called a permtation amuy if each row of A

called a permtation amuy if each row of A

Annals of Discrete Mathematics 30 (1986) 185-202
0 Elsevier Science Publishers B.V. (North.Holland)
185
ON PERMUTATION ARRAYS, TRANSVERSAL SEMINETS
AND RELATED STRUCTURES
M i c h e l Deza and Thomas I h r i n g e r
U n i v e r s i t i ! P a r i s V I I , U.E.R. de Math.,
P a r i s , France
Technische Hochschule, Fachbereich Mathematik,
Darmstadt, Federal R e p u b l i c o f Germany
E x p l o i t i n g some i d e a s o f C41, t h i s paper i s focused on t h e e q u i valence between s e t s o f m u t u a l l y o r t h o g o n a l p e r m u t a t i o n a r r a y s and
a s p e c i a l c l a s s o f seminets ( t h e s o - c a l l e d t r a n s v e r s a l s e m i n e t s ) .
Besides t h i s e q u i v a l e n c e , S e c t i o n 2 c o n t a i n s a c o n s t r u c t i o n method
f o r t r a n s v e r s a l seminets u s i n g groups. Nonsolvable p e r m u t a t i o n
groups o f p r i m e degree and t h e p r o j e c t i v e s p e c i a l l i n e a r groups
PSL(2,Zm) y i e l d examples f o r t h i s method. I n S e c t i o n 3 some upper
bounds a r e p r o v e d f o r t h e number o f m u t u a l l y o r t h o g o n a l p e r m u t a t i o n
a r r a y s , depending on t h e i n t e r s e c t i o n s t r u c t u r e o f t h e s e a r r a y s .
With t h e r e s u l t s o f S e c t i o n s 4 i t i s shown t h a t a l l examples o f
Section 2 are row-extendible. Section 5 deals w i t h several i n c i dence s t r u c t u r e s a s s o c i a t e d t o t r a n s v e r s a l seminets. The consequences a r e i n v e s t i g a t e d when t h e s e i n c i d e n c e s t r u c t u r e s have s p e c i a l
p r o p e r t i e s ( f o r i n s t a n c e , when t h e y a r e p a i r w i s e balanced d e s i g n s ) .
S e c t i o n 6 d i s c u s s e s b r i e f l y t h e r e l a t i o n s o f t r a n s v e r s a l seminets
w i t h o t h e r mathematical s t r u c t u r e s , e.g. w i t h t r a n s v e r s a l p a c k i n g s ,
g e n e r a l i z e d o r t h o g o n a l a r r a y s , and s e t s o f m u t u a l l y o r t h o g o n a l
p a r t i a l quasigroups.
1. INTRODUCTION
A v x r m a t r i x A = ( a . . ) w i t h e n t r i e s a . . f r o m t h e s e t I1,2 ,...,r l i s
1J
1J
c a l l e d a p e r m t a t i o n amuy i f each row o f A c o n t a i n s each o f t h e elements 1,2,
. . . ,r
e x a c t l y once, i . e . i f t h e rows o f
r l . The i n t e r s e c t i o n s tr u c tu r e o f
w i t h F. .,(A):= t j
I
A
A
represent permutations o f
i s d e f i n e d as t h e vxv m a t r i x
lsjsr, a..=a.
if, f o r a l l
aij
A
and
and
bij
B
,...,
F(A) =
1 . The v x r p e r m u t a t i o n a r r a y s
i'j
(Fi i , ( A ) )
1 1
IJ
A = ( a . . ) and B = ( b . . ) a r e c a l l e d e i r n . L Z a i * i f F ( A ) = F(B). C l e a r l y ,
1J
1J
a r e s i m i l a r i f and o n l y i f , f o r a l l i n d i c e s i , i ' , j ,
The p e r m u t a t i o n a r r a y s
{l,Z
A
and
B
a r e c a l l e d orthogonu2 i f t h e y a r e s i m i l a r and
i,i',j,j',
= ai,j,
= bi ,j, ==3 j = j ' .
T h i s concept o f o r t h o g o n a l i t y g e n e r a l i z e s t h e well-known i d e a o f o r t h o g o n a l l a t i n
r e c t a n g l e s . I t has been d e f i n e d and i n v e s t i g a t e d by B o n i s o l i and Deza i n C41.
M. Deza arzd T. lhringer
186
(See a l s o 1 7 3 f o r c l o s e l y r e l a t e d c o n s i d e r a t i o n s . )
S i m i l a r l y as f o r l a t i n r e c t a n g l e s and l a t i n squares, one w i l l be i n t e r e s t e d i n
sets
4=
{A1,A 2,...,At}
of
mutually orthogonal permutation arrays, w i t h
t
p o s s i b l y g r e a t e r t h a n 2 . I n t h i s case t h e i n t e r s e c t i o n s tr uc tur e
F(A) =
t
(Fi ,i(f)i
) o f A i s d e f i n e d as t h e common i n t e r s e c t i o n s t r u c t u r e o f t h e A k'
i . e . F(&):= F(A1) = F(A2) =
= F(At).
A p e r m u t a t i o n a r r a y A = ( a . .) i s c a l l e d standardized i f alj = j f o r a l l j
1J
...
.
Without loss of generality, i t w i l l be assumed i n t h i s paper t h a t each permut a t i o n array is standardized, and t h a t i t s a t i s f i e s the following n o n t r i v i a l i t y
conditions (CII and ( C 2 ) .
(C,)
The p e r m u t a t i o n a r r a y
A
A
has no c o n s t a n t column, i . e . each column o f
c o n t a i n > a t l e a s t two d i s t i n c t values,
any two rows o f
(C,)
X
Let
A
are d i s t i n c t .
be a nonempty s e t , and l e t
d i s j o i n t s e t s o f nonempty subsets o f
Lo,L1,
...,L t
(with
X. The elements o f
tzl)
X
u,,,,,
points and t h e elements o f
,..,, Lk Zines. Then
i s c a l l e d a seminet o r (more p r e c i s e l y ) a ( t t 1 ) - s e m i n e t i f
(S1)
any two d i s t i n c t l i n e s i n t e r s e c t i n a t most one p o i n t ,
(S,)
each c l a s s
Li
partitions the point set
be m u t u a l l y
w i l l be c a l l e d
3 :=(X;Lo,L1 ,...,L t )
X.
C o n d i t i o n (S2) j u s t i f i e s t h e t e r m p a r a l l e l c l a s s f o r each o f t h e l i n e s
Li.
The
n o t i o n o f a seminet g e n e r a l i z e s such well-known s t r u c t u r e s l i k e a f f i n e p l a n e s ,
n e t s and (more g e n e r a l l y ) t h e p a r a l l e l s t r u c t u r e s o f Andre 121. A subset o f
c a l l e d a transi.arsa2 o f t h e seminet
5
e x a c t l y one p o i n t . I f
p a r a l l e l class
Li
3
has a t r a n s v e r s a l c o n s i s t i n g o f
contains e x a c t l y
r
o f t h e seminet c o n s i s t s a l s o o f e x a c t l y
r:=
5
i f i t i n t e r s e c t s each l i n e o f
r
3
X
is
in
p o i n t s , t h e n each
l i n e s ( a n d hence each f u r t h e r t r a n s v e r s a l
r
p o i n t s ) . I f T1,T
*,...,TV
are trans-
versals o f
then
(X;Lo,L1 ,...,Lt;T1,T2 ,..., Tv) i s c a l l e d a transversa2
seminet ( o r transversal ( t t 1 , r ) - s e m i n e t i f each t r a n s v e r s a l c o n s i s t s o f r
p o i n t s ) . B o n i s o l i and Deza r41 p o i n t e d o u t t h a t t h e r e i s a c l o s e r e l a t i o n s h i p b e t ween set.s o f m u t u a l l y o r t h o g o n a l p e r m u t a t i o n a r r a y s and o t h e r mathematical s t r u c t u r e s . F o r i n s t a n c e , t h e y proved t h a t each s e t o f
p e r m u t a t i o n a r r a y s i s e q u i v a l e n t t o a 1-design w i t h
number
r
and
ttl
mutually orthogonal v x r
t
v
treatments, r e p l i c a t i o n
m u t u a l l y o r t h o g o n a l r e s o l u t i o n s (see S e c t i o n 5 ) . Moreover,
i t was shown t h a t any o f these a r e e q u i v a l e n t t o a t r a n s v e r s a l ( t t 1 , r ) - s c m i n e t
with
v
t r a n s v e r s a l s . T h e r e f o r e many o f t h e examples and r e s u l t s i n t h i s paper
can be t r a n s l a t e d i n t o analogous statements on c o m b i n a t o r i a l designs w i t h m u t u a l l y
ortogonal resolutions.
