数理解析研究所講究録
1190 巻 2001 年 61-81

61組み合わせ構造の研究への表現の利用法
画帳義塾大学理工学部

萩田真理子

Mariko Hagita

Department of Mathematics, Keio University
Abstract群が関わる組み合わせ構造で、 その存在性を調べるために群の表現を利用できるものをいくつかあげ、 それによって導かれた結果を紹介する。
’

$J$

1

Introduction

1.1

Difference Sets and Symmetric Designs

Let

$G$

be a group of order . A -subset $D$ in
$v$

$k$

$G$

is called a$(v, k, \lambda)$-difference set oflist of “differences” $\{xy^{-1}(x, y\in D)\}$ contains each nonidentity element of
Let $G:=Z/7Z=\{0,1,2, \ldots, 6\}$ and $D:=\{1,2. ’ 4\}\subset G$ .
(($\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{s}$”$G$exactly

Then the$1\mathrm{i}\mathrm{s}\mathrm{t}\backslash$$G$$\lambda$

if thetimes.

of nonzerois$2-1=1$ , $1-2=-1=6$ , $4-1=3$ ,$1-4=-3=4$ , $4-2=2$ , $2-4=-2=5$ .
Hence $D$ is a (7, 3, 1)-difference set of$G$.

We call $n:=k-\lambda$ the order of $D$ . Difference sets of order $n=0$ or 1 (i.e.
set, single sets, or their complements inany$(v, k, \lambda)$-difference set $D$ of order

$n,$

$G$

) are called$D’=G\backslash D$

the same order . Therefore we can assume
$n$

$D$

in some group$G$$k(k-1)=\lambda(v-1)$ .

$\zeta(\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{a}\mathrm{l})$$D=$ empty’difference sets. Note that foris a $(v, v-k, v-2k+\lambda)$ -difference set of$1<k \leq\frac{k}{2}$. If there exists a $(v, k, \lambda)$ -difference setof order , then by counting the number of nonidentity elements we have$v$

62
The main problem on difference set is which group has a nontrivial difference set. For
abelian cases, Jungnickel $(1992,[17])$ constructed a list of solutions of the problem for all possible
parameters with order$n$of the difference sets up to 30. Many constructions for difference sets

and many nonexistence theorems for difference sets are known. However, much work remains
to fill the gap between constructions and nonexistence theorems, even for abelian cases.

An incidence structure is a triple $(P, B, I)$ , where
and

$I$

is an incidence relation between $P$ andfor any set$T$symmetric$(v, k, \lambda)$

For any$G$of points, there are exactly$t$$\lambda$

$B$

.

$P$

is a set of points,

is a set of blocks,$B$

An incidence structure is called a -design if$t$blocks incident to all points in$T$for some$\lambda$.

A-design is a 2-design satisfying $|P|=|B|=v$ .

$(v, k, \lambda)$-difference set $D$ in$G$, we can define an incidence structure with point set, block set $\{Dg|g\in G\}$ and inclusion for incidence relation. Then the incidence structure isa symmetric$(v, k, \lambda)$-design.

The following theorem is the connection between difference sets and symmetric designs.
Theorem 1 $A(v, k, \lambda)$ -difference set inis equivalent to a symmetric$G$$(v_{\gamma}k, \lambda)$-design with aregular automorphism group.

1.2
Let

Group Rings and Characters$G$be a multiplicatively written group of order

is identified with$\sum_{s\in S}S$$D^{(-1)}= \sum_{d\in D}d^{-1}$. Also, for $A= \sum_{g\mathit{9}}\in c^{ag},$$v$.

In the group ring $ZG$ , a subset$\sum g\in c^{a_{g}}g^{t}$is denoted by$A^{(t)}$

.

$S$

of

$G$

So $D=\Sigma_{d\in D}d$ ,and $G= \sum_{g\in^{c}}g$ . Then a -subset $D$ is a $(v, k, \lambda)$ -difference set of $G$ if and$k$only if$DD^{(-1)}=n\cdot 1+\lambda G$

in $ZG$ , where 1 is the identity element of

We write$\xi_{t}$$G$.

for a primitive complex t-th root of unity and $[k]$ for the set $\{0,1, \ldots, k-1\}$ .

The cyclic group of order$k$is denoted as$Z_{k}$, and is often identified with $Z/kZ$ or $[k]$ without

explicitly mentioning it.

Let

$G$

be an abelian group anddenote the set of all$\chi\in G^{*}$$G^{*}$

be the character group ofwhich are trivial on

$U$

by

$U^{\perp}$.

$G$. For a subgroup $U$ of $G$ , we

We write $\exp(G)$ for the exponent

63$\mathrm{o}\mathrm{f}G$.

A prime

where
$n$

$e’$

$p$

is called self-conjugate mod$e$if there exists an , such that $p^{i}\equiv-1$ (mod$i$is the -free part of . If each prime divisor of
$e$

$p$

$n$

$e’$

)

is self-conjugate mod , then we say$e$is self-conjugate mod .
$e$

For any prime idealand$\wp^{i+1}\not\supset\Re$$\wp$

and any ideal$\Re$

in a ring, we write $\nu_{\wp}(\Re)=i$ if and only if$\wp^{i}\supset\Re$.

The following was used by Turyn in [31].

Lemma 2 Let
then$\wp=\overline{\wp}$$\wp$

be a prime ideal ofover a prime integer . If

$Z[\xi_{e}]$$p$is self-conjugate mod ,
$e$

$p$

holds, where $-denoteS$ the complex conjugation.

We have the following well-known result.
Lemma 3 (Inversion Formula) Let$G$

be a finite abelian group, and let $A= \sum_{g\in c^{a_{g}}}g\in ZG$ .

Then$a_{g}= \frac{1}{|G|}\sum_{G\chi\in*}x(A)x(B)$.

Hence, if $A,$ $B\in ZG$ satisfies $\chi(A)=\chi(B)$ for all characters

Then $D$ is a$(v, k, \lambda)$-difference set of abelian group$G$$\chi$of$G$, then $A=B$ holds.

if and only if$DD^{(-1)}=n\cdot 1+\lambda G$

in $ZG$ , where 1 is the identity element of$\chi(D)\overline{\chi(D)}=n$

If we assume that

$n$

$G$

, and this is equivalent tofor all nontrivial characters$\chi$

of$G$.

is a square, and is self-conjugate mod $\exp(G)$ , then the condition isequivalent to(1)$\frac{1}{\sqrt{n}}\chi(D)=u_{\chi}$for all nontrivial characters
because$\chi$$\nu_{\wp}(x(D))=\iota \text{ノ_{}\wp}(\overline{x(D)})$

of

$G$

, where$u_{\chi}$is a root of unity. This follows from Lemma 2

for all prime ideals

$\wp$

in

$Z[\xi_{e}]$over , where $e=\exp(c)$ .
$p$

64
Suppose that the cyclic decomposition of
to $Z/u_{i}Z$ for $i=1,2,$

is

$G$

$C_{1}\cross C_{2}\mathrm{x}\ldots\cross C_{t}$, where$C_{i}$

is isomorphic. Then the group ring $ZG$ is isomorphic to the ring$t$$\ldots,$$Z[x_{1}, x2, \ldots, xt]/(x_{1}^{u_{1}}-1, x_{2}^{u_{2}}-1, \ldots, xu_{t}t-1)$.

