Affine Geometries Obtained from Projective Planes and Skew Resolutions on AG(3,q)

Annals of Discrete Mathematics 18 (1983) 355-376 0 North-Holland Publishing Company

355

AFFINE GEOMETRIES OBTAINED FROM PROJECTIVE PLANES AND SKEW RESOLUTIONS ON AG( 3 ,q) R. Fuji-Hara and S . A . Vanstone

R

The main r e s u l t o f t h i s paper i s t o g i v e a construction o f AG(nt1,q) from n the p r o j e c t i v e plane PG(2,q ). This construction can be applied t o construct what we c a l l skew r e s o l u t i o n s o f the l i n e s i n AG(n,q)

and p a i r s o f orthogonal resolu-

tions.

Denniston has shown t h a t there e x i s t s a

skew r e s o l u t i o n o f the l i n e s i n

AG(3,q)

by constructing a r e s o l u t i o n o f the l i n e s i n PG(3,q). As an a p p l i c a t i o n of

the main r e s u l t , we g i v e a d i r e c t proof o f t h i s and show t h a t there i s a one t o one correspondence between skew r e s o l u t i o n s o f l i n e s i n AG(3,q) l i n e s i n PG(3,q).

and r e s o l u t i o n o f

A skew r e s o l u t i o n i n AG(n,q) along w i t h the n a t u r a l r e s o l u t i o n

o f l i n e s i n AG(n,q) obtained from p a r a l l e l i s m form a p a i r o f orthogonal resolutions.

Orthogonal r e s o l u t i o n s have r e c e n t l y a t t r a c t e d the i n t e r e s t o f various

authors.

1. INTRODUCTION Although we are p r i m a r i l y i n t e r e s t e d i n BIBDs, we begin by d e f i n i n g a s l i g h t l y more general design.

An (r,l)-design

D i s a c o l l e c t i o n B o f subsets

(blocks) from a f i n i t e s e t V ( v a r i e t i e s ) such t h a t (a) every element o f V i s contained i n p r e c i s e l y r blocks o f B. ( b ) every p a i r o f d i s t i n c t elements o f V i s contained i n e x a c t l y one block. I f a l l blocks o f D have the same c a r d i n a l i t y (block size), then D i s s a i d t o be a balanced incomplete block design (BIBD) w i t h parameters (v,k,l), v =

IVl,

and k i s the block size.

A (u,k,l)-BIB0

s e t V ' i s c a l l e d a subdesign o f a (v,k,l)-BIBD

where

w i t h block s e t B ' and v a r i e t y

w i t h block and v a r i e t y s e t B and V

r e s p e c t i v e l y i f V ' _C V and B ' _C B. An ( r , l ) - d e s i g n D i s s a i d t o be resolvable i f t h e blocks o f B can be p a r t i tioned i n t o classes ( r e s o l u t i o n classes) R,,

R2,

..., Rr

of V i s contained i n p r e c i s e l y one block o f each class.

such t h a t every v a r i e t y Such a p a r t i t i o n i n g i s

356

R . Fuji-Hara and S.A. Vanstone

c a l l e d a r e s o l u t i o n o f D. The f i n i t e a f f i n e geometry AG(n,q)

i s u s u a l l y o b t a i n e d from a v e c t o r space

Here, we s t a t e c o m b i n a t o r i a l axioms f o r AG(n,q)

over a G a l o i s f i e l d .

n

> 3,

taken

from Lenz 1101, as they a r e convenient f o r p r o o f s o f isomorphism which are d i s cussed l a t e r . L e t L be a c o l l e c t i o n o f subsets (blocks o r l i n e s ) o f a f i n i t e s e t o f p o i n t s P.

A paraZZeZisrn o f the system z = (P,L)

i s d e f i n e d t o be an equivalence r e l a -

tion // satisfying

(*) For any l i n e e and p o i n t , t h e r e i s a unique l i n e Let

z be

e'

such t h a t

e//e'

and PE e l .

a system w i t h a p a r a l l e l i s m // s a t i s f y i n g t h e f o l l o w i n g c o n d i t i o n s :

( i ) Any two p o i n t s Ply P2 are i n c i d e n t w i t h (contained i n ) e x a c t l y one l i n e

p1p2

( i i ) If

//

t h e r e i s a p o i n t P ' on

and P i s a p o i n t on

% and

F3 d i s t i n c t from P1 and P3,

( i i i ) I f no l i n e c o n t a i n s more than two p o i n t s and Ply

then

P2, P3 a r e d i s t i n c t

PP 1 2

have a

( v ) There e x i s t two l i n e s n e i t h e r p a r a l l e l nor w i t h a common p o i n t .