Oil
Pennutatioii Arrays
187
2. AN EQUIVALENCE AND A CONSTRUCTION METHOD
J :=
Let
1 2
{lo,l
o,...,lL).
(X;Lo,L1,..
. ,Lt;T1,T2,.
.. ,Tv)
Lo:=
be a t r a n s v e r s a l seminet w i t h
( I n f a c t , t h r o u g h o u t t h i s paper t h e s e t o f p a r a l l e l c l a s s e s , t h e
Lo o f any t r a n s v e r s a l seminet a r e
s e t o f t r a n s v e r s a l s and t h e s e t o f l i n e s of
assumed t o be l i n e a r l y ordered, by t h e numbering o f t h e i r elements.) One can now
k
d e f i n e t m u t u a l l y o r t h o g o n a l v x r p e r m u t a t i o n a r r a y s Ak = ( a . . ) , k=1,2, ...,t,
1J
i n t h e f o l l o w i n g way: F o r i E I 1 , 2 ,... ? v 1 , j E I 1 , 2 ,...,r l , k < I1,2 ,...,t ) l e t
Ti n l;, and l e t
be t h e unique p o i n t . c o n t a i n e d i n
x
x. L e t
be t h e u n i q u e p o i n t w i t h
y
t h r o u g h y. F i n a l l y , d e f i n e
a:j:=
be t h e
1;
L k - l i n e through
be t h e
Lo-line
c . From t h e p r o p e r t i e s o f t r a n s v e r s a l seminets
&J):=IA1,A2, . . . ,A t )
one can conclude t h a t
1
y c T 1 n 1, and l e t
i s , i n fact, a s e t o f mutually
orthogonal permutation arrays.
The c o n d i t i o n s (C,)
o f S e c t i o n 1 can be paraphrased i n terms o f
and (C,)
t r a n s v e r s a l seminets as f o l l o w s :
There i s no p o i n t o f t h e seminet w h i c h i s c o n t a i n e d i n a l l t r a n s v e r s a l s ,
(D1)
...,
. =
ni=1,2,
v Ti
1 , a n d v.2.)
1.‘.
line
(D2)
0.
Any two t r a n s v e r s a l s a r e d i s t i n c t , i . e .
Ti
i
J?-(r)
only
I n t h e c o n s t r u c t i o n procedure f o r
T
those p o i n t s o f
Ti.
r e s t r i c t o n e s e l f t o t h e reduced t r a n s v e r s a l seminet
( X ’ ;LA,Li,.
d e f i n e d by
X
have been
T h e r e f o r e one can always
,
. ,LC;T1,T2,. . . ,
L k’ : = ( 1 n X ’ j l c L k I .
Ti’
ui=1,2,...,v
XI:=
f o r each
iz j .
for
j
used which a r e c o n t a i n e d i n one o f t h e t r a n s v e r s a l s
Tv)
I1 I t 2
(This implies, i n particular,
I n t h e r e s t of this paper a l l transoersa2 seminets are assumed t o be reduced
arid t o s u t i s f g the ron,Zitions (0,) m d (DJ.
Y
The process of c o n s t r u c t i n g m u t u a l l y o r t h o g o n a l p e r m u t a t i o n a r r a y s f r o m t r a n s v e r s a l seminets can be r e v e r s e d : L e t
k
p e r m u t a t i o n a r r a y s Ak = ( a . . ) , k=1,2,
1J
+
r ; , and l e t
& be
...,t .
a set of
Define
be t h e e q u i v a l e n c e r e l a t i o n on
o n l y if j = j ’
and
Y
t
Y:=
with
j i Fi i ,(A).
Define the p o i n t s e t X
9, i . e . X:= Y / $ = “(i,j)llb
set o f equivalence classes o f
.
1 , 2 ,.... r
{I (i,j)ldj
~
...,
...,
( i , j ) $ ( i ’ , j ’ ) i f and
o f t h e seminet as t h e
1
(i,j)t:YI.
For
and k = 1,2,, .,t l e t lo:=
(I ( i , j ) l $ I j = c , i=1,2,...,v)
and
1 2
k
1 2
,...,1 L I .
a . . = c l , and d e f i n e L o : = {lo,lo
,..., lor}, L k : = Ilk,lk
Finally, l e t
(X;Lo,L1,..
mutually orthogonal v x r
{1,2.
v) x {l,Z,
1J
T.:=
1
.,Lt;Tl,T2,,
= A .Summarizing,
’,r
(i,j)l$
..
C
1
j = 1 , 2 , . ..,rI, i = 1 , 2
,...,v .
Then
J(R):=
,Tv) i s a (reduced) t r a n s v e r s a l seminet w i t h
one o b t a i n s
c =
C
lk:=
&(T(R))
M. Deza arid T. Ihririger
188
The existence of a s e t
2.1. PROPOSITION.
01
t
mutually orthogonal v x r perm-
tation arrays i s equivaZent t o the existence of a transversal (t+l,r)-seminet with
v
transversals.
T h i s e q u i v a l e n c e was a l r e a d y observed i n 141. One can show even a l i t t l e more.
Let
. . ,Lt;T1,T2,..
= (X;Lo,L1..
.,Tv)
be reduced t r a n s v e r s a l seminets w i t h
.
and ‘U = (Y;Mo,M1,. ..,Mt;U1,U2,.
.,Uv)
1 2 ,...,1 L I and Mo
1 2 ,...,
Imo,mo
Lo = ilo,lo
m i l , and assume R(’3’)= A(%).
D e f i n e a mapping $ : X 4 Y as f o l l o w s . F o r
X E Ti n 1:
l e t $ ( x ) be t h e unique element c o n t a i n e d i n Ui nm:.
Then 0 i s an
and U , i . e .
isomorphism o f
0 i s a b i j e c t i o n which maps p a r a l l e l l i n e s
i
i
0 s a t i s f i e s $Li = Mi, $Ti = Ui and $lo
= mo
onto p a r a l l e l l i n e s ( i n f a c t ,
for all
2.2.
i ) . This y i e l d s
PROPOSITION.
If
J , ( J )= L ( U ) then
y
&(J) =
T. nT. = 0
1
are reduced transversa2 seminets with
are isomorphic.
.
IA1,A 2,...,Atl
for a l l
J
u
.
= ( X;Lo,Ll,.
.,Lt;T1,T2,.
.,Tv) corresponds t o a
o f m u t u a l l y o r t h o g o n a l Zatin rectanglos e x a c t l y i f
The t r a n s v e r s a l seminet
set
and
and
i,j, i z j . The a r r a y s
A1,A 2,...,At
form a s e t o f m u t u a l l y
o r t h o g o n a l Zatin squares i f and o n l y i f
(X;Lo,L1,, ..,Lt,Lt+l),
w i t h Lt+l:=
I n o t h e r words, t h e e q u i v a l e n c e o f P r o p o s i t i o n 2.1
i s a net.
tT1,T2,,..,Tv\,
s p e c i a l i z e s t o t h e c l a s s i c a l correspondence o f m u t u a l l y o r t h o g o n a l l a t i n squares
w i t h nets.
The f o l l o w i n g theorem p r o v i d e s a c o n s t r u c t i o n method o f s e t s o f m u t u a l l y o r t h o gonal p e r m u t a t i o n a r r a y s v i a seminets, u s i n g groups.
2.3.
THEOREM.
G he a f i n i t e group with neutral eZement e . Let
Let
t and
be p o s i t i v e integers, and Zet So,S1,..
.,St
and F1,F2,.. .,F
be nontrivial
S
subgroups of G such that the foZZowing conditions are s a t i s f i e d f c r aZl i , j E
S
10,1,
..., tl,
k,l
E
11,2
,..., ~
4 Si n S j
(2)
i j
(2)
S. O F = {el,
(3)
k * 1 =$ Fk z F,,
(41
l F k I = CG:Sil.
i
= (el,
k
Then there e x i s t s a s e t o f
S
r : = I F I and v:= -./GI
.
1
r
Proof.
i.e.