In this ring, we can express the difference set $D$ as$D=D(X_{12\cdot\cdot t}, X,., x)= \sum_{a_{i}\in[ui]}d\cdot xa1a_{2}\ldots at1X^{a..a}a122.x_{t}t$with$d_{a_{1}\ldots a_{t}}=0$

or 1. Here, the condition (1.1) is equivalent to that$\frac{1}{\sqrt{n}}D(\zeta_{1}^{\alpha_{1}},$for all

except$(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{t})\in G$

Lemma 4 If an abelian group$exp(G)$ ,

then

$n$

Proof. Let
any character$p$$\chi$$(0,0, \ldots, \mathrm{o})$$G$

is a root of unity$(_{2}^{\alpha_{2}}, \ldots\}\zeta_{t}^{\alpha}t)$, where$\zeta_{i}=\xi_{u_{i}}$.

has a difference set whose orderis self-conjugate mod$n$

is a square.

be a prime divisor of . If there is a prime $q\neq p,$ $q|v$ , then from Lemma 2, for$n$of order

$q$

of

$G$

,
$\iota \text{ノ_{}\wp}(n)=2\nu_{\wp}(x(D))$

for any prime ideal

$\wp$

over

$p$

in

$Z[\xi_{q}]$. But since$p$is unramified in$\iota \text{ノ_{}\wp}(n)=\nu_{p}(n)$

Thus,

If

$\nu_{p}(n)$$\mathrm{t}\mathrm{h}\mathrm{e}\grave{\mathrm{r}}\mathrm{e}$$Z[\xi_{q}]$,.

is even.
is no prime$q$which satisfies $q|v,$ $q\neq p$ , then$v$

is a prime power

Since $v>n$ , we have $a>b$ . Then, since $p^{a}|\lambda v=k2-n$ and $a>b$ , we haveSo$b$$p^{a}$. Let $\nu_{p}(n)=b$ .

$b=\iota \text{ノ_{}p}(k2)=2\nu_{p}(k)$.

is an even integer.

Since

$n$

is positive,$n$is a square.

$\square$For nonabelian groups, we can use irreducible matrix representations instead of characters.
The following result sometimes helps us to extend some theorems on abelian groups to that ofnonabelian groups.
Theorem 5 (Brauer, see [8] (41.1)) Let

number field. Then

$Q(1^{\frac{1}{e}})$$e$be the exponent of

is a splitting field for$G$.

$G_{f}$and let$Q$

denote the rational

65

1.3

Famous‘ Constructions, Nonexistence Theorems and Conjec-tures
For any prime power$q$

and any integer $d>1_{\}}$ there exists a cyclic$(v, k, \lambda)$-difference set $D$with parameters$v= \frac{q^{d+1}-1}{q-1}$,$k= \frac{q^{d}-1}{q-1}\sim \mathrm{a}\mathrm{n}\mathrm{d}\lambda$$= \frac{q^{d-1}-1}{q-1}$.

.

This is equivalent to the symmetric design of points and hyperplains in the projective d-space$PG(d, q)$ over $GF(q)$ .

Let $q=4n-1$ be a prime power. Then the set $D$ of nonzero squares in$1,$$n-1)$ -difference set in the additive group of[Generalized$\mathrm{M}\mathrm{c}\mathrm{F}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{d}’ \mathrm{s}$. . . $+q^{2}+q+2$ )$\ldots$,. .

is a $(4n-1,2n-$$\backslash _{\mathrm{p}}$’$\mathrm{b}/$Difference Sets by Dillon] Let

be a group of order $q^{d+1}(q^{d}+$$G$which contains elementary abelian normal subgroup $E$ of order$r=q^{d}+\ldots+q^{2}+q$ , let $H_{0},$ $\ldots,$and iet go,$F_{q}$$F_{q}$$g_{r+1}$$H_{r}$$q^{d+1}$.

be the hyperplanes of $E$ (i.e., the subgroups of order

be a system of coset representatives of $E$ in

$G$

For
$q^{d}$),. Then the subset$D=g_{0}H_{0}+\ldots+g_{r}H_{r}$

of

$G$

is a difference set with parameters$v=q^{d+1}(q^{d}+\ldots+q+q+22),$ $k=q^{d}(q^{d}+\ldots+q+1),$$\lambda=q^{d}(q^{d-}+\ldots+q+11)$ .

Following two nonexistence theorems are due to Turyn (see Turyn [31], Lander [22]).
Theorem 6 Let $D$ be an abelian$p|n$-difference set in$G_{f}$and letand self-conjugate mod $exp(G)$ . Then the -Sylow subgroup of$p$Theorem 7 Let $D$ be an abelian$s$$(v, k, \lambda)$

and index$u$.

$(v, k, \lambda)$-difference set in

$G$

$G$

$p$

be a prime with $p|v$ ,is not cyclic.

, let $H$ be a subgroup of

Assume the existence of a positive integer $m$ with$m^{2}|n,$$G$$(m, u)\neq 1$of orderwhich isself-conjugate mod $exp(G/H)$ . If the -Sylow subgroup of $G/H$ is cyclic for every prime$p$$p|m,p|u$ ,then$m\leq 2^{r-1}s$, where$r$is the number of distinct primes dividing $(m, u)$ .

$p$

with

66

By the way, from these nonexistence theorems and known difference sets, Ryser and Lander
suggested the following conjectures.
[Ryser’s Conjecture] There is no cyclic$(v, k, \lambda)$-difference set with $(v, n)\neq 1$ .

[Lander’s Conjecture] Assume that there exists an abelian
is a prime with $p|v,$ $p|n$ , then -Sylow subgroup of
$p$

$G$

$(v, k, \lambda)$-difference set in

$G$

. If

$p$

is not cyclic.

In particular, these conjectures contain the following famous conjecture.
[Circulant Hadamard Conjecture] There exists no circulant Hadamard matrix of order $m>$$4$

. This is equivalent to that there is no cyclic $(4u^{2},2u^{2}-u, u^{2}-u)$ -difference set with $u>1$ .

Foldings of Difference Sets

2

Let $G,$ $G’$ be finite groups of the form $G=Z_{p^{m}}\cross H,$ $G’=U\cross H$ wherean abelian group with $p^{2}\exp H$ , and$U$

is any abelian group of order$p^{m}$$p$

is a prime, $H$ is. We exhibit a familyof bijections $Garrow G’$ which–extended to bijections $ZGarrow ZG’$ of the integral group ringsunder some conditions preserve the character values of group ring elements up to roots of unity.

We get similar results for related combinatorial structures like relative difference sets, building
sets and group invariant weighing matrices. The results can be generalized to cases where $H$

is nonabelian.
The content of this chapter has appeared in [16].

2.1Introduction

The main aim of this chapter is the study of the problem of switching groups for various typesof difference sets and related structures. That is, for groups $G,$ $H$ of the same order, we ask
whether we can find bijections between$G$

and $H$ preserving certain combinatorial propertiesof group ring elements like being a difference set. The main motivation is to gain more insight

into the existence of these combinatorial objects. If we can find bijections which work for many$H$

, then the construction of a single difference set, for instance, solves the existence problem

for difference sets in many groups simultaneously.

67

There are some known bijections between nonisomorphic groups preserving difference sets.
[Dillon, see [4]] Let $D$ be a $(n^{2}+n+1, n+1,1)$ -difference set in $G=Z_{v}$ with $v=n^{2}+n+1$ ,$n\equiv 1$

mod 3. (It always exists when$a>\mathrm{x}<b>$$v$

is a prime power.) Then we can write $G=Z_{3}\cross Z_{w}=<$

for $w=v/3$ since $3||v$ . Let
$H$

$f$

$:=<x,$ $b|x^{3}=b^{w}=1,$ $x^{-1}b_{X}=b^{n}>$ ,: $Garrow H$ with$f(a^{i}b^{i})=x^{i}\triangleright\dot{\uparrow}$for all$i,$ $j$.