Then

p o i n t s , then the l i n e

e2

and k3 such as P2 E t2 //

3'

P3 E k3 / /

p o i n t i n comnon. ( i v ) Some l i n e s c o n t a i n s e x a c t l y q 2 2 p o i n t s .

I = AG(n,q) f o r some n 2 3. I f E i s a BIBD, then i t i s e a s i l y seen t h a t every p a r a l l e l class, t h a t i s , every c l a s s o f t h e equivalence r e l a t i o n //, contains e q u a l l y many blocks.

But

p a r a l l e l i s m o f BIBD's may n o t be unique, since any r e s o l u t i o n i s a p a r a l l e l i s m . L

A f i n i t e a f f i n e plane i s more simply d e f i n e d t o be a ( q ,q,l)-BIBD.

It i s

w e l l known t h a t an a f f i n e plane has a unique r e s o l u t i o n ( p a r a l l e l i s m ) . I f / / i s a p a r a l l e l i s m o f an a f f i n e geometry s a t i s f y i n g t h e axioms, then

d i s j o i n t l i n e s which a r e n o t p a r a l l e l a r e c a l l e d skew Zines.

A resolution o f

l i n e s i n an a f f i n e geometry, i n which each c l a s s c o n s i s t s o f skew l i n e s , i s c a l l e d

a skew r e s o l u t i o n .

We d i s p l a y a c o n s t r u c t i o n f o r a skew r e s o l u t i o n o f AG(3,q)

in

Section 4.

'+'

2. CONSTRUCTION OF RESOLVABLE ((9 -1 ) / ( q - 1 ) ,1 )-DESIGNS 2 A p r o j e c t i v e plane o f o r d e r m i s an (m +m+l,m+l,l)-BIBD, v a r i e t i e s and b l o c k

blocks i n the design are c a l l e d p o i n t s and l i n e s o f t h e p r o j e c t i v e plane, respec-

357

Affine geometries obtained from projective planes tively.

L e t P be a p r o j e c t i v e p l a n e o f o r d e r q

n

,q

a p r i m e power.

L e t am and llo

be two l i n e s i n P i n t e r s e c t i n g a t a p o i n t z. U i s a s e t o f p o i n t s on am - { z } , n 2 Q I U I = u G q , L e t V(a), a E U, be a s e t o f l i n e s i n P-am p a s s i n g t h r o u g h a p o i n t a. We show c o n s t r u c t i o n f o r ((q V =

U

n+l

-l)/(q-l),l)-designs

on t h e v a r i e t y s e t

V(a).

aEU Since e v e r y l i n e o f V meets t h e l i n e

go

V(a) can be r e p r e s e n t e d b y a pair, V = {
-

Iz 1, a

a t a p o i n t o f a. x E

-

-

{z}, any element o f

Therefore,

{z}.

E U}.

L e t AG(n,q) be an a f f i n e geometry o f dimension n and o r d e r q. S i n c e l l 0 - I z } n n 1 correspondence c o n t a i n s q p o i n t s , t h e q p o i n t s i n AG(n,q) can be p u t i n t o 1

-

w i t h t h e p o i n t s on a o - { z } .

L e t L be a c o l l e c t i o n o f q-subsets i n a o - I d . w h i c h Hence, l e t G = ( a - { z l , L ) be an a f f i n e geometry

corresponds t o l i n e s o f AG(n,q). on L~-{Z}i s o m o r p h i c t o AG(n,q).

Now, we d e f i n e two t y p e s o f b l o c k s on V. BI(Y) where

3/a

=

{G; a

E

F o r each p o i n t y E P-a

,

u}, F o r e v e r y l i n e a E L and e v e r y

i s a l i n e p a s s i n g through p o i n t s y and a.

a E U BII(a,a)

= {;

x E a}.