1 :
F o r each
Li = {Sig
1
t
i E (O,l,
geG1. Then
mutualZy orthogonal v x r p e r m t a t i o n arrays, with
...,t l
let
(G;Lo,L l,...,Lt)
Li
c o n s i s t o f the r i g h t cosets o f
i s a (tt1)-seminet:
Si,
C o n d i t i o n (S1)
On Permu tutiori Arru-vs
189
i s t r i v i a l l y s a t i s f i e d w h i l e ( S 2 ) i s a consequence o f ( 1 ) . Each r i g h t c o s e t
Fkh
i s a t r a n s v e r s a l o f (G;Lo,L1, ..., L t ) : I t has t o be
Fk
n F k h l = 1 f o r a l l g,hsG. As a consequence o f ( 2 ) one o b t a i n s
o f one o f t h e subgroups
show that
lSig
1S.g n F k h l i 1. Assumption ( 4 ) then i m p l i e s u f C F k S i f = G, and hence
1
lSig n F k h l
2
1. Each t r a n s v e r s a l has
d i s t i n c t r i g h t c o s e t s . Thus t h e r e a r e
form
Fkh, w i t h
(D1) and
k t t1,2,
...,s l
and
r = lFll
elements, and each
v = :*IGI
Fk
IGI
has
d i s t i n c t transversals o f the
h e G. Finally, the n o n t r i v i a l i t y conditions
( D p ) a r e a consequence o f t h e n o n t r i v i a l i t y o f t h e subGroups Fk and o f
(3), r e s p e c t i v e l y . By P r o p o s i t i o n 2.1, t h e p r o o f i s complete.
The seminet
seminet, i . e .
n
,,...,
(G;Lo.L
L t ) o f t h e above p r o o f i s , i n f a c t , a transZation
i t has a t r a n s l a t i o n group o p e r a t i n g r e g u l a r l y on i t s p o i n t s : I n t h e
r i g h t r e g u l a r represeritation o f
G
each maoping
XH
xg,
g r G , maps e v e r y l i n e
o n t o a p a r a l l e l l i n e . On t h e o t h e r hand, each t r a n s l a t i o n seminet can be o b t a i n e d
i n t h i s way f r o m a group
G
and subgroups
So,S l....,St
satisfying condition
(1). Analogous group t h e o r e t i c c h a r a c t e r i z a t i o n s have been given, f o r i n s t a n c e ,
f o r t r a n s l a t i o n planes, t r a n s l a t i o n n e t s , t r a n s l a t i o n s t r u c t u r e s and t r a n s l a t i o n
group d i v i s i b l e designs ( s e e e.g.
rll,
1151. C221, [31 and 1201). Marchi r181 uses
s i m i l a r i d e a s f o r his c h a r a c t e r i z a t i o n o f r e g u l a r a f f i n e p a r a l l e l s t r u c t u r e s by
p a r t i t i o n l o o p s . P r o b a b l y one can f o r m u l a t e an analogue o f Theorem 2.3 u s i n g l o o p s
i n s t a e d of groups. The problem would be t o f i n d examples f o r such a g e n e r a l i z a t i o n . The r e s t o f t h i s s e c t i o n y i e l d s two c l a s s e s o f examples f o r Theorem 2.3.
C f . Huppert 1141 and W i e l a n d t 1241 f o r t h e group t h e o r e t i c n o t a t i o n s .
L e t G be a n o n s o l v a b l e t r a n s i t i v e p e r m u t a t i o n group o f p r i m e
degree p . L e t v:= p 2 , r:=-I,G I and l e t d be the p o s i t i v e i n t e g e r w i t h d < p - 1
P
and d = r (mod p ) . Then one can c o n s t r u c t a s e t o f t : =
1 mutually orthogonal
2.4. EXAMPLE.
i-
v * r p e r m u t a t i o n a r r a y s : Assume
tl,Z, . . . , p 1
define
FkzF,
k z
for
So,S l,...,St,
1
Fk
since
G
t o o p e r a t e on
*,.... a P I .
ial,a
t o be t h e s t a b i l i z e r o f
6
F o r each
k
6
ak i n G, i . e .
Fk:= Ga
Then
k'
i s d o u b l y t r a n s i t i v e ( c f . Theorem 11.7 o f r 2 4 1 ) . L e t
be t h e Sylow p-subgroups o f
G
(with
t ' 1 1 because
G
i s non-
s o l v a b l e ) . O b v i o u s l y , these subgroups s a t i s f y t h e assumptions o f Theorem 2.3.
m u t u a l l y o r t h o g o n a l v x r p e r m u t a t i o n a r r a y s . I t remains t o
r
t ' = t o r , e q u i v a l e n t l y , t h a t G has e x a c t l y
Sylow p-subgroups. L e t
Hence t h e r e a r e
show
P
t'
a
be a Sylow p-siibgroup o f
P
of
d"p-1
is
P
i t s e l f . Hence
G. The o n l y Sylow p-subgroup o f t h e n o r m a l i s e r
NG(P)
i s s o l v a b l e and t h u s o f o r d e r
pad'
NG(P)
with
( c f . r141, Satz 1 1 . 3 . 6 ) . T h e r e f o r e t h e number n o f Sylow p-subgroups
G
r
n=[G:N ( P ) l = p . d ' = a T .
From n = l (mod p ) one o b t a i n s d ' = r (mod p ) ,
G r
d = d ' and n =
satisfies
i.e.
a.
The n o n s o l v a b l e t r a n s i t i v e p e r m u t a t i o n groups o f p r i m e degree have been comp l e t e l y determined, due t o t h e c l a s s i f i c a t i o n o f f i n i t e s i m p l e qroups ( s e e
M. Deza Q
190
I I T.
~ Ihringer
C o r o l l a r y 4.2 o f F e i t C111).
N o t i c e t h a t solvabZe t r a n s i t i v e p e r m u t a t i o n groups o f p r i m e degree o cannot be
used i n t h e above c o n s t r u c t i o n : These groups have e x a c t l y one Sylow p-subgroup,
which would i m p l y
2.5. EXAMPLE.
t = 0.
F o r each i n t e g e r
mr2
one can c o n s t r u c t a s e t o f t m u t u a l l y
m-1 rn
( 2 - 1 ) - 1 9 v : = (2m+1)2 and r : =
m
2m(2m-l): Regard t h e p r o j e c t i v e s p e c i a l l i n e a r group G = PSL(2,q), w i t h q = 2 ,
orthogonal v x r permutation arrays, w i t h
t:= 2
as a p e r m u t a t i o n group o p e r a t i n g c a n o n i c a l l y on t h e
q+l
o f t h e p r o j e c t i v e l i n e o v e r t h e q-element f i e l d . F o r each
define
Fk
t o be t h e s t a b i l i z e r o f
ak
in
G, i . e .
be t h e ( m u t u a l l y c o n j u g a t e ) c y c l i c subgroups o f
=
=-,
t'
o f conjugates o f
ttl, i.e.
k
t
..,aq+l)
. . ,q+11
{al,a2,.
{1,2,.
Fk:= G
L e t So,S l,...,Stl
ak'
q + l . By t h e r e s u l t s
o f order
t h e s e subgroups s a t i s f y t h e assumptions o f Theorem 2.3.
i n 1141, pp. 191-193,
t h e number
G
points
one o b t a i n s
So
For
t ' + l= C G : N G ( S o ) l = w
t ' = t.
i n o r d e r t o conN o t i c e t h a t Hartman [121 used some o f t h e groups PSL(2,q)
s t r u c t designs w i t h m u t u a l l y o r t h o g o n a l r e s o l u t i o n s . F o r i n s t a n c e , f o r each q E
{19,31,431
t i o n s and
there e x i s t s a design w i t h
v=qtl
t r e a t m e n t s , r =?
replica-
t+l=q
mutually orthogonal resolutions.
3. BOUNDS FOR THE NUMBER
OF MUTUALLY ORTHOGONAL PERMUTATION ARRAYS
.
o f m u t u a l l y orthogonal p e r m u t a t i o n a r r a y s i s c a l l e d
{A1,A2,. . ,At)
maxima2 i f t h e r e e x i s t s no p e r m u t a t i o n a r r a y
which i s o r t h o g o n a l t o a l l
Attl
Ak, k = 1,2 ,..., t. A t r a n s v e r s a l seminet (X;L0,L1,
Lt;T1,T2
,Tv) i s c a l l e d
L-mo.ximaZ i f t h e r e e x i s t s no a d d i t i o n a l p a r a l l e l c l a s s Lt+l
such t h a t (X;Lo,L1,
A set
...,
..
.,Lt,Lttl;T1,T2,.
obvious.
3.1.
LEMMA.
. .,Tv)
i s a g a i n a t r a n s v e r s a l seminet. The f o l l o w i n g lemma i s
of mutually orthogond permutation arraps is maximal
A set
i f mid only if the associated transversal seminet
3.2. PROPOSITIOM.
,...