Then $f(D)$ is a $(n^{2}+n+1, n+1,1)$ -difference set in $H$ .

We also use a bijection from generalized quaternion groups to abelian groups in the previous
chapter.
However, these two constructions only work for very special types of groups. The aim of
this chapter is to find bijections preserving difference sets for more general groups.
But when we try to find such bijections for general groups, we have some troubles:
1) Difference sets with Singer parameters (see [29, 4]) exist in cyclic groups, but in many cases

it can be shown that they do not exist in any other abelian groups of the same order. So,

bijections from$G$

to $H$ where $H$ has lower exponent (and higher rank) than$G$

cannot work ingeneral.
2) Turyn, Davis and Kraemer [31, 9, 21] proved that an abelian 2-groupa difference set if and only if $\exp G\leq 2^{d+1}$ . So bijections from

$G$

$G$

of order$2^{2d}$has

to $H$ where $H$ has higher

exponent, also cannot work in general.
3) Arasu, Davis, Jedwab, and Sehgal $[2, 3]$ proved that an Hadamard difference set in$Z_{2}\cross Z_{3^{a}}\cross K$$Z_{2}\cross$with $K$ abelian, $|K|=3^{a}$ , exists if and only if $K$ is cyclic. So bijections fromto $H$ with $\exp H=\exp G$ , rank

$H>\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}G$$G$

also cannot work in general.

However, we can still exhibit that quite general difference sets are preserved by some bijec-

tions given by using appropriate lexicographic orderings together with some nonhomomorphic
permutations.

Let $G=C\cross K,$ $H=U\cross K$ be finite groups where$C$

is a cyclic -group,
$p$

$U$

is any abeliangroup with $|U|=|C|$ , and $K$ is any finite abelian group such that $p^{2}\Lambda w:=\exp H$ . Using alexicographic ordering of$U$, we define a bijection$f$: $ZGarrow ZH$ which we call a folding. We

68will show that for any difference set $D$ in

of

$G$

and for some root of unity$u_{\chi}$$G$such that$\chi(D)u_{\chi}\in Q[\xi_{p}, \xi_{w}]$for every character$\chi$, the folding $f(D)$ of $D$ is also a difference set in $H$ . We getsimilar results for several other combinatorial structures which can be described by group ringequations. It is interesting to note that the$‘(\mathrm{s}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}$field condition”, i.e.,$\chi(D)u_{\chi}\in Q[\xi_{p}, \xi_{w}]$, isoften satisfied automatically by a consequence of [28, Thm. 3.5] which we will state as Theorem
11 .

2.2

Preliminaries

Let $m>1$ be an integer, and let
define the lexicographic order on$p$be a prime. For any partition$[p^{r_{1}}]\cross[p^{r_{2}}]\ldots\cross[p^{r_{S}}]$by

for $t= \min\{i|b_{i}\neq b_{i}’\}$ .

$(b_{1}, b_{2}, \ldots, bS)>(b_{1}’, b’\ldots, b’)2’ s\Leftrightarrow b_{t}>b_{t}’$Using this ordering of
$f$

$U$

of $m$ , we$(r_{1}, r_{2}, \ldots, r)S$

, we can define a bijective map naturally:: $K:=Z/p^{m}Zarrow K’:=Z/p^{r_{1}}Z\cross Z/p^{r_{2}}Z\cross\ldots \mathrm{x}Z/p^{r_{s}}Z$

sending the element of $K=[p^{m}]$ to the i-th element of$i$$K’=[p^{r_{1}}]\cross[p^{r_{2}}]\ldots \mathrm{x}[p^{r_{S}}]$. We call $f$

the folding map from $K$ to $K’$ .

We extend$f$to a bijection between $G=K\cross H$ and $G’=K’\cross H$ for any group $H$ by

setting $f((x, y))=(f(x), y)$ . This map is also denoted by$f$and called the folding map from$G$

to $G’$ . For $T=\Sigma_{g\in K}Hgg\in ZG$ with $H_{g}\in ZH$ , we define the folding $f(T)\in ZG’$ by$f(T):= \sum_{k\in K^{l}}H1(k)kf^{-}$

We say $f(T)$ is the folding ofWrite $K=Z/p^{m}Z=\langle g\rangle,$$T$.

. Note that $f^{-1}(b_{1)}\ldots, b)s=\Sigma_{i=1}^{s}bip^{\sum^{S}r_{j}}j=i+1$ .

$K’=Z/p^{r_{1}}Z\cross\ldots\cross Z/p^{r_{s}}Z=\langle g_{1}\rangle\cross\ldots \mathrm{x}\langle g_{S}\rangle,$$\eta=\xi_{p^{m}},$ $\eta_{i}=\xi_{p^{r}i}$

The following correspondence between the absolutely irreducible representations of$G$.

and $G’$is essential in all our results on folding maps. For the background on representation theory, see[8].

Let $G=K\cross H$ and $G’=K’\cross H$ as above where $H$ is any finite group. For$\psi(g_{1}, \ldots, gS)=\eta_{1}^{a_{1}}\eta_{2\eta_{S}}^{a}2\ldots a_{S},$by$\omega_{\psi}(g)=\eta^{t}$$0\leq a_{i}<p^{r_{i}}$, we define the character$\omega_{\psi}\in K^{*}$

where $t=a_{1}+a_{2}p^{r_{1}}+a_{3}p^{r_{1}+r_{2}}+\ldots+a_{s}p^{r_{1}}+\ldots+r_{s-1}$ .

$\psi\in K’*\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}$corresponding to$\psi$

69
By Brauer’s theorem [8, Thm. 41.1], $F:=Q(\xi_{\exp G})$ is a splitting field for
Moreover, any irreducible $FG’$ representation$\psi$

of $K’$ and an irreducible $FH$ representation

representation corresponding toso we speak of the character

2.3$\chi_{\tau}$$\tau$.

Note that$\tau$

can be written as$\tau$$\varphi$. Thenandcorresponding to$\tau$$\chi_{\tau}$$G$

for a character$\tau=\psi\otimes\varphi$$\chi_{\tau}:=\omega_{\psi}\otimes\varphi$

as well as $G’$ .

will be called the $FG$

are actually characters if $H$ is abelian,in this case.

The Folding Theorem

In this section, we prove that folding maps preserve the character values of certain groupring elements up to roots of unity. As a consequence, we will be able to show that someconubinatorial properties of group ring elements like being a difference set, relative difference
set, building set, or group weighing matrix are preserved under folding. We shall introduce

these applications in Section 2.4.
Theorem 8 Let $G=Z_{p^{m}}\cross H$ where $H$ is an abelian group and

Let $K$ be an abelian group of order$m\geq 2$ .

map. Let $T\in ZG$ . Let $\tau\in c’*$ , and let

If$\chi_{\tau}u\in Q[\xi_{\exp H}, \xi p]$$p^{m}\mathrm{z}$$\chi_{\tau}\in G^{*}$

for some root of unity$u$and let

Proof.

$u’$$\tau\in K^{*},$: $Garrow G’=K\cross H$ be the folding$\tau$, then(2),.