These two t y p e s o f b l o c k s a r e s a i d t o be b l o c k s o f t y p e I and b l o c k s o f t y p e 11, respectively. By g e n e r a l i z i n g Lemma 1 i n Dickey and F u j i - H a r a [ 31, we have ( r , l ) - d e s i g n s on V. LEMMA 2.1:

IBI(y);

y E P-.to}

((q"+'-l)/(q-l),l)-design PROOF: L e t L~ and

B

{BII(xya);

a E L, a E U 3 is the

D on V w i t h block sizes q and

n2 be any two d i s t i n c t l i n e s o f V.

a2 i s i n U o r i n P a m . x and y.

U

set

of blocks of a

U.

The common p o i n t o f

t1

and

If a

n a = a E U, and L meet lo a t d i s t i n c t p o i n t s , 1 2 1 2 o f G which c o n t a i n s x and y. Then, t h e b l o c k There i s a u n i q u e l i n e

I 1 ( 6 , a ) i s a u n i q u e b l o c k c o n t a i n i n g a1 and a2.

If a n L ~ a= E P1

then

R. Fuji-Hara and S.A. Vanstone

358

B ( a ) i s the unique b l o c k c o n t a i n i n g I

and t2. 0

E~

We show t h a t t h e design D has a r e s o l u t i o n i f I U I

Q

qn.

Since AG(n,q)

has

( q n - l ) / ( q - l ) p a r a l l e l c l a s s o f l i n e s , G a l s o has p a r a l l e l classes, denoted by

..., St-l’

SO’ S1’

L e t U = (0,l 7...,u-l

t = (qn-l)/(q-l).

A c o l l e c t i o n of blocks o f type 11,

k.

T . = (BII(k,j); z E S 1

i’

i s a resolution class f o r 0 o f those classes.

Q

i s a p a r t i t i o n o f t h e s e t o f type I1 blocks. t-1 which i s n o t i n U, and l e t M be t h e s e t o f l i n e s i n P-im

krn

passing through t h e p o i n t c.

any l i n e m

Rm =

E

Every block o f type I1 i s contained i n one

i G t-1.

T O y Tl,...yT

L e t c be a p o i n t on

the l i n e s o f M.

j E U)

Every p o i n t i n P-tm i s contained i n e x a c t l y one o f

Any l i n e o f V meets each l i n e o f M a t a p o i n t .

Therefore, f o r

My

{BI(x); x

E in}

i R : m E M I forms a p a r t i t i o n o f t h e s e t of type I blocks. m i 4 t - 1 1 U R{;, m E M } i s a r e s o l u t i o n o f D.

i s a r e s o l u t i o n class. Hence, R = {T.;

1

0 G

We can s t a t e the f o l l o w i n g r e s u l t :

THEOREM: ?he ( ( 9

n+1 -l)/(q-l),l)-design

D of L e m 1 has a resoZution if / U I 4 q

n

.

(Note: The type I blocks form a t r a n s v e r s a l design).

3. ISOMORPHISM RESULTS I f u = q 7 t h e design D c o n s t r u c t e d i n t h e previous s e c t i o n has t h e same

parameters as a order q.

BIBD obtained from a f i n i t e a f f i n e geometry o f dimension n+l and

I n t h i s section, we show c e r t a i n c o n d i t i o n s under which D i s isomorphic

t o the a f f i n e geometry. L e t u = q.

For each p o i n t i E U, t h e s e t o f l i n e s V ( i ) i s i n 1 - 1 c o r r e -

spondence w i t h t h e s e t o f p o i n t s i n G = (I - [ z } , = (V(i),tBII(L,i);

e

E L } ) i s isomorphic t o G.

L).

Therefore H ( i ) =

Then, H(0)yH(l),...yH(q-l)

d i s j o i n t n dimensional a f f i n e geometries i n U, where U = tO,l,...,q-l}. every b l o c k o f type I meets any H ( i ) , 0 6 i G q-1,

2

any ( q ,q,l)-subdesign

are Since

a t one element o f V ( i ) , then

(plane) i n D c o n t a i n i n g a b l o c k o f type I meets H ( i ) i n a

block of t y p e I 1 f o r any i E U.