J (A)is
L-maximal.
of rnintualZy orthogoxu2 permutation arrays of
Eaeh s e t
Example 2 . 5 is maximal.
Proof.
Let
G = PSL(2,2m)
a s s o c i a t e d t r a n s v e r s a l seminet
be t h e group used f o r t h e c o n s t r u c t i o n of
7
has
G
as p o i n t s e t . The subgroups
...,St
a r e e x a c t l y t h e l i n e s t h r o u g h t h e n e u t r a l element
groups
F1,F2,.
. .,Fq+l
e
are e x a c t l y the transversals through
of
e
A.The
SO'S1,
G, and t h e sub( c f . the proof
191
Oir Permu tation Arrays
G. Hence
o f Theorem 2 . 3 ) . By Satz 11.8.5 o f Huppert L141, t h e s e subgroups c o v e r
t h e r e cannot be any a d d i t i o n a l l i n e through
e, and
i s t h e r e f o r e L-maximal.
J 2 J(&).
By Lemma 3 . 1 t h e p r o o f i s complete, s i n c e P r o p o s i t i o n 2.2 y i e l d s
0
A c t u a l l y , t h e a s s e r t i o n o f Propos t i o n 3 . 2 depends o n l y on t h e i n t e r s e c t i o n
F(&)
structure
of
a:L e t
3
,..., B t , I
B1,B2
F(JJ ) = F ( & ) .
gonal p e r m u t a t i o n a r r a y s w i t h
S e c t i o n 2 shows t h a t t h e t r a n s v e r s a l seminets
p o i n t s e t s and t h e same t r a n s v e r s a l s
n
p o i n t s , and
y @ ) As
.
for
u s e s oF1s
S E S ~ w, i t h
number
Y
e
7113)
7
G. T h e r e f o r e each l i n e o f
T(&)t h r o u g h
x
have t h e same
o f the proof
contains e x a c t l y
J(&)
t h a t t h e same i s t r u e f o r
a consequence, t h e r e i s a p o i n t
through
and
pairwise d i s j o i n t transversals
0
=
7 7(A)i m p l i e s
t'+l of lines o f
lines o f
')'(a)
The t r a n s v e r s a l seminet
n:= I S I = CG:F1l
o f Proposition 3.2 contains
F1s,
be a s e t o f m u t u a l l y o r t h o -
The c o n s t r u c t i o n procedure o f
x
of
J ( 3 ) such
and a l s o
t h a t the
c a n n o t exceed t h e number
( i n f a c t , t h i s i s t r u e f o r each p o i n t
x
t+l o f
y(2)) .
of
T h e r e f o r e P r o p o s i t i o n 3.2 can be improved as f o l l o w s .
3.3. PROPOSITION.
Get
a ={A1,A
2,...,Atl
be one of t h e s e t s ofmutuaZZy
3
orthogonal permutation arrays of Exnmple 2 . 5 . Let
= IB1,B2,.
of mutually orthogonal permutation arrays with F ( B ) = F(&).
. . ,Bt,
Then
be a s e t
1
t'
5
C..
The n e x t lemma g i v e s an upper bound f o r t h e number o f m u t u a l l y o r t h o g o n a l p e r m u t a t i o n a r r a y s depending on t h e i n t e r s e c t i o n s t r u c t u r e of t h e a r r a y s . The p r o o f
o f t h i s lemma i s a u n i f i e d v e r s i o n o f t h e p r o o f s o f s e v e r a l r e l a t e d r e s u l t s i n
'41 and 171.
3.4.
LEMMA.
Let
A =IA1,A2 ,..., At]
I
pemutatioil arrays, and l e t
..., r l
j o <{ l , Z ,
{ l , Z ,..., v},
i o c
1 1 , 2 ,..., vl,
J 5 {1,2,...,rI,
satisfg
II,
al
I
bi
t"il,i2t
I:
ai
'JiE I :
j o d F .1 i,
(A),
dl
V j c J
3i.I:
j + F i i
f
be a s e t of mutualZy orthogonu2 v x r
j O c Fil12'
(A),
0
(a.
I"he1i t ' r - ! J I - 1 .
-
P r o o f . L e t A = ( a . . ) be a v r p e r m u t a t i o n a r r a y w i t h F(A) = F ( & .
'J
t h e r e e x i s t s an element i l k
I . Define c:= a i
. From b ) one o b t a i n s
1" 0
f o r a l l i I , and c ) i m p l i e s aio,of c. F o r each j c J t h e r e e x i s t s an
witn
ai
OJ
= a i j , by d ) . Thus
jLj0
and
aijza..
'JO
= c . Hence
aioj
By a )
"jd
ic I
c . There-
192
M . Deza arid T. Iliringer
fore
aioj = c
f o r e x a c t l y one element
. . . ,rl \
(J
A,
with
c
kt+
j,
t J?-
mapping
{1,2,
o f the (r-iJl-l)-element set
j
I n p a r t i c u l a r , t h i s i s t r u e f o r each o f t h e p e r m u t a t i o n a r r a y s
{j,}).
IJ
and
j
r e p l a c e d by
ck
and
j,
.
By o r t h o g o n a l i t y ,
t s r - IJI
i s i n j e c t i v e . This implies
- 1.
the
0
The f o l l o w i n g c o r o l l a r y t r a n s l a t e s Lemma 3.4 i n t o t h e language o f t r a n s v e r s a l
seminets. N o t i c e t h a t t h i s c o r o l l a r y c o u l d have been used i n o r d e r t o prove t h e
P r o p o s i t i o n s 3.2 and 3.3.
COROLLARY.
3.5.
seminet, and 7.et
I
al
b)
,Lt;T1,T2,.
io
c {1,2¶,
,vl
. ., v l ,
..
..,Tv)
and
b e a transversa2
x
satisfy
X
E
6,
mi,, T ~ ,
Ti
x
el
Then
f
..
= (X;Lo,L1,.
Lat
I c {1,2,.
t sr-6
0
-
.
1, w i t h
3.6. COROLLARY.
Let
r : = ITl]
and
A={A1,A21...,A
uerrnutation arrays, and l e t
p: =
ITi n (uicI
6::
max { I Fi
0
t
Ti) I
.
1 be a s e t of rnutuaZZy orthogonal v x r
, (A)
I I i ,i'=1,2,.
. . ,v,
i*i' I . Then
t i r - U - 1 .
~
Proof.
Choose
.
i,iot.il,2
,..., v l
and
j o t {112]
..., r I 1 w i t h
j o & F i i (A)D e f i n e I : = t i 1 and J : = F i l o ( & )
0
s a t i s f y t h e assumptions of Lemma 3.4.
1-1
and
3.7. COROLLARY.
Let
pervnitution a r r a y s . Let
T'hen t s r
Proof.
-A-2 .
Choose
A=IAl,A2, ...,A t 1
h:=
min { I F . #(.+?-)I
li
. Then
IFi io(sZ)I =
IJ
I , io,
J , jo
be a s e t of mutzia1Zy orthogonal v x r
1
i,i'=1,2,..,,v),
io,i1,iz8
i 1 , 2 ,...,v l , il*i2$
and
joF11,2
and assume
,..., r l
X z l .
with
jo'
(A)
(A).
Fili2(&),
j o ~ F i l i o ( ~ ) and j o & F '2'0
. .
D e f i n e I : = {ilyi21,
J : = F.'1'0.
IJ F .
(f-). T h e n I , io,
J , j o s a t i s f y t h e assumptions o f Lemma 3.4. Hence t h e
12iO
proof i s complete i n t h e case IJI -Xtl. Assume now I J I = A . Then X = IF.
(&)I
A
1 l i O
IFi i (&)I
= IJ u tjo)l = X t l . T h e r e f o r e t h e case
1210
1 2
i s s e t t l e d by C o r o l l a r y 3.6.
U
= IF. .
(&)I,
and t h u s
The C o r o l l a r i e s 3.5,
IJI
=
3.6 and 3.7 y i e l d s l i g h t g e n e r a l i z a t i o n s for some o f t h e
r e s u l t s i n L41, 171 and r 1 7 1 (which a r e f o r m u l a t e d i n terms o f designs w i t h
mutually orthogonal r e s o l u t i o n s ) .
A v x r p e r m u t a t i o n a r r a y A i s c a l l e d row-transitive i f t h e rows o f
s e t o f p e r m u t a t i o n s o p e r a t i n g t r a n s i t i v e l y on t h e s e t { l Y 2 , . ..,rl.