Write $K=Z_{p^{r_{1}}}\cross\ldots\cross Z_{p^{r_{S}}}=<g_{1}>\mathrm{x}\ldots \mathrm{x}<g_{s}>$, let

the folding map and let$\rho\in H^{*}$, if$\eta=\xi_{p^{\mathrm{n}}},,$$\eta_{i}=\xi_{p^{r_{S}}}$and$p^{2}$

I $\exp H$ , forwhere$\chi_{\tau}(g)=\eta^{\beta}$$\chi_{\tau}\in Z_{p^{m}}^{*}$: $Garrow G’=K\mathrm{x}H$ befor some root of unity , then$u$$\tau\otimes\rho(f(T))=\chi_{\mathcal{T}}\otimes\rho(T)u$

$u’$

$f$

. Let $T\in ZG$ . All we have to prove is that for any$\chi_{\tau}\otimes\rho(T)u\in Q[\xi_{\exp H}, \xi_{p}]$for some root of unity

does not divide $\exp H$ ,be the corresponding character of .

$\tau(f(T))=\chi_{\mathcal{T}}(T)u’$for some root of unity$f$$p^{2}$

is the character corresponding to, we can write$\chi_{\tau}\otimes\rho(\tau)=\sum_{b\in[pm]}db$.

$\eta^{\beta}b$(3)’$\tau$.

Since$\rho\in H^{*}$,

70with some$d_{b}\in Q[\xi_{p}, \xi_{w}]$, and for$\tau(g_{1}, \ldots, gS)=\eta_{1\eta_{S}}^{\beta_{1}\ldots\beta S}$$\mathcal{T}\otimes\beta(f(\tau))(b1,\ldots,bs)\in[p]\mathrm{r}_{1}\cross\sum_{\cross \mathrm{r}p^{\Gamma_{S}}]}\ldots d_{f^{-}}1(b_{1},\ldots,bS)$, we can write.

$\eta^{\beta 1}11b\eta_{2}\beta 2b_{2.\eta_{s}}..\beta_{s}\cdot b_{s}$.

Hence Theorem 8 follows from the next lemma.
Lemma 9 Let$p^{2}\parallel w$.

For nonnegative integers$\beta,$let$\beta_{i_{f}}$$d( \beta):=\sum_{[b\in pm]}d_{b}\cdot\eta^{\beta}b$,$d’( \beta_{1}, \ldots, \beta_{S}):=\sum_{s(b_{1},\ldots,b_{s})\in[_{P}]\mathrm{r}_{P^{r}}}r1\cross\ldots\cross]d-f1(b1,\ldots,b_{s})$where all$d_{b}\in Q[\xi_{p}, \xi_{w}]$.

$\eta_{1}^{\beta 1}1b\eta_{2}..\eta^{\beta_{s}\cdot b_{S}}\beta 2b_{2}.s$’. Suppose$d(p^{i}a)u\in Q[\xi_{p}, \xi_{w}]$for some root of unity$u$where $(p, a)=1$ . Then, for any$\beta_{j+1},$$\ldots$,, there is a root of$\beta_{s}\in Z$

unity $u’=u’(\beta_{j+1,\ldots,\beta_{s}})$ such that$d’(\mathrm{o}, \ldots, \mathrm{o}, p^{e}a, \beta_{j1}+, .

, .

, \beta_{s})=d(p^{i}a)u’$

where $i=r_{1}+r_{2}+\ldots+r_{j-1}+e$ with $0\leq e<r_{j}$ .

Proof. Note that
$t$

since

$\beta_{1},$$\ldots,$$\eta_{j}^{p^{r}}j=1$$\beta_{s}\in Z$$d’(\mathrm{O}, , .

.

, 0, p^{e}a, \beta_{j+1}, .

.

, , \beta_{s})=d’(\mathrm{o}, \ldots, \mathrm{o}, p^{e}a+p^{r_{j}}t, \beta j+1, \ldots, \beta_{S})$, i.e., $d’(\beta 1, \ldots, \beta_{s})=d(f^{-1} (\beta 1, \ldots , \beta_{s}))u$’ for some root of unity $u’$ for any. Let$\omega=\xi_{p}=\eta^{p^{m-1}}=\eta_{i}^{p^{m_{i^{-1}}}}$$d(p^{i}a)u=$Now.

$\sum$$\omega^{at}\gamma r^{i}a\cdot bu\in Q[\omega, \xi_{w}]$$b \in[_{\mathrm{P}^{m-}}i-1]t\in[p]\sum_{i+1}db+tp^{m}-i-1$

But$\eta^{p^{i}a},$for any$(\eta^{p^{i}a})^{p^{m}}-i-1-1$$(\eta^{p^{i}a})^{2},$$\ldots,$

are independent over$Q[\omega, \xi_{w}]$.

So, since.

$d_{b}\in Q[\omega, \xi_{w}]$is a unique $b=b_{0}$ such that$\sum_{t\in[p^{i}]+1}d_{b\mathrm{o}+t}pm-i-1^{\cdot}\omega^{at}u\in Q[\omega, \xi_{w}]$and all the other sums are
$d’$

$:=$

$0$

.

,

Define$d’(0, \ldots 0, p)ae,$ $\beta_{j}+1,$ $\ldots,$$\beta_{s})$$=$$(b_{1}, \ldots,b)\in[p^{r_{1}}]\chi\ldots \mathrm{X}\mathrm{r}_{P^{r_{s}}}]\sum_{s}df-1(b1,\ldots,b_{S})$.

$\eta_{j}^{pb}6a\cdot j\eta_{j}^{\beta}+j+11bj+1\ldots\eta_{s}^{\beta s}b_{s}$, there71
and let$L:=f^{-1}(0, \ldots, 0, bj+1, \ldots, b_{S})+b_{1}\cdot p+m-r_{1}\ldots+bj.pm-r_{1}-\ldots-r_{j}$.

Then we have$d’=$$\sum_{1,b_{\mathrm{j}}’\in[p]^{b}r_{j}-e-1,\ldots,b\mathrm{j}-1}.\sum_{b\mathrm{j}+1\cdot.,b_{S}},,.\sum t\in[p]e+1d_{L}+t\cdot p^{m}.\eta-i-1\omega atj\mathrm{P}\mathrm{e}a\cdot b_{\mathrm{j}}’\eta_{j^{j+}}^{\beta\cdot b_{j+}}+1\ldots\eta_{S}11\beta_{s}\cdot b_{S}$because$(\eta_{j}^{p^{\mathrm{e}}a})p^{r_{j^{-e-1}}}=\omega^{a}$the coefficient ofand.

$p^{r_{\mathrm{j}}-}e-1$$\eta_{j}^{p^{\mathrm{e}}a\cdot b_{j}’}\eta_{j1}+\cdots\eta\beta j+1bj+1\beta_{S}\cdot b_{s}s$$p^{m-r_{1}-}\ldots-r_{j}=p^{m-i-1}$

in

$d’$

. Thus, if we fix$b_{j}’,$,$b_{j+}1,$$\ldots,$

$bs$

’

is

$\sum_{b_{1},\ldots,b_{j-}1}\sum_{t\in[p]\mathrm{e}+1}dL+t\cdot pm-i-1^{\cdot}\omega^{at}$.

