Generalized binary arrays from quasi-orthogonal cocycles

TL;DR

This paper introduces generalized optimal binary arrays (GOBAs) with specific energy properties, providing a construction method based on 2-cocycles and applying it to find negaperiodic Golay pairs from small-length sequences.

Contribution

It extends the concept of perfect binary arrays by defining GOBAs with new energy conditions and offers a construction method using 2-cocycles, also identifying related Golay pairs.

Findings

GOBAs can have even energy not divisible by 4.

A construction procedure for GOBAs using 2-cocycles is provided.

Negaperiodic Golay pairs are derived from small-length GOBAs.

Abstract

Generalized perfect binary arrays (GPBAs) were used by Jedwab to construct perfect binary arrays. A non-trivial GPBA can exist only if its energy is $2$ or a multiple of $4$. This paper introduces generalized optimal binary arrays (GOBAs) with even energy not divisible by $4$, as analogs of GPBAs. We give a procedure to construct GOBAs based on a characterization of the arrays in terms of $2$-cocycles. As a further application, we determine negaperiodic Golay pairs arising from generalized optimal binary sequences of small length.

Tables

Table 1. Table 1: NGPs ( ϕ 1 , ϕ 2 ) subscript italic-ϕ 1 subscript italic-ϕ 2 (\phi_{1},\phi_{2}) from quasi-orthogonal cocycles over ℤ 2 ​ k subscript ℤ 2 𝑘 \mathbb{Z}_{2k}

Table 2. Table 2: Enumeration of NGPs and their equivalence classes

Equations

$$R_{\varphi }(w):=\sum _{k=0}^{n-1}\varphi (k)\varphi (k+w)$$

$$\max_{0<w<n}|R_{\phi}(w)|\geq\left\{\begin{array}[]{cl}0&\hskip 10.0ptn\equiv 0\ \;\mbox{mod}\;4\\ 1&\hskip 10.0ptn\equiv 1\,\mbox{ or }\,3\ \;\mbox{mod}\;4\\ 2&\hskip 10.0ptn\equiv 2\ \;\mbox{mod}\;4.\end{array}\right.$$

$$|R_{\phi^{\prime}}(w)|=\left\{\begin{array}[]{cl}0&\,\,\mbox{$w$ odd}\\ 4&\,\,\mbox{$w$ even}\end{array}\right.$$

$$|R_{\phi^{\prime}}(w)|=\left\{\begin{array}[]{cl}4&\,\,\mbox{$w$ odd}\\ 0&\,\,\mbox{$w$ even}\end{array}\right.$$

$$\psi (g,h)\psi (gh,k)=\psi (g,hk)\psi (h,k)\forall g,h,k\in G$$

$$|R\cap xR|=\lambda \forall x\in E\setminus N.$$

$$|R\cap xR|=\left\{\begin{array}[]{ll}2t+1&\hskip 10.0ptx\in SZ\\ 2t\ \mathrm{or}\ 2t+2&\hskip 10.0pt\mathrm{otherwise.}\end{array}\right.$$

$$\begin{array}[]{l}H=\{h\in E\mid h_{i}=0\ \mbox{if}\ z_{i}=0,\ \mbox{and}\ h_{i}=0\ \mbox{or}\ s_{i}\ \mbox{if}\ z_{i}=1\},\\ K=\{k\in H\mid k\,\mbox{ has even weight}\}\end{array}$$

$$1⟶⟨-1⟩\stackrel{\iota }{⟶}E/K\stackrel{\beta }{⟶}G⟶0,$$

$$f_{\mathbf{z}}(x,y)={\iota }^{-1}(\tau (x)+\tau (y)-\tau (x+y)).$$

$$1⟶⟨-1⟩\stackrel{{\iota }^{\prime }}{⟶}{E}_{\psi }\stackrel{{\beta }^{\prime }}{⟶}G⟶0,$$

$$\begin{array}[]{ccccccccc}1&\longrightarrow&\langle-1\rangle&\stackrel{{\scriptstyle\iota^{\prime}}}{{\longrightarrow}}&E_{\psi}&\stackrel{{\scriptstyle\beta^{\prime}}}{{\longrightarrow}}&G&\longrightarrow&0\\[5.69054pt] &&\|&&{\tiny\mbox{$\Gamma$}}\big{\downarrow}\phantom{\Gamma}&&\|&&\\[5.69054pt] 1&\longrightarrow&\langle-1\rangle&\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}&E/K&\stackrel{{\scriptstyle\beta}}{{\longrightarrow}}&G&\longrightarrow&0\end{array}$$

$$\phi^{\prime}(g)=\left\{\begin{array}[]{rl}\phi(a)&\quad g\in a+K\\ -\phi(a)&\quad g\notin a+K\end{array}\right.$$

$$R_{{\varphi }^{\prime }}(g)=0\forall g\in E\setminus H.$$

$$R_{\varphi }(g)=0\forall g\in G\setminus \{0\}.$$

$$R_{\varphi }(g)=\pm 2\forall g\in G\setminus \{0\};$$

$$R_{\phi }(x)=|A|+4(d_{\phi }(x)-|N_{\phi }|)$$

$$\begin{array}[]{ccl}R_{\phi^{\prime}}(g)=-2|H|&\ \Leftrightarrow&\ d_{D}(g+K)=2t\\ R_{\phi^{\prime}}(g)=0&\ \Leftrightarrow&\ d_{D}(g+K)=2t+1\\ R_{\phi^{\prime}}(g)=2|H|&\ \Leftrightarrow&\ d_{D}(g+K)=2t+2.\end{array}$$

$$d_{D}(g+K)=\left\{\begin{array}[]{ll}2t+1&\quad g+K\in S+H/K\\ 2t\ \ {\rm or}\ \ 2t+2&\quad\mbox{otherwise}.\end{array}\right.$$

$$|S|=\frac{|\{g+K\in E/K|R_{{\varphi }^{\prime }}(g)=0\}|}{2}=|R_{{\varphi }^{\prime }}^{-1}(0)|/2|K|.$$

$${\footnotesize\left[\vspace{1pt}\begin{array}[]{lrrrrr}1&\phantom{-}1&\phantom{-}1&1&1&1\\ 1&\phantom{-}1&\phantom{-}1&1&1&1\\ 1&\phantom{-}1&\phantom{-}1&1&1&1\\ 1&\phantom{-}1&\phantom{-}1&-1&-1&-1\\ 1&\phantom{-}1&\phantom{-}1&-1&-1&-1\\ 1&\phantom{-}1&\phantom{-}1&-1&-1&-1\end{array}\vspace{1pt}\right]}\circ{\footnotesize\left[\vspace{1pt}\begin{array}[]{rrrrrr}1&1&1&1&1&1\\ 1&1&-1&-1&-1&-1\\ 1&-1&-1&1&1&1\\ 1&-1&1&1&-1&1\\ 1&-1&1&-1&1&1\\ 1&-1&1&1&1&-1\end{array}\vspace{1pt}\right]}={\footnotesize\left[\vspace{1pt}\begin{array}[]{rrrrrr}1&1&1&1&1&1\\ 1&1&-1&-1&-1&-1\\ 1&-1&-1&1&1&1\\ 1&-1&1&-1&1&-1\\ 1&-1&1&1&-1&-1\\ 1&-1&1&-1&-1&1\end{array}\vspace{1pt}\right]},$$