A
form a
193
Oil Permutatioii Arraj's
3.8. PROPOSITION.
A =IA1,A2, ...,At]
Let
be a s e t of mutuully orthogonal
v' r p e r m t a t i o n arrays. Assume one of these arrays (and hence a l l of them) t o be
Then t c min Ir-1 , m t l }
orti?ogonal Latin squares of order r .
row-transitive.
Proof.
R o w - t r a n s i t i v i t y i m p l i e s each l i n e o f t h e a s s o c i a t e d t r a n s v e r s a l semi-
T(&) =
net
line
1 cLi,
, with m t h e largest number of rm*tualZy
(X;Lo,L1
,..., Lt;T1,T2 ,...,T v )
i > l , i n t e r s e c t s each l i n e o f
t o have e x a c t l y
Lo
r
p o i n t s , s i n c e any
e x a c t l y once. Thus
(X;Lo,L
l,...,
i s a ( t - 1 ) - n e t o f o r d e r r . The e x i s t e n c e o f such a n e t i s e q u i v a l e n t t o t h e
Lt)
e x i s t e n c e o f t - 1 m u t u a l l y o r t h o g o n a l l a t i n squares o f o r d e r r Hence t - l s m .
.
A complete ( t t 1 ) - n e t o f o r d e r r has e x a c t l y
rtl
w i t h a t r a n s v e r s a l cannot be complete. T h e r e f o r e
p a r a l l e l classes. But a n e t
t + l c
rtl.
U
4. EXTENSION BY ROWS
A set
J1= IA1,A2,.
. .,At}
o f mutually orthogonal v x r permutation arrays i s
c a l l e d row-ertercdible i f i t i s p o s s i b l e t o a d j o i n a new row t o each o f t h e a r r a y s
such t h a t t h e r e s u l t i n g ( v t 1 ) x r p e r m u t a t i o n a r r a y s a r e a g a i n m u t u a l l y o r t h o g o n a l .
A t r a n s v e r s a l seminet
(X;Lo,L1,
...,Lt;TI,T2,
...,T V )
i s c a l l e d zrwsversal-
estendibZe i f t h e r e e x i s t s a t r a n s v e r s a l seminet (Y;Mo,M 1,...,Mt;T1,T2,...,TV,
Tvtl)
w i t h Y 2 X and Li = I m n X 1 m c M i l . As transversal-extendibility i s t h e
obvious t r a n s l a t i o n o f r o w - e x t e n d a b i l i t y i n t o t h e language o f t r a n s v e r s a l semin e t s , one o b t a i n s
4.1. LEMMA.
o f inutun1L;j orthogonaz permutation arrays i s
A set
extendible ij' and only i f t h e associated transversal seminet
J (&)
TOW-
i s transver-
sa 1-e.-cteadib l e .
The above d e f i n i t i o n o f r o w - e x t e n d a b i l i t y i s s t r i c t l y s t r o n g e r t h a n t h e one
g i v e n i n r 4 1 where t h e r e s u l t i n g a r r a y s were o n l y assumed t o be s i m i l a r . Both def i n i t i o n s coincide i f
Y =
X i n t h e t r a n s v e r s a l seminets used i n t h e d e f i n i t i o n o f
transversal-extendibility. I n p a r t i c u l a r , t h e d e f i n i t i o n s c o i n c i d e i f
(X;Lo,L1,
i s a n e t ( c f . P r o p o s i t i o n 4.4 o f r 4 1 ) . I n t h e case o f m u t u a l l y o r t h o g o n a l
...,L t )
l a t i n squares, r o w - e x t e n d i b i l i t y i s e q u i v a l e n t t o t h e e x i s t e n c e o f a common
colLumz-tj.ans?~ersaZ ( i .e. a u s u a l t r a n s v e r s a l o f l a t i n squares w i t h t h e c o n d i t i o n
"no two c e l l s a r e on t h e same row" r e p l a c e d by t h e weaker c o n d i t i o n " n o t a l l c e l l s
a r e on t h e same r o w " ) ; see 141, P r o p o s i t i o n 4.2.
Tv)
Let
,...,
(X;Lo,L1,...,Lt;T1,T2
be t h e t r a n s v e r s a l seminet a s s o c i a t e d t o t h e l a t i n squares
A1,A2,.
..,At.
194
M . Dezu urid T. Ilrringer
C l e a r l y , t h e l a t i n squares have a common column-transversal e x a c t l y i f t h e r e
Tvtl
e x i s t s a transversal
.. ,Lt)
(X;Lo,L1,.
with
a Lt+l:=
Tv+l
tT1,T2,.
I n g e n e r a l , i t i s an open q u e s t i o n whether such a t r a n s v e r s a l e x i s t s i f
...,Lttl)
..,
Tv+l
A1,A 2,...,At
if
(i.e.
of
i f t h e n e t (X;Lo,L1,...,Lt,Lt+l)
i s an a f f i n e p l a n e
f o r m a complete s e t o f m u t u a l l y o r t h o g o n a l l a t i n s q u a r e s ) .
T v I . There i s no such
(X;Lo,L1,
i s a transversal-free n e t i n t h e sense o f Dow C101.
A l t h o u g h t h e n e x t p r o p o s i t i o n i s easy t o prove, t h e r e s u l t i s q u i t e s u r p r i s i n q .
a
4.2. PROPOSITION. 411 s e t s
of muti*aZZy o?thogonaZ permutation arrays 0.7
the Examples 2.4 and 2.5 are row-extendible.
Proof.
G
Let
JL, and
of
let
associated t o
be t h e group used i n Example 2.4 o r 2.5 f o r t h e c o n s t r u c t i o n
r = (X;Lo,L1 ,...,Lt;TI,T2
G and t h e subgroups
,...,T v )
,..., S t
So,S1
X = G,
R e c a l l f r o m t h e p r o o f o f Theorem 2.3 t h a t
i E {O,l,
for all
...,tl,
4 . 1 and because o f
IT1,T2
r ( R ) ,i t
J
,...,T v }
and
,...,F,
F1,F2
of
G.
I gcG1 = Isif( f t F 1 }
k=1,2 ,...,s } . By Lemma
Li = ISig
= IFkg
1
gcG,
i s s u f f i c i e n t t o show t h a t t h e t r a n s v e r s a l
i s transversal-extendible.
seminet
with
and
be t h e t r a n s v e r s a l seminet
Let
Fl
{flfcF1)
=
b e a copy o f
F1
F1OX=O, and d e f i n e Y : = X u F 1 . A s a l l subgroups Si,
ao,al,
Mi:=
{Siaifut-f}
and d e f i n e
and
Li 2 tmnX
iaifl=Siaif2.
S o n F1
fore
I feF1},
I mcMi}.
I n o r d e r t o show
Then
le}
...,t,
i=O,l,
(a.S
1 0ay1)aifl
1
one o b t a i n s
Li = I m n X i mcMil.
i=O,l,...,t,
are
-1
w i t h S. = a . S a , L e t
1
i o i
Tv+l:=
F1. T r i v i a l l y t h e n Y g X
...,at E G
c o n j u g a t e t o each o t h e r , t h e r e e x i s t
Li = ImnX
= (a.S
1 0ar1)aif2
1
fl = f2. T h i s i m p l i e s
I mcMi},
assume
and t h u s
lLi I = ltmnX
The same argument i m p l i e s c o n d i t i o n
...,Mt).
I t remains t o show (S1).
satisfy
iz j, x r y
and
x,y
6
Let
(Siaifl
i,j
u
E
{O,l,
Ifl})
...,t } ,
n (S.a.f
3 5 2
fl,f2
E
I mcMi}
F 1 and
From
flf;'cSo.
I, and t h e r e -
(S2) f o r (Y;Mo,M1,
fl,f2
IJ IT2}).
E
F1 and
As
x , y ~X
(X;Lo,Ll,~..,
X. Assume y k X. Then
y=Tfl=f2
and x c S . a . f n S . a . f = (a.S aT1)a.f n (a.S aT1)a.f = aiSoflnajSofl.
1 1 1
J J 2
1 0 1
1 1
J O J
J 1
and thus Si = S
contraHence aiSo = a.S
T h i s i m p l i e s xf;' E a.S n a.S
1 0
JO'
J O
j'
d i c t i n g i* j .