359

Affine geometries obtained from projective planes L e t c ( x ) by a s e t o f p o i n t s on L example, c ( x ) = Iao,a2,

...,aq-1 1 i f

B ( x ) = {, ,...,
THEOREM 3.1:

-{z} a t

0

q-1

If D i s isomorphic to a n

any point x of P - ( i m

U Lo),

which l i n e s o f BI(x)

meet

a.

0)

for

,q-l>).

+

1 dimensional affine geometry, the, f o r

is a line of G or no three points in c ( x ) are

C(X)

collinear in G. PROOF: Suppose t h a t , f o r some x

E

and one p o i n t b n o t on t h e l i n e .

P-am, c ( x ) has t h r e e p o i n t s al,a2,a3

3 {,

L e t BI(x)

Consider a plane P i n D generated by BI(x)

on a l i n e

,y~,i4>}.

and B

(a, i4).

(a,

BI 1 3 i4) contains G3,i4>. Then P must i n c l u d e and , t h a t i s , t h e b l o c k

II 3

-

a ' ) i s also a l i n e o f P , since 2 3"2 a3,i2) c o n t a i n s because al is 1 2 on t h e l i n e a a and alyil a r e p o i n t s o f P. Then, B ( a ) = { i s a p o i n t o f P t h a t i s , P i n t e r s e c t s w i t h

B ( a ) = { and G2,i2>a r e p o i n t s o f P. BII(a2

H(i4) i n q

+1

points.

(a

0

The f o l l o w i n g t h r e e cases a r e p o s s i b l e when D

s isomorphic t o an a f f i n e

geometry. ( 1 ) For a l l x E P-(L-

U

x0),

c ( x ) i s a l i n e o f G.

( 2 ) For a l l x E P-(em u i 0 ) ,c ( x ) has no t h r e e c o l l i n e a r p o i n t s . (3) P has two types o f p o i n t s x, x ' , c ( x ) i s a l i n e o f G, c ( x ' ) has no t h r e e c o l l i n e a r points. Here we need t o d e f i n e a t r a n s v e r s a l design f o r t h e f o l l o w i n g lemma. t r a n s v e r s a l design TD(m,t) s e t s Gi,

i s a p a i r (X,B)

A

where X i s a union o f m u t u a l l y d i s j o i n t

1 G i G t (groups) o f elements ( v a r i e t i e s ) o f c a r d i n a l i t y m and B i s a

s e t o f t-subsets ( b l o c k s ) o f X w i t h t h e p r o p e r t y t h a t any B E B c o n t a i n s p r e c i s e l y one element from each Giy

1 G i G t, and two elements from d i s t i n c t groups a r e

contained i n e x a c t l y one b l o c k o f B.

TD(q,q)

= (X,B)

I t i s w e l l known t h a t a t r a n s v e r s a l design

i s isomorphic t o an a f f i n e plane w i t h p o i n t s e t X and l i n e s e t

U I G 1 U 8. 1 C i G q i I n t h e n e x t lemma and theorem, we show t h a t t h e case ( 1 ) i s a s u f f i c i e n t

c o n d i t i o n f o r isomorphism, when q 2 3.

360

R. Fuji-Hara and S.A. Vanstone

LEMMA 3.1:

&cn D kc: t h e property that

ti,en, f o r tmo LLacks ~ ~ ~ ( m and , i )B

C(X)

G P p ~mullei

L ),

X E P-(I?,U

i # j, there exists a (q2,q-1)-sd-

(mI,j),

I1 2ssign (Fiaiej P .z'n D containing BII(m,i)

i s a l i n e of G f o r any

and BII(m',j),

0

if and onZy if m and m '

ii-nes ic G.

PROOF: L e t F be a s e t o f p o i n t s i n P-I?, a t which l i n e s my b E m'.

G,i> i n t e r s e c t s < b,j> f o r

(m,i) and B

(m',j) a r e d i s j o i n t blocks belonging t o H ( i ) and I1 I f P i s a plane c o n t a i n i n g B (m,i) and B (rn',j) i n D, a H(j), respectively. I1 I1 block o f type I,BI(x), must be a l i n e o f P f o r any p o i n t x E F. From the r e l a -

a

E

B

I1

t i o n between an a f f i n e plane and a t r a n s v e r s a l design TD(q,q),

we can say t h a t

t h e r e e x i s t a plane i n D c o n t a i n i n g B (m,i) and B (m',j) i f and o n l y i f I1 I1 T = IBI(x); x E F } forms a t r a n s v e r s a l design w i t h groups B (m i),i = O,l,..., I 1 i' q-1 where mo,ml. ...,m a r e l i n e s o f G i n c l u d i n g m and m ' . q- 1 There i s a plane P ' i n G Suppose t h a t m i s p a r a l l e l t o m ' i n G b u t m # m'. which contains m and m ' .