$${\footnotesize\left[\vspace{1pt}\begin{array}[]{lrrrrr}1&1&1&\phantom{-}1&1&1\\ 1&1&-1&\phantom{-}1&1&-1\\ 1&-1&-1&\phantom{-}1&-1&-1\\ 1&1&1&\phantom{-}1&1&1\\ 1&1&-1&\phantom{-}1&1&-1\\ 1&-1&-1&\phantom{-}1&-1&-1\end{array}\vspace{1pt}\right]}\circ{\footnotesize\left[\vspace{1pt}\begin{array}[]{rrrrrr}1&1&1&1&1&1\\ 1&-1&-1&1&1&1\\ 1&-1&1&-1&-1&-1\\ 1&1&-1&1&1&-1\\ 1&1&-1&1&-1&1\\ 1&1&-1&-1&1&1\end{array}\vspace{1pt}\right]}={\footnotesize\left[\vspace{1pt}\begin{array}[]{rrrrrr}1&1&1&1&1&1\\ 1&-1&1&1&1&-1\\ 1&1&-1&-1&1&1\\ 1&1&-1&1&1&-1\\ 1&1&1&1&-1&-1\\ 1&-1&1&-1&-1&-1\end{array}\vspace{1pt}\right]},$$

$${\footnotesize\left[\vspace{1pt}\begin{array}[]{lrrrrr}1&1&1&\phantom{-}1&1&1\\ 1&1&-1&\phantom{-}1&1&-1\\ 1&-1&-1&\phantom{-}1&-1&-1\\ 1&1&1&-1&-1&-1\\ 1&1&-1&-1&-1&1\\ 1&-1&-1&-1&1&1\end{array}\vspace{1pt}\right]}\circ{\footnotesize\left[\vspace{1pt}\begin{array}[]{rrrrrr}1&1&1&1&1&1\\ 1&-1&-1&-1&-1&1\\ 1&-1&1&-1&1&1\\ 1&-1&-1&1&-1&-1\\ 1&-1&1&-1&-1&-1\\ 1&1&1&-1&-1&1\end{array}\vspace{1pt}\right]}={\footnotesize\left[\vspace{1pt}\begin{array}[]{rrrrrr}1&1&1&1&1&1\\ 1&-1&1&-1&-1&-1\\ 1&1&-1&-1&-1&-1\\ 1&-1&-1&-1&1&1\\ 1&-1&-1&1&1&-1\\ 1&-1&-1&1&-1&1\end{array}\vspace{1pt}\right]}.$$

$$a^i\mapsto x^{i+n{\delta }_{{\varphi }_1(i),-1}},a^{i}b\mapsto x^{i+n{\delta }_{{\varphi }_2(i),-1}}y$$

$$\small\left[\begin{array}[]{cr}A&A\\ B&-B\end{array}\right]$$

$$\begin{array}[]{ll}{\cal C}_{r}\in\{{\cal I}_{r}+t+\frac{1-r}{2},\,{\cal I}_{r}+t+\frac{3-r}{2}\}&\qquad r\,\,\mbox{odd}\\ \vspace{-12pt}&\\ {\cal C}_{r}={\cal I}_{r}+t+1-\frac{r}{2}&\qquad r\,\,\mbox{even}.\end{array}$$

$$\mathrm{Gr}(M_{\psi }{)}_{ij}=\psi (g_i{g}_j^{-1},g_j)\sum _{g\in G}\psi (g_i{g}_j^{-1},g)$$

$$\mathrm{Gr}(M_{\psi }^{\top }{)}_{ij}=\psi (g_i,g_i^{-1}{g}_j)\sum _{g\in G}\psi (g,g_i^{-1}{g}_j).$$

$$\small{\left[\begin{array}[]{rr}J&J\\ J&-J\end{array}\right]}.$$

$$\mathrm{Gr}(M_{\psi }{)}_{ij}=\psi (a^{i+j-2}b,a^{j-1}b)\sum _{g\in G}\psi (a^{i+j-2}b,g)=0$$

Full text

1 Introduction

Let $\phi=(\phi(0),\ldots,\phi(n-1))\in\{\pm 1\}^{n}$ be a binary sequence of length $n$. Reading arguments modulo $n$,

[TABLE]

is the periodic autocorrelation of $\phi$ at shift $w$. The expansion of $\phi$, denoted $\phi^{\prime}$, is the concatenation of $\phi$ and $-\phi$ (in that order). A pair $\phi_{1}$, $\phi_{2}$ of binary sequences, each of length $2t$, such that $R_{\phi_{1}^{\prime}}(w)+R_{\phi_{2}^{\prime}}(w)=0$ for $1\leq w\leq 2t-1$ (equivalently, for $1\leq\allowbreak w\leq 4t-1$ and $w\not=2t$), is a negaperiodic Golay pair (NGP). Note that the original definition of NGP in [4] coincides with the definition above by [8, Lemma 2].

We seek good sources of NGPs. This objective is connected to several existence problems in algebraic design theory. For example, Egan showed that NGPs of length $2t$ are equivalent to certain relative $(4t,2,4t,2t)$-difference sets in the dicyclic group $Q_{8t}$ of order $8t$ [8, Theorem 3]. Actually, there is a relative $(4t,2,4t,2t)$-difference set in a central extension $E$ of $\mathbb{Z}_{2}$ by a group $G$ of order $4t$, relative to $\mathbb{Z}_{2}$, if and only if there is a Hadamard matrix of order $4t$ whose expanded (group-divisible) design admits a special regular action by $E$: a cocyclic Hadamard matrix over $G$ [6, Theorem 2.4]. By way of [9, Theorem 3.3], Ito [13, p. 370] conjectured that $Q_{8t}$ contains such relative difference sets for all $t$. Schmidt [16] has verified Ito’s conjecture up to $t=46$. Our recent paper [3] initiated the study of quasi-orthogonal cocycles over groups $G$ of even order not divisible by $4$, in direct analogy with cocyclic Hadamard matrices. The present paper builds on [3].