17
Lt)
i s a seminet, e i t h e r
The a l t e r n a t i n g group
p r e c e d i n g s e c t i o n s . Since
x
or
y
cannot be c o n t a i n e d i n
f o r some o f t h e r e s u l t s o f t h e
A 5 y i e l d s an example
2
A 5 and PSL(2,2 ) a r e isomorphic as p e r m u t a t i o n
groups, b o t h o f t h e Examples 2.4 and 2.5 i m p l y t h e e x i s t e n c e o f a s e t
of
5
m u t u a l l y o r t h o g o n a l 25x12 p e r m u t a t i o n a r r a y s . By P r o p o s i t i o n 3.2, t h e s e t
59,
is
maximal, and by P r o p o s i t i o n 4.2
-9:
of
5
i s row-extendible, i . e . there e x i s t s a s e t
m u t u a l l y o r t h o g o n a l 2 6 x 1 2 p e r m u t a t i o n a r r a y s . I t i s unknown whether
i s row-extendible o r not.
195
Oit Pcrniutatiori Arra),s
5. TRANSVERSAL SEMINETS CARRYING DESIGNS
The concept o f t r a n s v e r s a l seminets comprises a l a r g e v a r i e t y o f d i f f e r e n t
mathematical s t r u c t u r e s . F o r d e t a i l e d i n v e s t i g a t i o n s one has t h e r e f o r e t o add
f u r t h e r r e s t r i c t i o n s . S e c t i o n s 2 and 4 and p a r t s o f S e c t i o n 3 t r e a t e d t r a n s v e r s a l
seminets w i t h a group o f t r a n s l a t i o n s o p e r a t i n g r e g u l a r l y on t h e p o i n t s . I n t h i s
s e c t i o n a d d i t i o n a l assumptions w i l l be imposed on t h e f o l l o w i n g i n c i d e n c e s t r u c = (X;Lo,L1,.
t u r e s which a r e a s s o c i a t e d t o each t r a n s v e r s a l seminet
T2,...,TV)
(let
o f l i n e s and
-
I(X,L)
I(T,X)
-
I(X,LUT)
LouL1u ... uLt
L:=
and
T:= IT1,T2,
the s e t o f transversals o f
T
J ):
i.e.
L
'j',
blocks a r e the l i n e s o f
t h e t r e a t m e n t s a r e t h e t r a n s v e r s a l s and t h e b l o c k s a r e t h e p o i n t s
J,
of
t h e t r e a t m e n t s a r e t h e p o i n t s and t h e b l o c k s a r e t h e l i n e s arid
The i n c i d e n c e s t r u c t u r e
J.
I(T,X)
exactly i f
J.I n
i s e s s e n t i a l l y t h e seminet o f
I(X,L
cases t h e in c i dence i s d e f ined n a t u r a l l y . For example,
I(T,X)
x r T..
1
x
E
and
X
x,yt X
x,y
E
or
1
.
are i n -
a r e d e f i n e d t o be
p a r a l l e l i n t h e i t h p a r a l l e l c l a s s i f and o n l y i f t h e r e e x i s t s a l i n e
I(T,X)
a l l three
Ti ,- T
i s the design w i t h mutually ortho-
gonal r e s o l u t i o n s mentioned i n S e c t i o n 1: The b l o c k s
with
i s the set
t h e t r e a t m e n t s o f t h i s i n c i d e n c e s t r u c t u r e a r e t h e p o i n t s and t h e
the transversa s o f
cident i n
. . . ,T V l ,
. . ,L t '- T 1'
1 t Li
Two i n t e r e s t i n g cases d i s c u s s e d l a t e r i n t h i s s e c t i o n o c c u r when
I(X,LuT)
a r e PBD's ( p a i r w i s e balanced d e s i g n s ) . F o r i n s t a n c e ,
has t h i s p r o p e r t y i f t h e a s s o c i a t e d s e t o f p e r m u t a t i o n a r r a y s i s a com-
I(X,LuT)
p l e t e s e t of m u t u a l l y o r t h o g o n a l l a t i n r e c t a f i g l e s ( t h e examples a f t e r P r o p o s i t i o n
5 . 3 show t h a t t h e converse o f t h i s s t a t e m e n t i s n o t t r u e ) . B e f o r e g o i n g i n t o det a i l , some d e f i n i t i o n s a r e necessary.
The seminet
3=
n
ILi I = r
f o r a l l i ), and
has
..., L t )
(X;Lo,L1,
exactly
's
p o i n t s . I n t h i s case
5
i s c a l l e d n - r e p Z a r i f each l i n e c o n t a i n s
i s a (tt1,r)-seniinet
i s a l s o c a l l e d an ( r , n ) - h n o
n s r, w i t h e q u a l i t y i f and o n l y i f
3
with
r = IXl/n
(i.e.
c o z f i p i m t i o z . One
i s a net or, equivalently, i f the
a s s o c i a t e d p e r m u t a t i o n a r r a y s a r e r o w - t r a n s i t i v e (see S e c t i o n 3 ) . An n - r e g u l a r
t r a n s v e r s a l seminet s a t i s f i e s
n
5
v
w i t h e q u a l i t y i f and o n l y i f t h e T i ' s
rv = C.
ITiI 2 1x1 = r n ) ,
1=1,2, . . . ,v
are pairwise d i s j o i n t . I n t h i s s i t u a t i o n
(because
t h e s e t o f t r a n s v e r s a l s can be c o n s i d e r e d as a new p a r a l l e l c l a s s
Lt+l,
associated permutation arrays are l a t i n rectangles. Therefore, i f
n = r = v, then
I(X,LIJT)
i s a (t+2,r)-net,
squares,
The t r a n s v e r s a l seminet
and t h e
and t h e a s s o c i a t e d p e r m u t a t i o n a r r a y s a r e l a t i n
r=
(X;Lo,L1
,..., Lt;T1,T *,.. .,Tv)
i s called
196
M . Deza and T, lhririger
q- uniform i f each p o i n t i s c o n t a i n e d i n e x a c t l y q t r a n s v e r s a l s . A q - u n i f o r m
t r a n s v e r s a l seminet i s n - r e g u l a r w i t h n = v/q. Each column o f each o f t h e assoc i a t e d p e r m u t a t i o n a r r a y s c o n t a i n s each element
i e
..., r l
{1,2,
either
q
or
0
t i m e s . An example o f an n - r e g u l a r ( w i t h n=4) b u t not q - u n i f o r m t r a n s v e r s a l seminet
i s p r o v i d e d by t h e o r t h o g o n a l p e r m u t a t i o n a r r a y s
(The s e t IA1,A21 i s a row-extension o f two o r t h o g o n a l l a t i n squares. I t i s easy
t o check t h a t A1,A2
have no o r t h o g o n a l mate.)
For each p o i n t
x
t r a n s v e r s a l s through
o f a t r a n s v e r s a l seminet, l e t
x, and d e f i n e
the incidence s t r u c t u r e
treatments
I(T,X)
i : ={: I
i s a 1-design
i s incident w i t h exactly
TicT
r
treatments. Notice t h a t
incident with
E ;
The b l o c k s o f
I(T,X)
have
ttl
Y
denote t h e number o f
X E X I . F o r each t r a n s v e r s a l seminet
Sr(l,i,v),
i . e . each o f t h e
b l o c k s , and each b l o c k
I(T,X)
x
v
is
may have repeated b l o c k s .
m u t u a l l y o r t h o g o n a l r e s o l u t i o n s ( i . e . any two
p a r a l l e l c l a s s e s o f d i s t i n c t r e s o l u t i o n s c o i n c i d e i n a t most one b l o c k ) . The r e s o Lo,L1 ,...,Lt.
I f t h e r e i s a nonnegative i n t e g e r
l u t i o n s a r e g i v e n by
i j
A
with
ITinTT.l
J
( i . e . i f t h e t r a n s v e r s a l s f o r m an ( r , h ) - e q u i d i s t a n t
f
= A
for all
code), t h e n
PBD w i t h any two d i s t i n c t t r e a t m e n t s c o n t a i n e d i n e x a c t l y
becomes a
i,j
with
I(T,X)
A
blocks.
E q u i v a l e n t l y , t h e a s s o c i a t e d p e r m u t a t i o n a r r a y s a r e equidistant w i t h Hammingdistance
r-A, 1 . e . any two rows o f each o f t h e p e r m u t a t i o n a r r a y s c o i n c i d e i n
exactly
I(T,X)
p o s i t i o n s . I f , moreover, t h e t r a n s v e r s a l seminet i s q-uniform,
A
i s a 2-design
if I ( T , X )
i s an
case, any
A'
t'
S,(2,q,v).