Putting$N:=f^{-1}(0, \ldots, 0, b_{j1,\ldots,s}+b)+b_{j}\cdot p^{m-r_{1}-r-\ldots-r_{\mathrm{j}}}2$,

we can write$L$for some$k$$=$$N+b_{1}\cdot p^{m-}+r1b_{2}\cdot p^{m}+\ldots+bj-1-r_{1}-r_{2}.p^{m-r-\Gamma_{2}}1-\ldots-r_{j-1}$$=$$N+k\cdot p^{e+}\cdot p^{m}1-i-1$. Since $\omega^{p}=1$ andvalues, the above coefficient$k$runs through$[p^{r_{1}+\ldots+r_{j-1}}]$when$\Sigma_{b_{1},\ldots,b_{\mathrm{j}-1}}\Sigma_{t\in}[p^{\mathrm{e}+1}]d_{Lp}+t\cdot m-i-1\omega^{at}$$b_{1},$$\ldots$,$b_{j-1}$

run through allis$k \in[p^{r_{1}+\ldots+}\sum_{1r\mathrm{j}-]t\in[}\sum_{p^{e}]}d_{N+}(kp^{\mathrm{e}+1}+t)p^{m-i-1}+1^{\cdot}\omega^{at}$$=$$\ldots\sum_{t\in[p^{r_{1}+}-1^{+}]}dN+t\cdot p-im-1+r_{j}\mathrm{e}+1^{\cdot}\omega^{at}$.

$=$$\omega^{at}$$\sum_{t\in[p^{i+}1]}dN+tp^{m}-i-1$

If$b_{j},$$b_{s}$$\ldots,$.

run through all values, $N=f^{-1}(0, \ldots, 0, b_{j}+1, \ldots, bs)+b_{j}\cdot p^{m-r_{1}-}\ldots-r_{j}$ runs

through all of $[p^{m-i-1}]$ . So the coefficients occurring in$d(p^{i}a)$

.

$d’$

coincide with the coefficients ofThus, with at most one exception, for each $(s-j+1)$ -tuple

occurring in$d’$is , and the only putative nonzero coefficient is$0$$t \in[p^{i+1}\sum_{]}d_{b\mathrm{o}+}tpm-i-1$,$(b_{j}, \ldots, b_{S})$, the coefficient72
i.e., $d’=d(p^{i}a)u$’ for some root of unity

$u’$

.

$\square$

For the sake of completeness, we mention that the folding theorem can be generalized to thecase where $H$ is nonabelian. The proof of the following theorem is a straightforward adaptationof the proof of Theorem 8 and will be omitted.
Theorem 10 Let $G=Z_{p^{m}}\cross H$ where $H$ is a (possibly nonabelian) group anddivide $\exp$ H. Let $K$ be an abelian group of orderfolding map. Let $F:=Q(\xi_{\exp}c)$ , let$\tau$$p^{m}$, and let$f$

$Q[\xi_{\exp H}, \xi p]$does not: $Garrow G’=K\cross H$ be thebe an irreducible $FG’$ matrix representation, and let

be the corresponding $FG$ representation. If, for some $T\in ZG$ , the matrixin$p^{2}$$\chi_{\tau}(T)u$$\chi_{\tau}$

has entries

only for some root of unity , then$u$$\tau(f(\tau))=\chi \mathcal{T}(\tau)u^{;}$

for some root of unity

$u’$

(4)

,

.

In the abelian case, the basic assumption necessary to make folding work is that the character
values of the group ring element we want to fold up to a root of unity lie in a rather small field.
For the combinatorial applications we have in mind, the character values will be cyclotomic
integers of prescribed absolute value. From the following consequence of [28, Theorem 3.5] we

see that the “small field assumption” is satisfied automatically in many cases.
Theorem 11 ([28]) Assume
and$m=p^{a}m_{f}’(p, m’)=1_{f}$each prime divisor

$q$

$p$

$X\overline{X}--n$$A(m, n,p)$ :$X\in Z[\xi_{m}]$is an odd prime. Letof define$n$for$m_{q}:=\Pi_{r\in P\backslash \{q}$$q^{\mathrm{O}\Gamma \mathrm{d}_{m_{q}}}(q)\not\equiv 1$

mod$p^{2}$} .
$r$$\prime \mathrm{p}$

where

$n$

and $m$ are positive integers,

be the set of prime divisors of $m$ . For

Consider the following assumption.

for all prime divisors $q\neq p$ of .
$n$

If $A(m, n, p)$ holds, then(5)

$X\xi_{m}^{j}\in Z[\xi_{\mathrm{P}^{m}}’]$

for some . In particular, (5) always holds if
$j$

$n$

is a power of .
$p$

73
2.4

Applications

In this final section, we present some applications of the folding and permutation theorems
to difference sets, relative difference sets, building sets, and group invariant weighing matrices.

We give the details for the nonabelian case only for difference sets. The other applications of
the folding theorem have similar nonabelian generalizations.

Differen.c Sets$\mathrm{e}$We begin with an application of the folding theorem to difference sets.
Theorem 12 Let $G=Z_{p^{m}}\cross H$ be an abelian group of orderthere is a$(v, k, \lambda)$-difference set

for some root of unity
is a$(v, k, \lambda)$$u_{\chi}$of order in such that
. Then for any partition-difference set in

Proof. For any character$u_{\tau}$

$D$

$n$

$G$

$v$

such that $p^{2}w:=\exp$ H. Suppose$\chi(D)u_{\chi}\in Q[\xi_{p}, \xi_{w}]$$(r_{1}, r_{2}, \ldots r)fs$$G’=Z_{p^{\mathrm{r}_{1}}}\cross Z_{p^{r_{2}}}\cross*\cdot$$\tau\neq id$of $G’$ , we have.

$\cross Z_{p^{r_{S}}}\cross H$

of

$m$

for any$\chi\in G^{*}$, the folding $f(D)$ of $D$.

$\tau(f(D))=\chi_{\tau}(D)u_{\mathcal{T}}$for some root of unity

by Theorem 8. Then,$\tau(f(D))\overline{\mathcal{T}(f(D))}=\chi_{\mathcal{T}}(D)u\tau\overline{\chi_{\tau}(D)}\overline{u_{\tau}}=\chi_{\tau}(D)\overline{\chi_{\mathcal{T}}(D)}=n$

concluding the proof.

$\square$Remark 13 By applying Theorem 12 to the known families of difference sets, we do not obtain

the existence of difference sets in any groups which previously had not been known to containdifference sets. However, we believe that Theorem 12 is important for the understanding of thephenomenon that difference sets with $(v, n)>1$ seem to”($‘ \mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{r}$groups of low exponent andhigh rank. Also, Theorem 12 certainly is of interest for the study of putative new families ofdifference sets and Lander’s conjecture, see Corollary 16.
The following result is useful.
Cororally 14 Let $G=Z_{p^{m}}\cross H$ be an abelian group with $p^{2}\exp$ H. Suppose that there is a-difference set in such that is self-conjugate mod $exp(H)$ . Then there is adifference set infor any abelian group of order .
$G$$(v, k, \lambda)$$U\cross H$

$n$

$U$

$p^{m}$$(v, k, \lambda)-$

74
Combining Theorem 12 with Theorem 11 we get the following result.
Cororally 15 Let $G=Z_{p^{m}}\cross H$ be an abelian group where

Suppose there is a$(v, k, \lambda)$-difference set of orderof Theorem 11 holds. Then there is aof order .

$U$$(v, k, \lambda, n)$

$n$

in

$G$

$p$

is an odd prime and $p^{2}\exp H$ .

and that the assumption $A(\exp G, n, p)$-difference set in$U\cross H$

for any abelian group$p^{m}$

An important unsolved conjecture of Lander [22] asserts that the Sylow -subgroup of an$p$abelian group containing a $(v, k, \lambda, n)$ -difference set with $p|(v, n)$ cannot be cyclic. In view ofLander’s conjecture, the following special case of Corollary 15 is of particular interest. Note thatin this situation, folding works without any assumptions besides the existence of a difference

set.
Cororally 16 Let $G=Z_{p^{m}}\cross H$ be an abelian group where

If there is adifference set in$(v, k, \lambda, n)$$p$

is an odd prime and $(p, |H|)=1$ .