It is easy to see that

[TABLE]

The sequence $\phi$ is optimal if equality holds in (1). In particular, $\phi$ is perfect if $R_{\phi}(w)=0$ for $0<w<n$. No perfect binary sequence of length $n>4$ is known. Attention consequently turns to the larger class of perfect binary arrays (PBAs). Jedwab [14] introduced generalized perfect binary arrays (GPBAs) to aid in the construction of PBAs. Hughes [11] subsequently demonstrated the cocyclic nature of GPBAs.

A generalized perfect binary sequence (GPBS) is a $1$-dimensional GPBA; such $\phi$ have $R_{\phi^{\prime}}(w)=0$ for all $w$. Each pair of GPBSs is obviously an NGP. However, a GPBS exists only if $n=2$ [14, Result 4.8]. So let $n>2$ be even; since $R_{\phi^{\prime}}(w)$ is divisible by $4$, and not every $R_{\phi^{\prime}}(w)$ is [math], some $|R_{\phi^{\prime}}(w)|$ must be at least $4$. Thus, we will say that $\phi$ of length $2t$ is a generalized optimal binary sequence (GOBS) if $\max_{0<w<2t}|R_{\phi^{\prime}}(w)|=4$. Equivalently, $\phi$ is a GOBS if, for $0<w<2t$,

[TABLE]

when $t$ is odd, and

[TABLE]

when $t$ is even. We propose searching for NGPs in the set of GOBs of length $2t$, $t$ odd.

Just as the notion of GPBA extends that of GPBS to dimensions greater than $1$, a GOBA (generalized optimal binary array) is a higher-dimensional version of a GOBS. Section 3 treats GPBAs and GOBAs from the perspective of [3]. We prove a one-to-one correspondence between GOBAs, quasi-orthogonal cocycles over abelian groups, and abelian relative quasi-difference sets. In Section 4, we outline and apply a method to find NGPs among GOBSs that correspond to quasi-orthogonal cocycles over cyclic groups. The concluding Section 5 looks at an important question for cocyclic designs prompted by the analysis in Section 4.

2 Quasi-orthogonal cocycles and related

combinatorial structures

Let $G$ and $U$ be finite groups, with $U$ abelian. A map $\psi:G\times G\rightarrow U$ such that $\psi(1,1)=1$ and

[TABLE]

is a (normalized) cocycle over $G$. If $\phi:G\rightarrow U$ is any map that is normalized (i.e., $\phi(1)=1$) then $\partial\phi(g,h)=\phi(g)^{-1}\phi(h)^{-1}\phi(gh)$ defines a cocycle $\partial\phi$, called a coboundary. The set of all cocycles over $G$ forms an abelian group $Z^{2}(G,U)$, whose quotient by the subgroup $B^{2}(G,U)$ of coboundaries is the second cohomology group $H^{2}(G,U)$. We display $\psi\in Z^{2}(G,U)$ as a cocyclic matrix $M_{\psi}=[\psi(g,h)]_{g,h\in G}$. If $U=\mathbb{Z}_{2}=\langle-1\rangle$ and $M_{\psi}$ is Hadamard then $\psi$ is said to be orthogonal.

The row excess $RE(M)$ of a cocyclic matrix $M$ indexed by $G$ is the sum of the absolute values of all row sums, apart from row $1_{G}$. The cocycle equation (2) guarantees that $\psi$ is orthogonal if and only if $RE(M_{\psi})$ is optimal, i.e., zero.

For the rest of this section, $|G|=4t+2>2$.

Proposition 1

• (i)

If $\psi\in Z^{2}(G,\mathbb{Z}_{2})$ then $RE(M_{\psi})\geq 4t$.

• (ii)

If $\psi\in B^{2}(G,\mathbb{Z}_{2})$ then $RE(M_{\psi})\geq 8t+2$.

Proof

See [3, Proposition 1]. $\Box$

In analogy with the definition of orthogonal cocycles, we say that $\psi$ is quasi-orthogonal if its matrix has least possible row excess: by Proposition 1, either $\psi\not\in B^{2}(G,\mathbb{Z}_{2})$ and $RE(M_{\psi})=4t$, or $\psi\in B^{2}(G,\mathbb{Z}_{2})$ and $RE(M_{\psi})=8t+2$ (coboundaries were excluded from the notion of quasi-orthogonality in [3]).

Lemma 1

Let $X_{m}=\{g\in G\ |\ {\textstyle\sum}_{h\in G}\psi(g,h)=\allowbreak m\}$. Then $\psi$ is quasi-orthogonal if and only if $|X_{2}\cup X_{-2}|=4t+1$ for $\psi\in B^{2}(G,\mathbb{Z}_{2})$, or $|X_{0}|=\allowbreak 2t+1$ and $|X_{2}\cup X_{-2}|=2t$ for $\psi\not\in B^{2}(G,\mathbb{Z}_{2})$.

Proof

See [3, Lemma 2.4]. $\Box$

It is not known whether quasi-orthogonal cocycles always exist. Indeed, we do not know of a group $G$ such that $Z^{2}(G,\mathbb{Z}_{2})$ does not contain a quasi-orthogonal element (in contrast, there are several non-existence results for orthogonal cocycles, e.g., due to Ito [12]). We have found quasi-orthogonal coboundaries over many abelian $G$, but none over non-abelian $G$ such as dihedral groups, apart from the dihedral group of order $6$. Thirdly, for all $t$ such that $4t+1$ is a sum of two squares that we tested, we always found a quasi-orthogonal cocycle $\psi$ over some group of order $4t+2$ with $|\mathrm{det}(M_{\psi})|$ attaining the maximum $2(4t+1)(4t)^{2t}$ established by Ehlich-Wojtas. These existence questions all merit deeper investigation.

Let $E$ be a group with a normal subgroup $N$ of order $m$ and index $v$. A relative $(v,m,k,\lambda)$-difference set in $E$ relative to $N$ (the forbidden subgroup) is a $k$-subset $R$ of a transversal for $N$ in $E$ such that

[TABLE]

Relative $(2s,2,2s,s)$-difference sets are especially interesting. If $s$ is even then they are equivalent to cocyclic Hadamard matrices [6, Corollary 2.5], whereas none exist if $s$ is odd [10]. In the latter case there is a natural analog of relative difference set. Suppose that $|E|=8t+4$ and let $Z\cong\mathbb{Z}_{2}$ be a normal (hence central) subgroup of $E$. A relative $(4t+2,2,4t+2,2t+1)$-quasi-difference set in $E$ with forbidden subgroup $Z$ is a transversal $R$ for $Z$ in $E$ containing a subset $S\subset R\setminus\{1\}$ of size [math] or $2t+1$ such that, for all $x\in E\setminus Z$,

[TABLE]

We call $R$ extremal if $S=\emptyset$. (This modifies the original definition in [3] of relative quasi-difference set, to allow quasi-orthogonal coboundaries).