SA,(t',q,v),
As a consequence,
with
t ' 22, then
then
r ( q - 1 ) = A ( V - 1 ) . Analogously,
(:,--\)
r (?,--ll)
= h'
.
I n this
d i s t i n c t rows o f each o f t h e p e r m u t a t i o n a r r a y s c o i n c i d e i n e x a c t l y
positions.
The t r a n s v e r s a l seminets c o n s t r u c t e d i n t h e p r o o f o f Theorem 2.3 a r e n - r e g u l a r
and q-uniform, w i t h n = -I G I
and q = s . F o r none o f these examples I ( T , X ) i s a
r
PBD.
Many examples o f t r a n s v e r s a l seminets a r e g i v e n i n r171, w i t h
I(T,X)
an
o r a g r o u p - d i v i s i b l e d e s i g n ( i n t h i s case ITi n T . 1 < 1 f o r a l l d i s J
t i n c t i , j ) . For i n s t a n c e , by Theorem 1 . 5 o f r171 t h e r e i s a p o s i t i v e i n t e g e r v1
SA,(t',q,v)
such t h a t , f o r
v z vl,
the condition
tence o f a t r a n s v e r s a l seminet
S(2,3,v).
with
v = 3 (mod 12)
t
2
i s equivalent t o the exis-
and t h e p r o p e r t y t h a t
I(T,X)
i s an
There a r e s i m i l a r r e s u l t s o f o t h e r a u t h o r s ; some o f them a r e l i s t e d here:
I97
On Pwmictation Arrays
1) D i n i t z l81.
F o r each p r i m e power o f t h e f o r m
a positive
c > l an odd i n t e g e r , t h e r e e x i s t s a t r a n s v e r s a l seminet such t h a t
t=c-1
I(T,X)
and
I(T,X)
3)
k
i n t e g e r and
i s an
Kramer e t a l . C161.
2)
k
q = 2 c t 1, w i t h
i s an
S(2,2,qtl).
There e x i s t s a t r a n s v e r s a l semient such t h a t
t = 12
and
5(5,8,24).
Hartman C121.
There e x i s t t r a n s v e r s a l seminets such t h a t
S(3,4,v)
( v , t ) = (20,3),(32,5),(44,7).
for
I(T,X)
i s an
T r a n s v e r s a l seminets can be r e p r e s e n t e d by t h e f o l l o w i n g ( t t 1 ) - d i m e n s i o n a l
array.of side r : the c e l l
(io,il,.
,,,it)
contains the s e t
1
ITicT
xcTi}
if
l ~ o n l ~ l n . . . n l ~ t= { X Iand
, i t i s empty o t h e r w i s e ( t h e l i n e s o f each p a r a l l e l
1 2
r
Lk = {lk,l
ky...,lk}).
I f I(T,X)
c l a s s a r e assumed t o be l i n e a r l y ordered, i . e .
i s an
S(2,2,v),
t h e n t h i s a r r a y i s c a l l e d a Room ( t t 1 ) - c u b e o f s i d e
such a r r a y i s e q u i v a l e n t t o
size
(v-l)x(v-l)
(see 191). I f
(t+l)-dimensional
S(5,8,24)
then
i s an
S,,(t',q,v),
t h e n t h e above
a r r a y i s a ( t + l ) - d i m e n s i o n a l Room design; f o r i n s t a n c e ,
I 1I
Sttl(l,;
I(X,L)
I
I(X,L)
o f t h e seminet
I(X,L)
l c L } ,I X I )
becomes an
purullel structure) i f
ments o f
I(T,X)
y i e l d s a 13-dimensional Room d e s i g n o f s i d e 253, see [161.
The i n c i d e n c e s t r u c t u r e
1-design
r = v - 1 . Each
p a i r w i s e o r t h o g o n a l symmetric l a t i n squares o f
ttl
5
Sttl(l,n,rn).
I(X,L)
3=
(X;Lo,L l,...,Lt)
w i t h o u t repeated blocks. I f
3
is a
i s n-regular,
i s c a l l e d an An&& seminet ( o r a f f i n e
i s a l i n e a r space, i . e . i f any two d i s t i n c t t r e a t -
a r e i n c i d e n t w i t h e x a c t l y one b l o c k . O b v i o u s l y , AndrG seminets
do n o t a d m i t t r a n s v e r s a l s . A t r a n s v e r s a l seminet i s c a l l e d aZmost-Andr6
( o r com-
p l e t e ) i f any two d i s t i n c t p o i n t s a r e e i t h e r connected by a l i n e o r by a t r a n s v e r s a l . As each almost-Andre t r a n s v e r s a l seminet i s L-maximal, Lemma 3.1 y i e l d s
5.1, REMARK.
raiatecl s e t
Let t h e warisverstil seniinet
r)
fL(
0.fm : i t i t a
be aZmost-An&&.
Then. the asso-
2Zy mtho<gontrl perrrmtation arrays is mcLcima2.
A c t u a l l y , P r o p o s i t i o n 3.2 was proved e s s e n t i a l l y hy showing t h a t t h e i n v o l v e d
t r a n s v e r s a l seminets a r e almost-AndrG.
Recall t h a t a s e t o f
c a l l e d .?ompZete i f
5.2. PROPOSITISN.
sols. Then
t .. r-1,
t
mutually orthogonal l a t i n rectangles o f s i z e
vxr
is
t = r-1.
Let
be a trwi?suersiiZ ( t t 1 , r ) - s e m i n s t
wit;z e p n l i t y if
U ~ L oxLy
!
with
if t h e associated s e t
v
trcrnsver-
A(3')
of
pemnutation arrags is a z o ! p i e t e se% of ni!tunllLg orthogonal L i t i n r e c t a n g l e s . I n
this i?me
I(X,LuT)
is dr:
S(P,jr,v},rv)
.
M. Dezu and T. Iliringiv
198
Proof. The inequality t 5 r-1 i s a t r i v i a l consequence of Corollary 3.5.
Assume now t =r-I. By Corollary 3.6 then u = O f o r Jt(T) o r , equivalently,
& ( T ) i s a s e t o f l a t i n rectangles. The completeness of fL(J) implies 7
t o be complete, i . e . any two elements of X are connected by a b l o c k from LuT.
As T i r i T . = O for a l l i , j with i t j , one obtains t h a t I ( X , L u T ) i s a n
J
S(Z,{r,v),rv). n
Notice that complete sets of mutually orthogonal l a t i n rectangles have been
constructed in Quattrocchi, Pellegrino C191 f o r a l l v n o t exceeding the smallest
prime divisor of r
For a transversal seminet (X;Lo,L1,.. .,Lt;T1,T2 ,...,T v ) and M c {1,2, ...,v l ,
M Z ~ ,l e t
tM:=
T ~ I, and define d : = E
~
~
~
,
~
,Mz0
- ~ ,. .. ,v}
Then
.
inicM
with equality if the transversal seminet i s almost-Andr6.
5 . 3 . PROPOSITION. Let
s a k . Let
Thev t
=
w7
J
be a transversaZ (ttl,r)-semiMet i i ) i t h v
transuer-
be n-regular, and asswne I(X,LuT) to bc an S ( 2 , { r , n l , r n ) .
V
1 , anti there are eractly transversals through. each x
n n-
n
t
X,
Proof. As the transversal seminet i s almost-Andr6, ( 1 ) i s valid with equality.
-
tli',)
Since I T i nT.1 5 1 for i z j , one o b t a i n s o = El:.
J
'-1 I V
for a l l 1 - L and 1x1 = rn, ( 1 ) turns into
=
v(;).
With
lll=n
This can easily be transformed into the claimed equality for t . Let Q. be the
number o f transversals through some x c X. Then v - l = r n - l = a ( r - 1 ) +( t + l ) ( n - 1 )
together with the equality j u s t proved yields Q. .=:
3
A class of examples satisfying the assumptions of Proposition 5 . 3 can be obtained as follows. Let (X;Lo.L1, ...,L r ) be an affine plane of order r . For
kz2, consider the transversal seminet J = (X;Lo,L1,. . . ,Lr-k;T1,T2,.
.. ,Trk)
where T1 ,T2,. . . ,Trk are the lines contained in Ur-k<iir
L i . Then
satisfies the assumptions of Proposition 5 . 3 , with n = r and v = rk . I n t h i s s i t u a tion, Corollary 3.6 yields the bound t s r-2 (since ~ = 1 )while the exact value
i s t = r-k .