-difference set in and is a power offor any abelian group of order .
$G$$U\cross H$

$n$

$U$

$p_{f}$

then there is a $(v, k, \lambda, n)-$$p^{m}$

Now we are going to describe the nonabelian version of Theorem 12. We encounter slighttechnical difficulties here, since, for an arbitrary nonlinear matrix representation$G$$\rho$

of a groupand $D\in ZG$ , the matrix $\rho(D^{(1)}-)$ is slightly more difficult to obtain from $\rho(D)$ than in thelinear case where just$\rho(D^{(-1)})=\overline{\rho(D)}$.

However, the following lemma is enough to escape all

trouble.
Lemma 17 Let $H$ be a finite group, let $F$ be a subfield of the complex numbers, and let$\rho$

bean $FH$ matrix representation. Then$\rho(h^{-1})=E^{-1}\overline{p(h)}^{t}E$

for all $h\in H$ where
Proof. Note that

$E=\Sigma_{g\in H}\overline{\rho(g)}^{t}\rho(g)$

$E$

(6)

.

is nonsingular since it is a positive hermitian matrix. Thus (6) follows

from$\overline{\rho(h)}^{t}E\rho(h)=g\in H\sum\overline{\rho(gh)}^{t}\rho(gh)=E.\square$

75
Theorem 18 Let $G=Z_{p^{m}}\cross H$ whereis a $prime_{f}$ and $H$ is a (possibly nonabelian) group$p$with $p^{2}\exp$ H. Let $F:=Q(\xi_{\exp H})$ , and letany$\omega\in Z_{p^{m}}^{*}$

and$u(\omega, \varphi)[\omega\otimes\varphi(D)]$$f(D)$$\varphi\in T_{f}$lie inbe any complete set of nonequivalent irreducible

Let $D$ be a $(v, k, \lambda, n)$ -difference set in G.

matrix representations.

$FH$

$T$

there is a root of unity$Q(\xi_{p}, \xi\exp H)$$u(\omega, \varphi)$such that all entries of the matrix. Then, for any partition

is a $(v, k, \lambda, n)$ -difference set in $G’=K\cross H$ where

Suppose that, for$(r_{1}, r_{2}, \ldots, r_{s})$

of

$m$

, the folding$K=Z_{p^{r}1}\cross Z_{p^{r}2}\mathrm{x}\ldots \mathrm{x}Z_{p}r_{s}$.

Proof. By a standard result on difference sets (see [25], for instance), it suffices to show$\tau(f(D))_{\mathcal{T}}(f(D)^{(-1}))=nI$

for every nontrivial irreducible $FG’$ representation
appropriate size. Note that any such representation

form$\psi\otimes\varphi$

where$\psi\in K^{*}$

and$\varphi\in T$$\tau$$\tau$

.

Here

$I$

is an identity matrix of theis equivalent to a representation of the. Now, by Theorem 10, we have(7)$\psi\otimes\varphi(f(D))=u\omega_{\psi}\otimes\varphi(D)$for some root of unity$u$.

Since $D$ is a $(v, k, \lambda, n)$ -difference set in$[\omega_{\psi}\otimes\varphi(D)][\omega_{\psi^{\otimes\varphi}}(D(-1))]=nI$$G$, we know that.

From (6) we get$\psi\otimes\varphi(X(-1))$

$=$

$=$

$=$

$\sum_{x\in X}\psi(x-1)\varphi(x-1)$$\sum_{x\in X}\overline{\psi(_{X)\varphi}}E-1\overline{(x)}^{t}E$$E^{-1}[\overline{\psi_{\otimes\varphi}(X)}]tE$

for any $X\subset G’$ and similarly$\omega_{\psi}\otimes\varphi(Y(-1))=E^{-1}[\overline{\omega_{\psi}\otimes\varphi(Y)}]tE$

for any $Y\subset G$ . Thus, using (7), (8) and$[\psi\otimes\varphi(f(D))][\psi\otimes\varphi(f(D)(-1))]$$u\overline{u}=1$,$=$$[\psi\otimes\varphi(f(D))]E^{-1}[\overline{\psi_{\otimes\varphi}(f(D))}]^{t}E$$=$$[u\omega_{\psi^{\otimes}}\varphi(D)]E^{-1}[\overline{u\omega_{\psi^{\otimes}}\varphi(D)}]^{t}E$$=$$[\omega_{\psi}\otimes\varphi(D)][\omega_{\psi}\otimes\varphi(D(-1))]$

$=$

$nI$

(8)

76
concluding the proof.

$\square$Remark 19 The small field conditions or the assumption thatin the theorems cannoi be dropped. \‘Inof order 40 (see [17] Example$Z_{4}\cross Z_{2}\cross Z_{5}$

or

$n$

is self-conjugate mod $exp(G)$$fact_{f}$there is a (40, 13, 4)-difference set in the cyclic group$\mathit{1}_{f}\mathit{1}PG(\mathit{3}_{J}\mathit{3})$) while there is no (40, 13, 4)-difference set in either$f$$Z_{2}\mathrm{x}Z_{2}\cross Z_{2}\cross Z_{5}$(see [17] Example 7,2). Note that 3 is not self-conjugate

mod 40.
Remark 20 In general, the designs generated by a difference set and by the folded difference

set are not isomorphic.
For$examp\iota e_{J}$

let $G=Z_{4}\cross Z_{3}\cross Z_{3}--<a>\cross<b>\mathrm{x}<c>and$$G’=Z_{2}\dot{\mathrm{x}}Z_{2}\mathrm{X}z_{3}\mathrm{x}z_{3}$.

Then$D=\{a, a^{2}, a^{3}, c, ac, bc, a^{2}bC, b^{2}c, ab32ac, C^{2},c, b_{C^{2}}, a^{3}b_{C}22, b222bc, ac\}22$is a (36, 15, 6)-difference set of$G$

and the designs generated by $D$ and its folding to $G’$ are notisomorphic.

On the other $hand_{f}$ let $G=Z_{8}\cross Z_{2}=<a>\mathrm{x}<b>and$ $D=\{1, a, a^{2}, a^{5}, b, ba^{6}\}$ . Then $D$
is a (16, 6, 2)-difference set inand to$Z_{2}\cross Z_{2}\cross Z_{2}\mathrm{x}Z_{2}$$G$, and the designs generated by $D$ and its foldings to$Z_{4}\cross Z_{2}\cross Z_{2}$are all isomorphic.

Relative Difference Sets

Let

$G$

be a group of order $mn$ with a normal subgroup $N$ of order$n$.

A -subset $D$ of$k$called an $(m, n, k, \lambda)$ -relative difference set relative to $N$ if the list of quotients $xy^{-1},$

contains each element of$G\backslash N$

exactly$\lambda$

terms of the group ring, a -subset $D$ of$k$$\chi(D)\overline{x(D)}=\{$

is$x,$ $y\inD$,times and contains no nonidentity element of $N$ . In$G$

is a $(m, n, k, \lambda)$ -difference set in$DD^{(-1)}=k\cdot 1+\lambda(G\backslash N)$

in $ZG$ where 1 is the identity element of

$G$

$G$

. This is equivalent to$k+\lambda(mn-n)$for$k-\lambda n$

for $\chi|_{N}=id,$

$k$

for

$\chi=\chi_{0}$$\chi|_{N}\neq id$$\chi\neq\chi_{0}$$G$

if and only if

77
The proof for the folding of relative difference sets is slightly more difficult than for difference
sets because of the special role of the forbidden subgroup $N$ . But in this case, the explicit

correspondence of characters gets us out of trouble.