The next result is mostly Proposition 4.3 in [3]. For each $\psi\in Z^{2}(G,{\mathbb{Z}}_{2})$ we have a canonical central extension $E_{\psi}$ with element set $\{(\pm 1,g)\,|\,g\in G\}$ and multiplication defined by $(u,g)(v,h)=(uv\hskip 0.5pt\psi(g,h),gh)$.

Proposition 2

The cocycle $\psi$ is quasi-orthogonal if and only if $D=\{(1,g)\ |\ g\in G\}$ is a relative $(4t+2,2,4t+2,2t+1)$-quasi-difference set in $E_{\psi}$ with forbidden subgroup $\langle(-1,1)\rangle$, where $D$ is extremal for $\psi\in B^{2}(G,\mathbb{Z}_{2})$.

Remark 1

The requisite subset $S$ of $D$ corresponds to the rows of $M_{\psi}$ with zero sum.

3 Generalized binary arrays with optimal autocorrelation

Jedwab [14] showed that a GPBA is equivalent to an abelian relative difference set, and Hughes [11] identified its underlying orthogonal cocycle. In this section we carry over these ideas into the setting of quasi-orthogonal cocycles.

We start with an adaptation of some material from [11] and [14]. The cyclic group of order $m$ will be written additively, i.e., as $\mathbb{Z}_{m}=\{0,1,\ldots,\allowbreak m-1\}$ under addition modulo $m$. Let ${\bf s}=(s_{1},\ldots,s_{r})$ be an $r$-tuple of positive integers greater than $1$, and let $G={\mathbb{Z}}_{s_{1}}\times\cdots\times{\mathbb{Z}}_{s_{r}}$. A binary ${\bf s}$-array is just a set map $\phi:G\rightarrow\{\pm 1\}$; it has energy $n:=\prod_{i=1}^{r}s_{i}=|G|$. We view a binary sequence as an $\bf s$-array with $r=1$.

Given $\bf s$ and a type vector ${\bf z}=(z_{1},\ldots,z_{r})\in\{0,1\}^{r}$, let $E={\mathbb{Z}}_{(z_{1}+1)s_{1}}\times\cdots\allowbreak\times{\mathbb{Z}}_{(z_{r}+1)s_{r}}$. Then

[TABLE]

are elementary abelian $2$-subgroups of $E$. Note that $E$ is a (central) extension of $H$ by $G$. For ${\bf z}\neq{\bf 0}$ we obtain the short exact sequence

[TABLE]

where $\iota$ maps $-1$ to the generator of $H/K$ and $\beta(g+K)=g\,\;\mbox{mod}\;\,{\bf s}$. This sequence determines a cocycle $f_{\bf z}\in Z^{2}(G,\langle-1\rangle)$ after choice of a transversal map $\tau:\allowbreak G\rightarrow\allowbreak E/K$. Specifically, set $\tau(x)=x+K$; then

[TABLE]

We can express $f_{\bf z}$ as a product of cocycles on cyclic groups. Define $\gamma_{m}\in Z^{2}(\mathbb{Z}_{m},\langle-1\rangle)$ by $\gamma_{m}(j,k)=(-1)^{\lfloor(j+k)/m\rfloor}$, evaluating the exponent as an ordinary integer.

Proposition 3 ([11, Lemma 3.1])

• (i)

$f_{\bf z}(x,y)=\prod_{z_{i}=1}\gamma_{s_{i}}(x_{i},y_{i})$.

• (ii)

$f_{\bf z}\in B^{2}(G,\langle-1\rangle)$* if and only if $s_{i}$ is odd for all $i$ such that $z_{i}=1$.*

Each cocycle $\psi\in Z^{2}(G,\langle-1\rangle)$ has an associated short exact sequence

[TABLE]

where $\iota^{\prime}(u)=(u,0)$ and $\beta^{\prime}(u,x)=x$. The following is standard.

Proposition 4

If $\psi$ and $f_{\bf z}$ are cohomologous, say $\psi=f_{\bf z}\partial\phi$, then (3) and (4) are equivalent short exact sequences: the isomorphism $\Gamma$ defined by $(u,x)\mapsto\iota(u\phi(x))+\tau(x)$ makes the diagram

[TABLE]

commute.

We broaden concepts defined earlier only for sequences. The expansion of a binary $\bf s$-array $\phi$ with respect to a type vector ${\bf z}$ is the map $\phi^{\prime}$ on $E$ given by

[TABLE]

where $a$ denotes $g$ modulo ${\bf s}$. For any array $\varphi:A\rightarrow\{\pm 1\}$ and $x\in A$, let $R_{\varphi}(x)=\sum_{a\in A}\varphi(a)\varphi(a+x)$.

Lemma 2

If $h\in H\setminus K$ then $\phi^{\prime}(h+g)=-\phi^{\prime}(g)$, and if $h\in K$ then $\phi^{\prime}(h+g)=\allowbreak\phi^{\prime}(g)$.

Corollary 1

$R_{\phi^{\prime}}(g)=|H|\sum_{x\in T}\phi^{\prime}(x)\phi^{\prime}(x+g)$* where $T$ is any transversal for $H$ in $E$.*

Lemma 3

The isomorphism $\Gamma$ in Proposition 4 maps $\{(1,x)\ |\ x\in G\}\subseteq E_{\psi}$ onto $\{g+K\in E/K\mid\phi^{\prime}(g)=1\}$.

Proof

(Cf. [11, p. 330].) Let $\phi^{\prime}(g)=1$ and write $a$ for $g$ modulo ${\bf s}$; then $g+K=\allowbreak\iota(\phi(a))+a+K=\Gamma((1,a))$. Conversely, $\Gamma((1,x))=h+x+K$ where $h+K$ is the generator of $H/K$ if $\phi(x)=-1$ and $h=0$ otherwise. By Lemma 2, $\phi^{\prime}(h+x)=\allowbreak 1$. $\Box$

The $\bf s$-array $\phi$ is a GPBA$({\bf s})$ of type ${\bf z}$ if

[TABLE]

When ${\bf z}=0$, this condition becomes (by Corollary 1)

[TABLE]

In the latter event $\phi$ is a PBA; which is equivalent to $\partial\phi$ being orthogonal (we return to this case later in the section). More generally, a GPBA$({\bf s})$ is equivalent to a relative difference set in $E/K$ relative to $H/K$, hence equivalent also to a cocyclic Hadamard matrix over $G$: see [11, Theorem 5.3] and [14, Theorem 3.2]. So a GPBA can exist only if its energy $n$ is $2$ or a multiple of $4$. Theorems 3.1 and 3.2 below are analogous results for $n\equiv 2\;\mbox{mod}\;4$.