However, there also exist transversal seminets f o r which I ( X , L u T ) i s a linear
space S ( Z , { r , n } , r n ) with r z n . I n terms of 1131 these transversal seminets
correspond exactly t o the partiully resolvable 2-partitions PRP 2 - ( n , r , v ; t t l )
'
On Permurariorr A F T U ~
199
v = n r . Together w i t h P r o p o s i t i o n 5.3, t h e
w i t h the additional property that
r e s u l t s o f [131 and [51 on t h e s e designs i m p l y
5.4.
PROPOSITION.
S (2, I r,nl, r n )
A n n - r e g u l a r transversa2 ( t + l , r ) - s e m i n e t
I ( X,LuT)
with
an
ezists
a)
i n t h e case
n = 2, r ?3
if a n d onZy if e i h t e r t = 0, r = 3 o r t = r-1,
bi
in the case
n = 3, r = 2
if and onZy if
t = 0,
ci
in t h e case
n=3, r = 4
if
and onZy if
t = O
or
t=3.
N o t i c e t h a t a l l seminets i n t h e above p r o p o s i t i o n a r e e i t h e r complete s e t s o f
m u t u a l l y o r t h o g o n a l ? a t i n r e c t a n g l e s ( t = r-1) or t r a n s v e r s a l d e s i g n s (t=O).
6. SOME STRUCTURES RELATED TO TRANSVERSAL SEMINETS
I t i s well-known t h a t n e t s a r e e q u i v a l e n t t o s e t s o f m u t u a l l y o r t h o g o n a l l a t i n
squares, t o t r a n s v e r s a l d e s i g n s ( v i a d u a l i t y ) , t o o r t h o g o n a l a r r a y s , t o o p t i m a l
codes, e t c .
P r o p o s i t i o n 2.1 i s a g e n e r a l i z a t i o n o f t h e f i r s t o f t h e s e e q u i v a l e n -
ces. Next, t h e second e q u i v a l e n c e i s g e n e r a l i z e d t o t r a n s v e r s a l seminets.
Each t r a n s v e r s a l s e v i n e t
. . ,Lt;T1,T2,.
= (X;Lo,L1,.
a ~ P ~ ~ ~ ~ J C Ip aW cJk Li n g , v i a t h e i n c i d e n c e s t r u c t u r e
each o f t h e gyi"7u:s
t i n c t groups
Li,L.
L.
1
J
i n e x a c t l y one t r e a t m e n t
. . ,Tv)
I(L,X).
lcLi,
i s equivalent t o
Each b l o c k X C X h i t s
two t r e a t m e n t s f r o m d i s -
a r e j o i n e d b y a t most one b l o c k , and t h e r e i s no such b l o c k
L . =L.. Moreover, t h e r e a r e a d d i t i o n a l mzin tr.eatmeurts Ti;
1
$1
t i o n s t h e treatments o f I(L,X)
i n t o d i s j o i n t blocks.
if
and
each
Ti
parti-
...
Supoose now X and each Li t o be l i n e a r l y o r d e r e d , i . e .
X = Ix1,x2,
I
1 2
r
Li = { l i 7 1 .,..., l j 1 . L e t t h e m a t r i x B = ( b . . ) o f s i z e ( t t 1 ) X I X I be d e f i n e d
1J
1
b . . = k :0 x. i
. T h i s m a t r i x i s an OA (orthogomZ a r r a y ) i f t h e seminet
1J
J
i s a n e t ( s e e e.g. 191). 11.1 t h e general case, t h e s e t o f a l l 1x1 columns o f B
by
forms a code o f ZeilgLh
t+l o v e r t h e a l p h a b e t
{1,2,
...,r l
with
1x1
w r d s and
distance t , s i n c e any two d i s t i n c t columns c o i n c i d e i n a t most one p o s i T. corresponds t o a f a m i l y o f codewords (columns o f 6 )
J
such t h a t , f o r a l l k ' 11,2
~
,...,r I and a l l i c [0,1, . . . ,t l , t h e r e i s e x a c t l y one
codeword i n t h i s f a m i l y w i t h v a l u e k i n row i , The t r a n s v e r s a l T . can a l s o
miriimaZ
t i o n . Each t r a n s v e r s a l
be regarded as an ii!je,:tioe
, i i a g o n r l subset o f
r words o f l e n g t h t + l o v e r
{l,Z,
..., r ?
or, equivalently, l e t
Doints
xcX
Ti i . , T . = 0
J
and o f t h e l i n e s
li.Li
for a l l
(i21)
. . . ,rIttl
J
( i . e . as a s e t o f
which d i f f e r i n an-y c o o r d i n a t e ; see
161).
L e t the associated permutation arrays
{1,2,
A1,A2,
...,At
now b e l a t i n r e c t a n g l e s
i , j , i z j . Assume t h e numbering o f t h e
now be more s p e c i a l t h a n above: L e t
2 00
M . Deza and T. Ihririger
k
k
l = l k if lnT1zlo, andlet x=x
with j = ( i - 1 ) r t k i f x c T i n l o .
i
j
these assumptions t h e f i r s t row o f B s a t i s f i e s b = k i f j = k (mod r )
m.1,
for
t h e m t h row o f
i.e.
'm-19
(bm,(i-l)r+l
9
bm,(i-1)r+2
.. . , i r
(i :;)
of
Q(A,A')
of
A
3..
,
A
and
A'
bm,ir)
i s t h e i t h row o f
Am-l.
(i-l)r+l
,
a r e s i m i l a r e x a c t l y i f , f o r a l l j , column
can be o b t a i n e d f r o m column
A'
3
corresponds now t o t h e s e t o f columns
Ti
o f B . F o r example, c o n s i d e r i n g t h e s i m p l e complete s e t A1 =
1 2 3
2) o f m u t u a l l y o r t h o g o n a l l a t i n r e c t a n g l e s , one o b t a i n s
A2 = (3
The p e r m u t a t i o n a r r a y s
j
becomes a " l i n e a r i z a t i o n " o f t h e l a t i n r e c t a n g l e
B
Notice t h a t the transversal
(i-l)rt2,
U
Under
and,
be t h e
become
Q(
r x r
k's
matrix
(q..)
i n column
[; :;)' (;
'J
J
:23)
j
of
d e f i n e d by
of
=
A ' , and
A by renaming t h e symbols. L e t
q i j = k i f t h e i ' s i n column j
qij=*
o t h e r w i s e . For example,
(; ;:).
* 1 3
As a l l p e r m u t a t i o n a r r a y s a r e assumed t o be s t a n d a r d i z e d ,
qii
=i
for all
Q ( A , A ' ) . I n no column o f t h i s m a t r i x a symbol a p e a r s t w i c e . T h e r e f o r e
i in
Q(A,A')
can be c o n s i d e r e d as ( t h e m u l t i p l i c a t i o n t a b l e o f ) a p a r t i a l l e f t - c a n c e l l a t i v e
g r o u p o i d d e f i n e d on
{1,2,
..., r l .
The p e r m u t a t i o n a r r a y s
A,A'
are orthogonal i f
and o n l y i f t h i s g r o u p o i d d l s o i s r i g h t - c a n c e l l a t i v e , and
a p a r t i a l quasigroup. Moreover,
Q(A,A')
the permutation arrays are row-transitive.
Q ( A , A ' ) t h e n becomes
i s a complete quasigroup i f and o n l y i f
Let
A1,A2¶.
. . ,At
be s i m i l a r permuta-
t i o n a r r a y s . By P r o p o s i t i o n 1.2 o f 141, these a r r a y s a r e m u t u a l l y o r t h o g o n a l
exactly i f
Q(A1,A2j,
Q(A1,A3),
. .. , Q(A1,At)
are mutually orthogonal p a r t i a l
quasigroups. N o t i c e t h a t t h e a s s o c i a t e d t r a n s v e r s a l seminet i s n - r e g u l a r i f and
o n l y i f each
Q(Ai,A.),
i z j , has t h e f o l l o w i n g p r o p e r t i e s : Each i E (1,2, ...,r }
J
n times, and t h e r e a r e e x a c t l y n d i s t i n c t symbols i n each rcw
appears e x a c t l y
and i n each column.
There a r e many i n c i d e n c e s t r u c t u r e s b u i l t f r o m n e t s . Examples a r e t r a n s v e r s a l
geometries 161, d-dimensional n e t s 1211, and e x t e n s i o n s o f d u a l a f f i n e planes
C231. I t w i l l be i n t e r e s t i n g t o s t u d y s i m i l a r s t r u c t u r e s w i t h n e t s r e p l a c e d by
t r a n s v e r s a l seminets.
On Permutation Arrays
101
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