L e t S be a p a r a l l e l c l a s s o f P ' c o n t a i n i n g m and m'.

i c ( x ) ; x E F ; i s the s e t o f l i n e s o f P ' e x c l u d i n g S.

o f S, then l i n e

i n G meets m and m',

that is,

5 i s a c(a)

a block c o n t a i n i n g Q,i> and must be o f type 11. block o f type 11.

Si = BII(mi,i),

(x); x € F } ,

Therefore,

i E U.

f o r some a E F.

So any Si,

But

f o r i E U, i s a

T i s a t r a n s v e r s a l design

When m = m ' ,

T i s a t r a n s v e r s a l design w i t h groups

Suppose t h a t m and m ' a r e skew l i n e s .

There a r e no two d i s t i n c t l i n e s L and

mi E S.

w i t h groups BII(mi,i),

I1

I

I f Si has two elements Q,i> and where x and y a r e i n d i s t i n c t l i n e s

i E U.

B

L e t Si={EB

(m,i),

i= 0

,...,q-1.

r ' o f G such t h a t

I? and P' b o t h meet in and m',

n o t on m nor m ' .

Therefore, t h e r e a r e no two d i s t i n c t b l o c k o f t y p e I , BI(xl),

BI(x2)

such t h a t BI(x,)

Bi(xl)

and BI(x2)

and BI(x2)

and I? and R ' have a common p o i n t

have a comnon p o i n t i n H(k), k # i,j, and

b o t h i n t e r s e c t s BII(m,i)

and BII(m',j).

(m,i) and B (m',j). I1 I1 Suppose m and m ' have a p o i n t c i n comnon.

So, t h e r e i s no a f f i n e

subdesign i n D c o n t a i n i n g B

Let

P be a plane i n D c o n t a i n i n g

B

(in,:) and BII(m',J), i # j. L e t xa and yb be p o i n t s i n P-am a t which I1 meets d,i>,a E m-Ic I, and meets 4,j>, b E m ' - ( c l , r e s p e c t i v e l y . For any a

E

-

m - { c l and b

cb = m',

m ' - l c l , c ( x a ) = m and c ( y ) = m ' , r e s p e c t i v e l y , since b r e s p e c t i v e l y . L e t Ak = Iz, < r , D E BI(xa), a E m-{c}) and Bk =

i z ;
b E m'-(cIl,

yb # yb, if b # b ' , then A

k

where k # i,j.

= m - { c I and B

k

=

m and

Since xa # xal i f a # a ' , and

= m'-(c).

L e t d be a p o i n t i n P-a_

at

Affine geometries obtained from projective planes which meets 4,j>,a Bl(d),

+,k>E

i s not i n m

ments i n H ( k ) , k BII(m',j). LEMMA 3.2:

E

m-{c},

U

m'.

361

Then p o i n t e o f G, where

b E m'-{c}.

Therefore, P has a t l e a s t 2(q

-

1)

+

2

>q

ele-

# i,j. So t h e r e i s no p l a n e o f D c o n t a i n i n g BI 1(m,i) and

0

Suppose D has the property ( 1 ) f o r c ( x ) .

For any bZock B and any

L

v a r i e t y e # B of D, there e x i s t a unique ( q ,q-1)-subdesign

(pZmel o f D contain-

ing B and e. PROOF: F i r s t , we show t h a t i f a (q2,q-1)-subdesign

I b l o c k B = {aSi,i>; H ( i ) , i E U.

(plane) S i n D contains a type

i E U } , t h e n S c o n t a i n s e x a c t l y one t y p e I 1 b l o c k f r o m each

There; i s a v a r i e t y i n S b u t n o t on By say e = .