Assume that $|G|=4t+2>2$ unless stated otherwise. Let $s_{1}/2,s_{2},\ldots,s_{r}$ be odd. Thus, if $z_{1}=0$ then $E$ splits over $H$ by Proposition 3, and so $R_{\phi^{\prime}}$ is never zero by Corollary 1 and Lemma 2.

Definition 1

A GOBA$({\bf s})$ of type ${\bf z}$ is a binary $\bf s$-array $\phi$ such that

• (i)

$R_{\phi^{\prime}}(g)\in\{0,\pm 2|H|\}$ $\ \,\forall\hskip 1.0ptg\in E\setminus H$,

and if $z_{1}=1$ then

• (ii)

$|\{g\in E\ |\ R_{\phi^{\prime}}(g)=0\}|=|E|/2$.

A GOBS as defined in Section 1 is a GOBA($\bf s)$ with $r=z_{1}=1$. When ${\bf z}={\bf 0}$, Definition 1 reduces to

[TABLE]

we call $\phi$ satisfying this condition an optimal binary array (OBA).

Lemma 4 ([14, Lemma 3.1])

For any array $\varphi:A\rightarrow\{\pm 1\}$,

[TABLE]

where $N_{\varphi}=\{a\in A\,|\,\varphi(a)=-1\}$ and $d_{\varphi}(x)=|N_{\varphi}\cap(x+N_{\varphi})|$.

Proof

Routine counting. $\Box$

Theorem 3.1

Let $\phi$ be a binary ${\bf s}$-array, ${\bf z}$ be a non-zero type vector, and $D=\{g+K\in E/K\mid\phi^{\prime}(g)=-1\}$. Then $\phi$ is a GOBA$({\bf s})$ of type ${\bf z}$ if and only if $D$ is a relative $(4t+2,2,4t+2,2t+1)$-quasi-difference set in $E/K$ with forbidden subgroup $H/K$; furthermore, $D$ is extremal if $z_{1}=0$.

Proof

We continue with the notation of Lemma 4. By Lemma 3, $D$ is a full transversal for $H/K$ in $E/K$. Also, $|N_{\phi^{\prime}}|=\allowbreak|E|/2$ by Lemma 2; thus $|D|=\allowbreak|N_{\phi^{\prime}}|/|K|$.

For each $g\not\in H$, denote $|D\cap(g+K+D)|$ by $d_{D}(g+K)$: this is the number of $x+K\in D$ such that $x-g+K\in D$. Since $d_{D}(g+K)=d_{\phi^{\prime}}(g)/|K|$, Lemma 4 implies that

[TABLE]

Let $S=\{g+K\in D\mid R_{\phi^{\prime}}(g)=0\}$. According to (5), Definition 1 (i) holds if and only if

[TABLE]

Lemma 2 yields

[TABLE]

Thus $|S|=2t+1$ for $z_{1}=1$ if and only if Definition 1 (ii) holds. $\Box$

Remark 2

Theorem 3.1 remains valid when $D$ is replaced by its complement $\{g+K\in E/K\;|\;\allowbreak\phi^{\prime}(g)=\allowbreak 1\}$.

Theorem 3.2

A (normalized) binary ${\bf s}$-array $\phi$ is a GOBA$({\bf s})$ of type ${\bf z}\not={\bf 0}$ if and only if $f_{\bf z}\partial\phi$ is quasi-orthogonal.

Proof

This is a consequence of Theorem 3.1, Remark 2, Proposition 2, and Lemma 3. $\Box$

We proceed to formulate ‘base’ cases of Theorems 3.1 and 3.2. Let $\partial\phi\in B^{2}(G,\mathbb{Z}_{2})$. Since $M_{\partial\phi}$ is Hadamard equivalent to a group-developed matrix, and such a matrix has constant row sum, $\partial\phi$ can be orthogonal only if $|G|$ is square. This situation has been extensively studied.

Theorem 3.3

Let $|G|=4u^{2}$, and let $D$ be a subset of $G$ of size $2u^{2}-\allowbreak u$. Define $R=\{(\phi(g),g)\,|\,g\in G\}\subset{\mathbb{Z}}_{2}\times G$ where $\phi:G\rightarrow\{\pm 1\}$ is the characteristic function of $D$. Then the following are equivalent.

• (i)

$\partial\phi$* is orthogonal.*

• (ii)

$D$* is a Menon-Hadamard difference set in $G$.*

• (iii)

$R$* is a relative $(4u^{2},2,4u^{2},2u^{2})$-difference set in ${\mathbb{Z}}_{2}\times G$ with forbidden subgroup ${\mathbb{Z}}_{2}\times\{1_{G}\}$.*

• (iv)

$\phi$* is a perfect nonlinear function.*

If $G$ is abelian then (i) – (iv) are further equivalent to

• (v)

$\phi$* is a PBA.*

Proof

See [15, Theorem 1] for (iii) $\Leftrightarrow$ (iv). The other equivalences are given by Theorem 2.6 and Lemma 2.10 of [6]. $\Box$

Remark 3

In Theorem 3.3 and Theorem 3.4 below we may assume that $\phi$ is normalized, by taking the complement of $D$ (and thus also of $R$) if necessary.

The next theorem is an analog of the previous one for $|G|\equiv 2\;\mbox{mod}\;4$ (recall that we have not found quasi-orthogonal coboundaries over non-abelian $G$ at orders greater than $6$).

Theorem 3.4

Let $G$ be abelian of order $4t+2$, and let $D$ be a $k$-subset of $G$ with characteristic function $\chi:G\rightarrow\mathrm{GF}(2)$. Define $R=\{(\phi(g),g)\,|\,g\in G\}\subset{\mathbb{Z}}_{2}\times G$ where $\phi(x)=(-1)^{\chi(x)}$. Then the following are equivalent.

• (i)

$\partial\phi$* is quasi-orthogonal.*

• (ii)

$D$* is a $(4t+2,k,k-(t+1),(4t+1)(k-t)-k(k-1))$-almost difference set in $G$.*

• (iii)

$R$* is an extremal relative $(4t+2,2,4t+2,2t+1)$-quasi-difference set in ${\mathbb{Z}}_{2}\times G$ with forbidden subgroup $\mathbb{Z}_{2}\times\{1_{G}\}$.*

• (iv)

$\phi$* is an OBA.*

If a difference set with parameters $(n,\frac{n\pm\sqrt{3n-2}}{2},\frac{n+2\pm 2\sqrt{3n-2}}{4})$ does not exist, then (i) – (iv) are further equivalent to

• (v)

$\chi$* has optimal nonlinearity $(t+1)/(2t+1)$.*

Proof

Put $|G|=n$.

