2 \end{array}\right] = \left[ \begin{array}{ccccccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 1 &{} 2 &{} 1 &{} 0 &{} 2 &{} 2 \\ 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 2 &{} 2 &{} 1 &{} 2 \\ 0 &{} 1 &{} 0 &{} 2 &{} 1 &{} 1 &{} 2 &{} 2 &{} 0 \\ 0 &{} 2 &{} 0 &{} 1 &{} 2 &{} 2 &{} 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 2 &{} 1 &{} 2 &{} 0 &{} 2 &{} 0 &{} 1 \\ 0 &{} 0 &{} 2 &{} 2 &{} 1 &{} 2 &{} 0 &{} 1 &{} 1 \\ 0 &{} 2 &{} 1 &{} 2 &{} 1 &{} 0 &{} 1 &{} 0 &{} 2 \\ 0 &{} 2 &{} 2 &{} 0 &{} 0 &{} 1 &{} 1 &{} 2 &{} 1 \end{array} \right] . \end{aligned}$$

The expansion $\phi ' :{\mathbb {Z}}_{9}^2 \rightarrow {\mathbb {Z}}_{3}$ is defined by

$$ \begin{aligned} \left[ \begin{array}{ccccccccc} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 2 &{} 2 &{} 2 \\ 0 &{} 1 &{} 0 &{} 1 &{} 2 &{} 1 &{} 2 &{} 0 &{} 2 \\ 2 &{} 2 &{} 1 &{} 0 &{} 0 &{} 2 &{} 1 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 &{} 2 &{} 2 &{} 2 &{} 0 &{} 0 &{} 0 \\ 1 &{} 2 &{} 1 &{} 2 &{} 0 &{} 2 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 2 &{} 1 &{} 1 &{} 0 &{} 2 &{} 2 &{} 1 \\ 2 &{} 2 &{} 2 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \\ 2 &{} 0 &{} 2 &{} 0 &{} 1 &{} 0 &{} 1 &{} 2 &{} 1 \\ 1 &{} 1 &{} 0 &{} 2 &{} 2 &{} 1 &{} 0 &{} 0 &{} 2 \end{array} \right] , \end{aligned} $$

with $L=\{(0,0),(0,3),(0,6),(3,0), (3,3),(3,6),(6,0),(6,3),(6,6)\}$,

$$ AC_{\phi '}((v_1,v_2))= \left\{ \begin{array}{cl} 81 \,\zeta _3^{-(v_1+v_2)/3} &{} \quad (v_1,v_2)\in L\\ 0 &{} \quad (v_1,v_2)\notin L, \end{array}\right. $$

$\mathcal{F}=\{(1,1),(1,4),(1,7),(4,1), (4,4),(4,7), (7,1),(7,4),(7,7)\}$, and

$$ \begin{aligned}\Big |\sum _{x\in {\mathbb {Z}}_{9}^2} \zeta _3^{\phi^{\prime}(x)} \zeta _9^{-v\cdot x} \Big |^2= \left\{ \begin{array}{cl} 729 &{} \quad v\in \mathcal{F}\\ 0 &{} \quad v\notin \mathcal{F}. \end{array}\right. \end{aligned}$$

Thus $\phi '$ is a GPBF. Also $\phi '$ is a 4-generalized plateaued function (see Remark 5).

It may be checked that the sufficient condition (1) is satisfied in each of the Examples 1–3.

We now recite a bit more algebraic design theory in preparation for our penultimate result, which provides an infinite family of GPhAs of type $\textbf{1}$ arising from Example 1.

Proposition 5

(cf. [3, Theorem 15.8.4]) Let $G_{\textbf{s}} = \mathbb {Z}_{s_1} \times \cdots \times \mathbb {Z}_{s_{m}}$, $G_{\textbf{t}} = \mathbb {Z}_{t_1} \times \cdots \times \mathbb {Z}_{t_{n}}$, and $G = G_{\textbf{s}} \times G_{\textbf{t}}$. Suppose that $\psi \partial \phi _{\textbf{s}} \in Z^2( G_{\textbf{s}}, \langle \zeta _{k_{1}} \rangle )$ and $\rho \partial \phi _{\textbf{t}}\in Z^2( G_{\textbf{t}}, \langle \zeta _{k_{2}} \rangle )$ are orthogonal. Let $k = \textrm{lcm}(k_{1},k_{2})$. Define $\varphi \in Z^2(G, \langle \zeta _{k} \rangle )$ by

$$ \varphi (g_{s}g_{t},h_{s}h_{t}) = \psi (g_{s},h_{s})\rho (g_{t},h_{t}), $$

and define a map $\phi$ on G by

$$ \phi (g_{s}g_{t}) = \phi _{\textbf{s}}(g_{s}) \phi _{\textbf{t}}(g_{t}). $$

Then $\varphi \partial \phi \in Z^2(G,\langle \zeta _{k} \rangle )$ is orthogonal, with cocyclic matrix $[\psi \partial \phi _{\textbf{s}}]\otimes [\rho \partial \phi _{\textbf{t}}]$.

Corollary 2

Let $h = q$ be prime. If there exist symmetric cocyclic matrices in ${B\!H}(q^m,h)$ and ${B\!H}(q^n,h)$, corresponding to a GPhA$(q^m)$ of type $\textbf{1}$ and a GPhA$(q^n)$ of type $\textbf{1}$, respectively, then there exists a symmetric cocyclic matrix in ${B\!H}(q^{m+n},h)$ corresponding to a GPhA$(q^{m+n})$ of type $\textbf{1}$.

Proof

Since the Kronecker product of symmetric matrices is symmetric, it remains only to observe that the product matrix in ${\text {BH}}(q^{m+n},h)$ corresponds to an array of type $\textbf{1}$. This also follows from the construction and Theorem 6. $\square$

Example 1 furnishes a symmetric orthogonal cocycle $\mu _{\textbf{1}}\partial \phi \in Z^{2}({\mathbb {Z}}_{2}^{3},{\mathbb {Z}}_{2})$ with nontrivial coboundary $\partial \phi$. By iteration of Proposition 5 (Kronecker multiplying $\mu _{\textbf{1}}\partial \phi$ by powers of $\mu _{\textbf{1}}\in Z^2({\mathbb {Z}}_{2},{\mathbb {Z}}_{2})$), we get a symmetric orthogonal cocycle $\mu _{\textbf{1}}\partial \chi \in Z^{2}({\mathbb {Z}}_{2}^{k},{\mathbb {Z}}_{2})$. Then $\chi$ is a $\textrm{GPBA}(2^k)$ of type $\textbf{1}$. Thus, for all $k\ge 3$ there exists a map ${\mathbb {Z}}_2^k\rightarrow {\mathbb {Z}}_2$ with expansion a GPBF; whereas for odd k, recall that no GBF—i.e., no bent function—can exist. We note that for odd k this map is a Boolean near-bent function [12, Section 16.1.1]. Chapter 16 of [12] may be consulted for similar existence results, e.g., on plateaued and partially bent functions.