Theorem 21 Let $G=Z_{p^{l}}\cross H$ be an abelian group such that $p^{2}w=\exp H,$ $l>1$ . Suppose $D$

is a $(m, n, k, \lambda)$ -relative difference set inroot of unity$(m, n, k, \lambda)$$G$such that for any$\chi$of. Then for any partition $(r_{1}, r_{2}, \ldots, r)S$ of , the folding $D’=f(D)$ of $D$ is a$l$$u_{\chi}$-relative difference set in$G’=Z_{p^{r_{1}}}\cross Z_{p^{r_{2}}}\cross\ldots\cross Z_{p^{r_{S}}}\cross H$

Proof. Since $DD^{(-1)}=k+\lambda(G\backslash N)$ for some $N\leq G$ of order$k+\lambda(mn-n),$ $\chi(D)\overline{\chi(D)}=k-\lambda n$Let$G=Z_{p^{l}}\cross;$.

$\eta_{i}=\xi_{p^{r_{i}}}$$H=\langle g\rangle\cross H,$. We have$\chi(\alpha)=\xi_{p}$for $\chi|_{N}=id,$then$\chi\neq\chi_{0}$$n$.

, we have$\chi_{0}(D)\overline{\chi 0(D)}=$and $\chi(D)\overline{\chi(D)}=k$ for $\chi|_{N}\neq id$ .

$G’=z_{p^{r_{1}}}.\cross\ldots\cross Z_{p^{r_{S}}}\mathrm{x}H=\langle g_{1}\rangle\cross\ldots \mathrm{x}\langle g_{s}.\rangle\cross H$$\tau(D’)\overline{\tau(D’)}=\chi_{\mathcal{T}}(D)\overline{x\tau.(D)}$(1) if $\chi(\alpha)=1$ then(2) iffor a$G_{f}\chi(D)u_{\chi}\in Z[\xi_{p}, \xi_{w}]$$\tau_{\chi}(f(\alpha))=1$.

$\mathrm{L}\mathrm{e}\mathrm{t}.\eta=\xi p^{\iota}$

by Theorem 8. And we see that for any$\alpha\in Z_{p^{l}}$

’

,

,

$\tau_{\chi}(f(\alpha))=\xi_{p}$.

Thus, for any $x\in G$ , if $\chi(x)=1$ , then $\tau_{\chi}(f(X))=1$ since $\chi(h),$ $\tau(h)\in Z[\xi_{p}, \xi\exp H]$ for any$\chi$

of $G,$

that$\tau$

of $G’,$ $h\in H$ . Let $N’=f(N)$ . Then $N’$ is a subgroup in $G’$ of order$\chi_{\tau}|_{N}=id$

if and only if $\tau|_{N’}=id$ for any$\tau$$n$, and we seeof $G’$ .

Claim: $D’D^{\prime(-1)}=k+\lambda(G’\backslash N’)$ .

Proof of the claim:

Case 1: $\tau=\tau_{0}=id$ . Then $\tau_{0}(D’)\overline{\tau 0(D’)}=k+\lambda(mn-n)$ .
Case 2: $\tau|_{N’}=id,$Case 3:$\tau|_{N’}\neq id$$\tau\neq\tau_{0}$. Then, since $\chi_{\tau}|_{N}=id,$. Then, since$\chi_{\tau}|_{N}\neq id$$x_{\tau}\neq\chi 0$, we haveThis proves the claim and hence the theorem.

, we have$\tau(D’)_{\mathcal{T}}\overline{(D’)}=k$$\tau(D’)\overline{\mathcal{T}(D’)}=k-\lambda n$.

.

$\square$

Building Sets
Davis and Jedwab [10] introduced building sets and covering extended building sets (CEBSs)

as a powerful tool for the construction of difference sets. In this section, we apply the folding
nuethod to CEBSs. Similar results also can be obtained for all other types of building sets.

78

An $(a, m, h, \pm)$ -covering extended building set (CEBS) in an abelian group$\{D_{1}, \ldots , D_{h}\}$of subsets of

$G$

$G$

is a familywith the following properties.

1) $|D_{1}|=a\pm m$ and $|D_{i}|=a$ for $i=2,$2) For every nonprincipal character$\chi$, .
$h$$\ldots$of$G$

there is exactly one$i$with $|\chi(D_{i})|=m$ andif $j\neq i$ .

$\chi(D_{j})=0$

As was shown in [10], a CEBS ingroups which contain

$G$

$G$

can be used to construct difference sets in many abelian

as a subgroup.

Theorem 22 Let $G=Z_{p^{l}}\cross H$ be an abelian group such that $p^{2}\parallel w=\exp H_{\mathrm{Z}}l>1$ . Suppose$\{D_{1)}\ldots, D_{h}\}$is an $(a, m, h, \pm)$ covering $EBS$ in

for a root of unity .
Then for an.y partition$G$such that for any$\chi$

of$G,$ $\chi(D)u_{\chi}\in Z[\xi_{p}, \xi_{w}]$$u_{\chi}$$(r_{1}, r_{2}, \ldots, rS)$is also an $(a, m, h, \pm)$ covering $EBS$ inof , the $folding.\{f(D_{1}), \ldots , f(D_{h})\}$ of$l$$G’=Z_{p^{r_{1}}}\mathrm{x}Z_{p^{r_{2}}}\mathrm{x}\ldots \mathrm{x}Z_{p^{r_{S}}}\mathrm{x}H$$\{D_{1}, \ldots , D_{h}\}$.

Proof. 1) Since is a bijection, we have $|f(D_{1})|=|D_{1}|=a\pm m$ and $|f(D_{i})|=|D_{i}|=a$ for$f$$i=2,$$h$$\ldots,$.

2) For every nonprincipal character$\tau$

of $G’$ , we have $\tau(f(D_{i}))=\chi_{\tau}(D_{i})$ from Theorem 8. Thus

we see that there is exactly one with $|\tau(f(Di))|=|\chi_{\mathcal{T}}(D_{i})|=m$ and $|\tau(f(Dj))|=|\chi_{\mathcal{T}}(D_{j})|=0$$i$

if $j\neq i$ .

$\square$Group Invariant Weighing Matrices

A weighing matrix $W(m, n)$ is an $m\cross m$ matrix $H$ with entries-l, $0,1$ such that $HH^{t}=nI$
where

$I$

is the identity matrix. The integer$n$

is called the weight of $H$ . Weighing matrices havebeen studied intensively, see [14] for a survey and [6, 7, 15] for some more recent results. The
proofs of the following results on weighing matrices are similar to those of the previous sections
and will be omitted. Let
$G$

$G$

be a group of order $m$ . We say that a matrix $H=(h_{f,g})_{f.g\in}G$ is-invariant if $h_{fk,gk}=h_{f,g}$ for all $k\in G$ . If we identify athe element$\Sigma_{g1.g}\in c^{hg}$$G$-invariant weighing matrix $H$ withof $ZG$ we get the following useful criterion, see [28], for example.

79

Lemma 23 Let$G$

be an abelian group of order $m$ , and let $H$ be a$G$-invariant $m\cross m$ matrix

with entries $-1,0,1$ . Then $H$ is a weighing matrix $W(m, n)$ if and only if$\chi(H)\overline{x(H)}=n$

for all characters$\chi$

of$G$

where $H$ is viewed as an element of $ZG$ .