(i) $\Leftrightarrow$ (iv): Lemma 1 and the fact that $\phi(g)R_{\phi}(g)$ is the sum of row $g$ in $M_{\partial\phi}$.

(i) $\Leftrightarrow$ (ii): by Lemma 4, $R_{\phi}(g)=2$ or $-2$ if and only if $d_{\phi}(g)=\allowbreak k-t-1$ or $k-t$, respectively. Identity (19) of [5] then accounts for this part.

(i) $\Leftrightarrow$ (iii): Proposition 2 together with the isomorphism $E_{\partial\phi}\rightarrow\mathbb{Z}_{2}\times G$ defined by $(u,g)\mapsto(u\phi(g),g)$; cf. Proposition 4.

(ii) $\Leftrightarrow$ (v): see [5, Theorem 25]. $\Box$

Remark 4

The condition attached to (v) is only needed for (v) $\Rightarrow$ (ii). No difference sets with the stated parameters are known; see [5, Remark II, p. 224].

We end this section with a discussion of calculating GOBAs. Label the elements of $G$ as $g_{1}=0,g_{2},\ldots,g_{4t+2}$, and let $\delta_{k}\colon G\rightarrow\{\pm 1\}$ be the characteristic function of $\{g_{k}\}$. Up to relabeling, $\{\partial_{2},\ldots,\partial_{4t+1}\}$ is a basis of $B^{2}(G,\langle-1\rangle)$, where $\partial_{k}:=\partial\delta_{k}$ is an elementary coboundary. Choose ${\bf z}\neq{\bf 0}$. We first try to find quasi-orthogonal $\psi\in Z^{2}(G,\langle-1\rangle)$ such that $f_{\bf z}\psi\in B^{2}(G,\langle-1\rangle)$. Straightforward linear algebra gives the decomposition $\psi=f_{\bf z}\prod_{k}\partial_{k}^{i_{k}}$. Then $\phi=\prod_{k}\delta_{k}^{i_{k}}$ is a GOBA(s) of type ${\bf z}$ over $G$.

Example 1

The maps \phi_{1}={\small\left[\begin{array}[]{rrr}1&-1&\phantom{-}1\\ 1&1&\phantom{-}1\end{array}\right]}, \phi_{2}={\small\left[\begin{array}[]{rrr}1&\phantom{-}1&-1\\ 1&\phantom{-}1&1\end{array}\right]}, \phi_{3}={\small\left[\begin{array}[]{rrr}1&1&-1\\ 1&-1&\phantom{-}1\end{array}\right]} on ${\mathbb{Z}}_{6}={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{3}$ are GOBA($2,3$)s of type ${\bf z}_{1}=(1,0)$, ${\bf z}_{2}=(0,1)$, ${\bf z}_{3}=(1,1)$, respectively. We display each quasi-orthogonal cocycle $f_{{\bf z}_{i}}\partial\phi_{i}$ as a Hadamard (componentwise) product:

[TABLE]

[TABLE]

[TABLE]

Note that $f_{{\bf z}_{2}}\partial\phi_{2}$ is a quasi-orthogonal coboundary; as are all the $\partial\phi_{i}$.

Example 2

The map {\small\left[\begin{array}[]{rrrrrrrrr}1&\ -1&1&\ -1&\ -1&\phantom{-}1\\ 1&1&\ -1&1&1&1\\ 1&1&1&1&1&1\end{array}\right]}^{\top} on ${\mathbb{Z}}_{6}\times{\mathbb{Z}}_{3}={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$ is a GOBA($6,3$) of type ${\bf z}=(1,0)$. Its quasi-orthogonal cocycle is $f_{\bf z}\partial_{4}\partial_{8}\partial_{10}\partial_{13}$.

4 Negaperiodic Golay pairs

In this section we explore how GOBSs can be used to construct NGPs.

Proposition 5 ([8, Theorem 3])

Binary sequences $\phi_{1}$, $\phi_{2}$ of length $2t$ form an NGP if and only if $\{x^{i}\mid\phi_{1}^{\prime}(i)=1\}\cup\{x^{i}y\mid\phi_{2}^{\prime}(i)=1\}$ is a relative $(4t,2,4t,2t)$-difference set in the dicyclic group $Q_{8t}=\langle x,y\,|\,x^{2t}=y^{2},\allowbreak y^{4}=1,\,y^{-1}xy=x^{-1}\rangle$.

Remark 5

By Proposition 5 and [2, Theorems 5.6 and 5.7], NGPs of length $(q+1)/2$ exist for all prime powers $q\equiv 3\ \;\mbox{mod}\;4$.

Proposition 5 ties NGPs into the mainstream theory of cocyclic Hadamard matrices: by [9, Proposition 6.5], existence of a $(4t,2,4t,2t)$-difference set in $Q_{8t}$ is equivalent to existence of certain orthogonal cocycles over the dihedral group $D_{4t}$ of order $4t$. (Incidentally, this gives another justification of Remark 5, via Ito’s Hadamard groups of quadratic residue type [12, pp. 986–987].) These cocycles lie in a single cohomology class, with representative labeled $(A,B,K)=(1,-1,-1)$ in [9]; $A$, $B$ are ‘inflation’ variables and $K$ is the ‘transgression’ variable in a Universal Coefficients theorem decomposition of $H^{2}(D_{4t},\mathbb{Z}_{2})$.

The next theorem makes Proposition 5 more explicit. It shows how to translate directly between cocycles and NGPs. When the latter are complementary GOBSs, this implies existence of orthogonal cocycles if there exist quasi-orthogonal cocycles at half the order (unfortunately, the process does not reverse).

Theorem 4.1

Let $G=\langle a,b\,|\,a^{n}=b^{2}=1,a^{b}=a^{-1}\rangle\cong D_{2n}$ with elements ordered as $1,a,\ldots,a^{n-1},b,\allowbreak ab,\ldots,a^{n-1}b$. Also let $\phi_{1}$, $\phi_{2}$ be binary sequences of length $n$, and define $j_{k,i}$ to be $1$ or [math] depending on whether $\phi_{i}(k)=-1$ or $1$, respectively. Then $(\phi_{1},\phi_{2})$ is an NGP if and only if $\lambda\prod_{k=1}^{n}\partial_{k}^{j_{k,1}}\partial_{n+k}^{j_{k,2}}$ is an orthogonal cocycle over $G$, where $\lambda$ is the cohomology class representative labeled $(A,B,K)=(1,-1,-1)$ in [9, Section 6].