Then, a t y p e

I 1 b l o c k C = B ( b , k ) must be a b l o c k i n S. There i s a u n i q u e b l o c k B ' i n S I1 k c o n t a i n i n g e and d i s j o i n t t o B. I f B ' i s a b l o c k o f t y p e 11, which i s a b l o c k i n H ( k ) , t h e n two d i s t i n c t b l o c k s B', C h a v i n g a common v a r i e t y e a r e c o n t a i n e d i n S.

B ' and C g e n e r a t e a p l a n e i n H ( k ) , which i m p l i e s t h a t S i s i n H(k). p o i n t s n o t i n H(k).

i E U}.

T h e r e f o r e B' must be a b l o c k o f t y p e

-

I,

B u t S has

say B' =

{a!,i>; 1

i n each H ( i ) , i E U.

Then S has a t l e a s t one t y p e I 1 b l o c k BII(bibiyi)

By t h e same manner above, S cannot have two t y p e I 1 b l o c k s f r o m a H ( i ) . Lemma 3.1 says t h a t t h e r e i s a p l a n e c o n t a i n i n g BII(e,i) and o n l y i f L i s p a r a l l e l t o L' i n G.

i # j if

and BII(e',j)

Then, b l o c k s o f t y p e I 1 BII(eiyi),

c o n t a i n e d i n S have t h e p r o p e r t y t h a t t h e ki's

Further,

iE U

a r e p a r a l l e l i n G.

Suppose B i s a b l o c k o f t y p e I {Oiyi>; i E U } and e =
j E U, t h e r e i s u n i q u e l i n e

p r o o f o f Lemma 3.1,

eelJ

Suppose B = B ( e , i ) and e = , i # j. I f t h e r e i s a p l a n e c o n t a i n i n g I1 B and e, i t must c o n t a i n a b l o c k o f t y p e I, say C, such t h a t e, 4 , i > E C, where b i s a p o i n t o f L.

There i s a u n i q u e l i n e L' i n G which c o n t a i n s a and p a r a l l e l t o

e.

t h e r e i s a p l a n e c o n t a i n i n g B and BI1(e',j),

By Lemma 3.1,

which i s t h e u n i q u e

p l a n e b y t h e s t r u c t u r e o f S mantioned above. Suppose B = B ( e , i ) and e = , t h e n t h e r e i s a unique p l a n e i n H ( i ) I1 0 c o n t a i n i n g B and e s i n c e H ( i ) i s i s o m o r p h i c t o AG(n,q). S i n c e G i s an a f f i n e geometry, G has t h e p a r a l l e l i s m on l i n e s s a t i s f y i n g t h e axioms o f an a f f i n e geometry.

We denote p a r a l l e l i s m i n G by / /

a r e l a t i o n //,, on b l o c k s o f D u s i n g //G.

G'

Here, we d e f i n e

362

R , Fuji-Hara and S.A. Vanslone For any blocks B,B'

o f 0, B//DB' i f and o n l y i f

( a ) B and B ' a r e both type I 1 blocks, say B = BII(ayi),

B = B

or

I1

(t',i'),

i,i'E U, such t h a t a / / G

1'.

( b ) B and B ' are b o t h type I blocks, say B = tGiyi>; i E U 1 and 8 ' = {G'.,i>; i E U], 1

(b,) B = B ' ,

satisfying:

or

(b2) B n B ' =

0

-

//

such t h a t bibii

G

b b ' f o r any i,j E U. J j

i t i s c l e a r t h a t t h e design D w i t h the r e l a t i o n s

From Lemma 3.1 and 3.2,

//D s a t i s f i e s the axioms o f an a f f i n e geometry i f //D i s a p a r a l l e l i s m . n e x t two lemmas, we show t h a t //D

I n the

i s a p a r a l l e l i s m on blocks o f D.

For any A i n G, a subset Z = {; 3 U ) o f V and t h e blocks o f D contained i n Z form a ( q , q - 1 ) - s u b d e s i g

L e t A be an a f f i n e plane i n G.

x

E

A, i E

o f 0, say S ( A ) .

This f o l l o w s from t h e p r o o f o f Lemma 2.1.

We r e q u i r e t h e

following result.

LEMMA 3.3:

S ( A ) is isomorphic to AG(3,q) for any affine pZane A in G, q 2 3.