Theorem 24 Let $G=Z_{p^{l}}\cross K$ be an abelian group such that $p^{2}w=\exp K,$ $l>1$ . Suppose $H$ isa$G$$u_{\chi}$.

-invariant weighing matrix such that for any

Then for any partition

weighing matrix where$\chi$

of$G,$$\chi(H)u_{\chi}\in Z[\xi_{p}, \xi_{w}]$of , the folding $H’=f(H)$ of$(r_{1}, r_{2}, \ldots, r_{s})$$l$$G’=Z_{p^{r_{1}}}\cross Z_{p^{r}2}\cross\ldots\cross Z_{p^{r_{S}}}\cross K$

for a root of unity$H$

is a $G’$ -invariant.

参考文献[1] Arasu,K.T., Davis,J.A., Jedwab,J., Ma,S.L., andfamily of abelian difference sets, inby$\mathrm{K}.\mathrm{T}$.Arasu,$\mathrm{J}.\mathrm{F}$.Dillon, K.Harada,(($\mathrm{G}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}_{\mathrm{S}}$$\mathrm{S}.\mathrm{K}$$\mathrm{M}\mathrm{c}\mathrm{F}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{d},\mathrm{R}.\mathrm{L}.$: Exponent bounds for a, Difference Sets, and the Monster” Edited.Sehgal,$\mathrm{R}.\mathrm{L}$.Solomon,$\mathrm{D}\mathrm{e}\mathrm{G}\mathrm{r}\mathrm{u}\mathrm{y}\mathrm{t}\mathrm{e}\mathrm{r}$Verlag, Berlin,

New York, (1996),129-143.
[2] Arasu, K. T., Davis, J. A., Jedwab, J., Sehgal, S. K.: New constructions of Menon differ-ence sets. J. Comb. Theory A 64 (1993), 329-336.
[3] Arasu, K. T., Davis, J. A., Jedwab, J.: A nonexistence result for abelian Menon difference
sets using perfect binary arrays. Combinatorica 15 (1995), 311-317.
[4] Beth, T., Jungnickel, D., Lenz, H.: Design theory (2nd edition). Cambridge University

Press (in press).
[5] Bruck,$\mathrm{R}.\mathrm{H}.$: Difference sets in a finite group. Trans. Amer. Math. Soc. 78 (1955), 464-481.

[6] Craigen, R.: The structure of weighing matrices having large weights. Designs, Codes andCryptography 5 (1995), 199-216.

80[7] Craigen, R., Kharaghani, H.: Hadamard matrices from weighing matrices via signed

groups. Designsf Codes and Cryptography 12 (1997), 49-58.
[8] Curtis, C.W., Reiner, I.: Representation theory of finite groups and associative algebras.
Wiley Classics Library. Wiley, New York, 1988.
[9] Davis,J. : Difference sets in abelian 2-groups. J. Comb. Theory A 57 (1991), 262-286.
$\mathrm{A}$$[10.],$$.\cdot \mathrm{D}.\mathrm{a}\mathrm{v}\mathrm{i}_{\mathrm{S}}.’.

\mathrm{J}.\mathrm{A}$$\cdot,...\mathrm{J}.\mathrm{e}\mathrm{d}_{\mathrm{W}\mathrm{a},\mathrm{l}}\mathrm{b}:.$$|80(.1997’)$’

J.: A uni.fying$\mathrm{c}\mathrm{o}..\mathrm{n}.$struction.of.ce$\mathrm{d}\mathrm{i}.\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}.$, 13-78.

Variations on a scheme of[11] Dillon,J.F.:$\mathrm{M}\mathrm{c}\mathrm{F}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{d}$sets. J. Combin. Theory$A$

for noncyclic difference sets,

J. Combin. Theory ser.A 40(1985), 9-21.

[12] Enomoto,H., Hagita,M., Matsumoto,M.: A note on difference sets, J. Combin. Theory ser.
..

$A$

84(1998), 133-144.

[13] Fan,C.T., Siu,M.K., Ma,$\mathrm{S}.\mathrm{L}.$: Difference sets in dihedral groups and interlocking difference

sets, $Ars$ . Combin. $20\mathrm{A}(1985)$ , 99-107.

[14] Geramita, A.V., Seberry, J.: Orthogonal designs III. Weighing matrices. Utilitas Math. 6(1974), 209-236.

[15] Gysin, M., Seberry, : On the weighing matrices of order $4n$ and weight $4n-2$ and $2n-1$ .
$\mathrm{J}$Australas. J. Combin. 12 (1995), 157-174.
[16] Hagita,M.:

Foldings of difference sets in abelian groups. Graphs and Combinatorics

15(1999), 187-193.

[17] Jungnickel,D.: Difference sets, $in$ “ Contemporary Design Theory: A Collection of Surveys,”

Edited by Jeffrey H.Dinitz and Douglas R.Stinson, (1992),241-324.

[18] Jungnickel, D., Schmidt, B.: Difference sets: An update. In: Geometry, combinatorial designs and related structures (Eds.

$\mathrm{J}.\mathrm{W}$.P. Hirschfeld,

Cambridge University Press, 1997, 89-112.

$\mathrm{S}.\mathrm{S}$. Magliveras and$\mathrm{M}.\mathrm{J}$. de Resmini).

81[19] Jungnickel, J., Schmidt, B.: Difference Sets: A Second Update. Rend. Circ. Palermo Serie
II, Suppl. 53 (1998), 89-118.

[20] Kibler,$\mathrm{R}.\mathrm{E}.$: A summary of noncyclic difference sets, $k<20$ , J. Combin. Theory ser.

$A$

25(1978), 62-67.

[21] Kraemer,$\mathrm{R}.\mathrm{G}.$: Proof of a conjecture on Hadamard 2-groups. J. Comb. Theory A 63

(1993), 1-10.
[22] Lander,E.S.:$‘(\mathrm{S}\mathrm{y}\mathrm{m}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}$Designs: An Algebraic Approach,” London Math. Soc. Lecture

Note Series 74, Cambridge Univ. Press, Cambridge,1983.
[23] Lang,S.:“

Algebraic Number Theory,” Addison-Wesley Series in Mathematics, 1970.

[24] Leung,K.H., Ma,S.L., Wong,V.L.: Difference sets in dihedral groups, Designs, Codes andCryptography 1 (1991), 333-338.
On difference sets in certain 2-groups, in[25] Liebler,R.:($‘ \mathrm{C}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}$Theory, Design Theory,

Group Theory,” Proceedings of the Marshall Hall Conference, ed. by D.Jungnickel, Jphn
Wiley and Sons, 1993.

[26] Ma,S.L.:

Polynomial addition sets and polynomial digraphs,

Linear Algebra and its

Applications 69(1985),213-230.

[27]$\mathrm{M}\mathrm{c}\mathrm{F}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{d},\mathrm{R}.\mathrm{L}.$:A fanlily of difference sets in noncyclic groups. J. Combin. Theory A

15 (1973),1-10.
[28] Schmidt, B.: Cyclotomic integers and finite geometry. J. Amer. Math. Soc., to appear.
[29] Singer, J.: A theorem in finite projective geometry and some applications to number
theory. Trans. Amer. Math. Soc. 43(1938), 377-385.

[30] Smith,K.W.:

Nonabelian Hadamard difference sets. J. Combin. Theory ser.A 70(1995),

144-156.
[31] Turyn,$\mathrm{R}.\mathrm{J}.$: Character sums and difference sets, Pacific J. Math. 15(1965),319-346.