Proof

The center of $\langle x,y\,|\,x^{n}=y^{2},\allowbreak y^{4}=1,\,y^{-1}xy=x^{-1}\rangle\cong Q_{4n}$ is $\langle x^{n}\rangle$. Since $G\cong Q_{4n}/\langle x^{n}\rangle$, we may define a transversal map $\sigma:G\rightarrow Q_{4n}$ by

[TABLE]

where $\delta$ is the Kronecker delta. Assuming that $\phi_{1}$ and $\phi_{2}$ are normalized, let $\psi$ be the cocycle for $\sigma$, i.e., $\psi(g,h)=\sigma(g)\sigma(h)\sigma(gh)^{-1}$. By Proposition 5 and [6, Corollary 2.5], $\psi$ is orthogonal if and only if $(\phi_{1},\phi_{2})$ is an NGP.

Set $\varphi(a^{i})=\phi_{1}(i)$ and $\varphi(a^{i}b)=\phi_{2}(i)$. Then $\lambda=\psi\hskip 0.5pt\partial\varphi$ has matrix

[TABLE]

where $A=[(-1)^{\lfloor(i+j)/n\rfloor}]_{0\leq i,j\leq n-1}$ is back negacyclic, and $B$ is $A$ with rows $r$ and $n-r+1$ swapped for $1\leq r\leq n$. Furthermore, $\partial\varphi=\prod_{k=1}^{n}\partial_{k}^{j_{k,1}}\partial_{n+k}^{j_{k,2}}$ under the stipulated ordering of $G$. $\Box$

We now undertake a case study of quasi-orthogonal cocycles over cyclic groups. Let $G=\mathbb{Z}_{4t+2}$ and index matrices by $1,\ldots,4t+2$ in this order. The set ${\cal B}=\{\gamma,\partial_{i}\,|\,\allowbreak 2\leq i\leq 4t+2\}$ where $\gamma=\gamma_{4t+2}$ (as defined before Proposition 3) is a basis of $Z^{2}(G,\mathbb{Z}_{2})$. We get an elementary coboundary matrix $M_{i}:=M_{\partial_{i}}$ by normalizing the back circulant matrix whose first row is $1$s except for the $i$th entry. Also, $M_{\gamma}$ is the back negacyclic matrix $N$ of order $4t+2$.

Lemma 5

Let $\psi\in Z^{2}(G,\mathbb{Z}_{2})\setminus B^{2}(G,\mathbb{Z}_{2})$, say $M_{\psi}=M_{i_{1}}\circ\cdots\circ M_{i_{w}}\circ N$. Then

• (i)

*up to sign, $M_{\psi}$ has *$i$*th row sum equal to its *$(4t+4-i)$th row sum.

• (ii)

*The *$(2t+2)$th row sum of $M_{\psi}$ is [math].

• (iii)

$\psi$* is quasi-orthogonal if and only if the *$i$th row sum of $M_{\psi}$ is [math] for even $i$ and $\pm 2$ for odd $i>1$.

Proof

If $\psi\in\mathcal{B}$ then row $i>2t+2$ of $M$ or its negation is row $(4t+4-i)$ cycled $4t+4-i-1$ positions to the right. Part (i) then follows. For (ii), observe that row $2t+2$ in $N$ is $[1\stackrel{{\scriptstyle 2t+1}}{{\cdots}}1$ $-1\stackrel{{\scriptstyle 2t+1}}{{\cdots}}-1]$, whereas the first half of row $2t+2$ in $M_{i}$ is identical to the second half. Finally, (iii) holds because the number of $-1$s in any row of $M_{i}$ is even; and the rows of $N$ indexed by an even (respectively, odd) integer have an odd (respectively, even) number of $-1$s. $\Box$

We use an approach borrowed from [1] to count the negative entries in a $G$-cocyclic matrix. Negating row $i$ of $M_{i}$ gives a generalized coboundary matrix $\overline{M}_{i}$, with exactly two $-1$s in each non-initial row $r$: these are in columns $i$ and $[i-r+1]_{4t+2}$, where $[m]_{n}\in\{1,\ldots,n\}$ denotes the residue of $m$ modulo $n$. (Although $\overline{M}_{i}$ is not cocyclic, row negation preserves row excess.) Hence the two generalized coboundary matrices with $-1$ in position $(r,c)$ are $\overline{M}_{c}$ and $\overline{M}_{[r+c-1]_{4t+2}}$.

A set $\{\overline{M}_{i_{j}}\colon 1\leq j\leq w\}$ defines an $r$-walk if there is an ordering $\overline{M}_{l_{1}},\ldots,\overline{M}_{l_{w}}$ of its elements such that $\overline{M}_{l_{i}}$ and $\overline{M}_{l_{i+1}}$ both have $-1$ in row $r$ and column $l_{i+1}$, for $1\leq i\leq w$. The walk is an $r$-path if its initial (equivalently, final) element shares a $-1$ in row $r$ with a generalized coboundary matrix not in the walk itself. Clearly, the number of $-1$s in row $r$ of $\overline{M}_{i_{1}}\circ\cdots\circ\overline{M}_{i_{w}}$ is $2{\cal C}_{r}$ where ${\cal C}_{r}$ is the number of maximal $r$-paths in $\{\overline{M}_{i_{1}},\ldots,\overline{M}_{i_{w}}\}$. To calculate $\mathcal{C}_{r}$ we set up a bipartite graph on vertex sets $S=\{i_{1},\ldots,i_{w}\}$ and $T=\{[i_{1}-r+1]_{4t+2},\ldots,[i_{w}-r+1]_{4t+2}\}$. Draw an edge between $i_{j}\in S$ and $l\in T$ if $i_{j}=l$ or $l=[i_{j}-r+1]_{4t+2}\in S$. The number of maximal paths in this bipartite graph is $\mathcal{C}_{r}$.

Next, let ${\cal I}_{r}$ be the number of columns where $N$ and $\overline{M}_{i_{1}}\circ\cdots\circ\overline{M}_{i_{w}}$ share a $-1$ in row $r$. These column indices comprise the intersection of $\{4t+4-r,\ldots,4t+2\}$ and the set of endpoints of the previously calculated maximal $r$-paths.

Theorem 4.2 (cf. [1, Proposition 1])

A $\mathbb{Z}_{4t+2}$-cocyclic matrix $M_{i_{1}}\circ\cdots\circ M_{i_{w}}\circ N$ is quasi-orthogonal if and only if, for $2\leq r\leq 2t+1$,

[TABLE]

Proof

The number of $-1$s in row $r$ of $\overline{M}_{i_{1}}\circ\cdots\circ\overline{M}_{i_{w}}\circ N$ is $2{\cal C}_{r}+r-1-2{\cal I}_{r}$, so Lemma 5 gives the result. $\Box$

Corollary 2

Let $\psi=\gamma\prod_{j=1}^{w}\partial_{i_{j}}$ with $\partial_{i_{j}}\in{\cal B}$. If $\psi$ is quasi-orthogonal then $t\leq w\leq 3t+1$.