PROOF: L e t n be t h e s e t o f (q',q,l)-subdesigns

( p l a n e ) of D contained i n Z.

As a

step o f the proof, we show t h a t a p a i r E(A) = (Z,n) i s a (q3,q2,q + 1)-BIBD. It 3 2 i s c l e a r t h a t l Z / = q and every plane of n c o n t a i n s q v a r i e t i e s o f V. L e t us show t h a t , f o r any a r b i t r a r y p a i r o f d i s t i n c t v a r i e t i e s v ,v o f Z, t h e r e are q t l 1 2 planes i n n c o n t a i n i n g v1v2. There i s a unique b l o c k B c o n t a i n i n g v1,v2 and a plane c o n t a i n i n g v1,v2 must c o n t a i n B. Suppose B i s a block o f type 11, say BII(a,k),

k

E U.

From the p r o o f o f Lemma 3.2,

type I block.

where

r

i s a l i n e o f A and

A plane c o n t a i n i n g B i s i n H(k) o r contains a

Any plane o f t h e f i r s t type c o n s i s t s o f t; a E A } , say Pk.

i s u n i q u e l y determined by k.

It

Any plane o f second type c o n t a i n s e x a c t l y one type

I 1 block from each H ( i ) , i E U.

From Lemna 3.1 and 3.2,

k ' f k i f and o n l y i f z / / G a ' .

c o n t a i n i n g B and B ' = BII(a',k'),

k ' ( # k ) , t h e r e a r e q blocks B

t h e r e i s unique plane

I1

(x',k')

such t h a t a ' / / G a.

By f i x i n g

Hence, i n t o t a l ,

t h e r e a r e q + 1 planes c o n t a i n i n g B i n Z. Suppose B i s a type I b l o c k o f D, and E

must c o n t a i n a block B

I1

(a,i)

such t h a t a E a.

B.

Any p l a n e i n D c o n t a i n i n g There a r e q

t

1 lines i n A

363

Affine geometnes obtained from projective planes which c o n t a i n t h e p o i n t a.

By Lemma 3.2,

f o r each l i n e a such t h a t a E a , B and

B (a.,i) g e n e r a t e a u n i q u e p l a n e i n D which i s c o n t a i n e d i n Z. I1 t h a t E(A) = (Z,n) forms a (q3,q2,q t 1)-BIBD.

n.

Next, we show t h a t E(A) has a p a r a l l e l i s m on

a.

E

C, i E U}, where C i s a p a r a l l e l c l a s s o f A.

Hence, we can say

L e t Xc = {BII(k,i);

Xc can be p a r t i t i o n e d i n t o

( k , i ) ; R E C} f o r i = O,l,...,q-1. L e t Bc be t h e s e t o f p l a n e s i n E(A) I1 which have a t l e a s t one b l o c k o f Xc and a t l e a s t one b l o c k o f t y p e I. Then, f r o m G. = {B 1

Lemma 3.1,

any p l a n e o f Bc has e x a c t l y one b l o c k f r o m each Gi.

i # j, t h e r e i s a u n i q u e p l a n e i n B , which i m p l i e s jy C forms a t r a n s v e r s a l d e s i g n T(q,q). It follows t h a t there are q p a r a l l e l

two b l o c k s B (Xc,Bc)

F u r t h e r , f o r any

Gi,

E

B' E G

c l a s s e s i n Bc f o r each p a r a l l e l c l a s s C o f A, which we t a k e as p a r a l l e l c l a s s e s o f Also, {Pi;

E(A).

i E U } i s a n o t h e r p a r a l l e l c l a s s o f E(A).

q(q t 1) t 1 d i s j o i n t p a r a l l e l classes i n

n.

There are, i n t o t a l ,

I t f o l l o w s t h a t E(A) has a p a r a l l e l -

ism. We now appeal t o a r e s u l t due t o Dembowski [ 1 ] (pp. 74). The f o l l o w i n g p r o p e r t i e s ( a ) , ( b ) o f a 'BIBD F w i t h a p a r a l l e l i s m such t h a t

m

> 3 (m

(a)

i s t h e number o f b l o c k s i n a p a r a l l e l c l a s s o f

F)

a r e equivalent:

F i s i s o m o r p h i c t o t h e system o f p o i n t s and hyperplanes o f an a f f i n e geometry.