Proof

We have ${\cal I}_{2}=0$, and ${\cal C}_{2}=t$ by Theorem 4.2. Thus $t\leq w$. On the other hand, since the basis of coboundaries forms a $2$-path, at least $t-1$ coboundaries must be removed to get $t$ $2$-paths. Hence $w\leq 4t-(t-1)$. $\Box$

Corollary 2 is equivalent to

Lemma 6

If $\phi\colon\mathbb{Z}_{4t+2}\rightarrow\{\pm 1\}$ is a GOBS containing $w$ occurrences of $-1$ then $t\leq w\leq 3t+1$.

Proof

Negating all odd index entries or all even index entries of a GOBS produces another GOBS. So it may be assumed that $\phi(0)=\phi(4t+1)=1$. $\Box$

We search for NGPs in the set of quasi-orthogonal cocycles over ${\mathbb{Z}}_{4t+2}$, motivated by the ubiquity of these cocycles and the optimal autocorrelation of each map in the resulting pair. Computer-aided searches found the NGPs in Table 1.

Each sequence in Table 1 starts with $1$ and is designated by an integer string, where $i$ in the string means a run of $i$ identical entries in the sequence, and $1^{j}$ is an alternating subsequence of length $j$. There are no NGPs among the sequences coming from quasi-orthogonal cocycles over $\mathbb{Z}_{22}$ (however, as we know, NGPs of length $22$ exist). This gap could be related to the maximal determinant problem: the Ehlich-Wojtas bound is not attainable because $21$ is not a sum of two squares.

Egan [8] classified NGPs of length $2k$ for $k\leq 10$ up to equivalence with respect to five elementary operations as defined in [4]. The set of NGPs that come from GOBSs is invariant under each elementary operation. Table 2 records the number $\hat{n}(k)$ of such NGPs of length $2k$, and the number $\hat{d}(k)$ of their equivalence classes. To compare against [8, Table 2], we have included the total number $n(k)$ of NGPs of length $2k$ and the number $d(k)$ of their equivalence classes.

5 Normal cocyclic matrices

This section is essentially independent of the main thrust of the paper. Nonetheless, it addresses a fundamental question in algebraic design theory, which we answer in special cases that were the focus of Section 4.

A matrix $M$ is normal if it commutes with its transpose (possibly up to row or column permutations), i.e., $\mathrm{Gr}(M)=\mathrm{Gr}(M^{\top})$, where $\mathrm{Gr}(M)$ denotes the Grammian $MM^{\top}$. Many kinds of pairwise combinatorial designs are normal matrices (the defining pairwise constraint on rows implies the same constraint on columns; see [7, Chapter 7]). We also note that the matrix of a quasi-orthogonal cocycle is normal [3, Remark 6]. Thus, by the following lemma derived from (2), a cocycle $\psi$ is quasi-orthogonal if and only if $M_{\psi}$ has optimal column excess.

Lemma 7

For any group $G$ and $\psi\in Z^{2}(G,\mathbb{Z}_{2})$,

[TABLE]

and

[TABLE]

We use Lemma 7 to prove that cocyclic matrices for two familiar classes of indexing groups are normal.

Proposition 6

Let $G$ be abelian or dihedral of order $2m$, $m$ odd, and let $\psi\in Z^{2}(G,\mathbb{Z}_{2})$ where $\psi\not\in B^{2}(G,\mathbb{Z}_{2})$ if $G$ is dihedral. Then $M_{\psi}$ is normal (under the same indexing of rows and columns by the elements of $G$).

Proof

We suppose that $G$ is generated by $a$ and $b$, with $a^{m}=b^{2}=1$, and index rows and columns by the elements of $G$ under the ordering $1,a,\ldots,a^{m-1},\allowbreak b,ab,\ldots,a^{m-1}b$. A representative $\beta$ for the non-identity element of $H^{2}(G,\mathbb{Z}_{2})$ has matrix

[TABLE]

Thus, if $G$ is abelian then $M_{\psi}$ is symmetric and so trivially normal.

Henceforth $G$ is dihedral. Let $\psi=\beta\partial\phi$. We collect together some basic properties of $M_{\psi}$.

1. (i)

For each $i$, $\{\partial\phi(a^{i}b,a^{j})\;|\;1\leq j\leq m\}=\{\partial\phi(a^{i}b,a^{j}b)\;|\;1\leq j\leq m\}$; and for each $j$, $\{\partial\phi(a^{i},a^{j}b)\;|\;1\leq i\leq m\}=\{\partial\phi(a^{i}b,a^{j}b)\;|\;1\leq i\leq m\}$. Thus, if $k>m$ then the $k$th row sum and $k$th column sum of $M_{\psi}$ are zero. 2. (ii)

Since $\{\partial\phi(a^{i},a^{j}b)\;|\;1\leq j\leq m\}=\{\partial\phi(a^{j}b,a^{i})\;|\;1\leq j\leq m\}$, the $k$th row sum of $M_{\psi}$ equals its $k$th column sum for $k\leq m$.

Now we consider the Grammian quadrants in turn.

If $1\leq i\leq m$ and $m+1\leq j\leq 2m$ then

[TABLE]

by Lemma 7 and (i); $\mathrm{Gr}(M_{\psi}^{\top})_{ij}=0$ similarly.

Let $1\leq i\leq m$ and $1\leq j\leq m$. Then

[TABLE]

and

[TABLE]

These entries are equal by the identity $\sum_{k=1}^{m}\phi(a^{k})\phi(a^{k+1})=\allowbreak\sum_{k=1}^{m}\phi(a^{k})\phi(a^{k-1})$.

Finally, let $m+1\leq i,j\leq 2m$. Then

[TABLE]

and

[TABLE]

Since $\psi(a^{i-1}b,a^{i-j})=\partial\phi(a^{i-1}b,a^{i-j})=\partial\phi(a^{i-j},a^{j-1}b)=\psi(a^{i-j},a^{j-1}b)$, we are done by (ii). $\Box$

Remark 6

There are plenty of examples of non-normal cocyclic matrices $M_{\psi}$ for $\psi\not\in B^{2}(G,\mathbb{Z}_{2})$ and $|G|$ divisible by $4$.

Acknowledgments.

The authors thank Kristeen Cheng for reading the manuscript, and Víctor Álvarez for his assistance with computations. We are also grateful to Ronan Egan, who shared his insights on NGPs with us. Remark 5 and reference [2] were kindly provided by one of the referees. This research has been supported by project FQM-016 funded by JJAA (Spain).