( b ) Every l i n e c o n s i s t s o f m p o i n t s , where a l i n e i n F i s d e f i n e d t o be a l l p o i n t s i n c i d e n t w i t h e v e r y b l o c k t h r o u g h two d i s t i n c t p o i n t s o f F.

l i n e s o f F w i l l be a b l o c k o f D c o n t a i n e d i n Z, and a b l o c k o f F,

I n E(A),

o f course, i s a p l a n e o f

n.

T h e r e f o r e , by t h e Dembowski r e s u l t above, S(A) i s

i s o m o r p h i c t o t h e a f f i n e space AG(3,q).

LEMMA 3.4:

0

The r e l a t i o n /ID on blocks o f D i s a parallelism o f D, uhen q

PROOF: From Lemma 3.2, show t h a t t h e r e l a t i o n

/ID /ID

s a t i s f i e s t h e c o n d i t i o n (*) o f p a r a l l e l i s m .

Here, we

The symmetric and r e f l e c -

i s an e q u i v a l e n c e r e l a t i o n .

t i v e p r o p e r t y o f t h e r e l a t i o n /ID a r e e a s i l y seen.

> 3.

We show t h a t

/ID possesses

the t r a n s i t i v e property. L e t A , By C be b l o c k s such t h a t A /ID B and A

/ID

C.

blocks a r e type I 1 blocks, then i t i s c l e a r l y t r u e t h a t B

I f a l l o f these C.

I f A, B, C a r e

d i s t i n c t b l o c k s ( o f A = B o r A = C, t h e n t r i v i a l l y B / /

-

-

ab / /

G

a ' b ' and

,

E C

/ I Ga", where ,

EA,

C ) o f t y p e I, t h e n D G,i>, QI',~'>E B and

i # i ' . The f o l l o w i n g two cases must be considered.

364

R. Fuji-Hara and S.A. Vanstone

- -

- -

( 1 ) ab # a ' b ' and ac # a ' c ' ,

ab = a". generated by ab and-,

( 2 ) a t l e a s t one o f t h e two p a i r s i s t h e same l i n e , say Consider the case ( 1 ) . A

by

2 and

a'c'.

r

spectively).

be a plane i n 0

the set o f points i n

:(c r e s p e c t i v e l y ) i n t e r s e c t Lines i n

r

and

c o n t a i n d the blocks A and B, and A contains A and C.

t h e p r o o f o f Lemna 3.1,

tively).

Let

'G

L e t ga,

.to induces

From

G a t which v a r i e t i e s contained i n

an a f f i n e plane o f

G, say rG

(AG

respec-

G r e s p e c t i v e l y ) n a t u r a l l y correspond t o blocks i n r(A r e zbb' kc be l i n e s o f G corresponding t o A, B, C, r e s p e c t i v e l y . (A

a

r C

( F i g u r e 3.4)

Now, we have

pa

/IG

-

Desarguesian a f f i n e geometry AG(n,q), Consider, next, t h e case ( 2 ) ;

- _ _ - -

ab = a ' b ' = aa' = bb' i n G. planes

r,

- -

-

/ I GgC, ab / I -Ga ' b ' , ac / I Ga ' c ' i n G. then

ab =

-

bc /IG b ' c ' ,

-

a ' b ' and

L e t A be a plane o f

//

Since

G i s the

which i m p l i e s B //

- --

G

a'c'.

G

C.

ab = a ' b ' i m p l i e s

G c o n t a i n i n g 2 and

z.

The

.i i n D a r e b o t h contained i n t h e a f f i n e space S(A) d e f i n e d i n Lemna 3.3.

I t f o l l o w s t h a t t h e same c o n f i g u r a t i o n as F i g u r e 3.4 occurs i n t h e a f f i n e space S ( A ) , t h a t i s B / I DC.

0

Now, we can s t a t e the f o l l o w i n g r e s u l t : THEOREM 3.2: If C ( X ) is a line of G for any point x in P-(nm U go), ntl ,q,l)-BIBD D is isomorphic to AG(n+l,q), when n 2 2, q 2 3. (q

then the

Next, we show t h a t t h e design D which i s isomorphic t o an a f f i n e geometry i s c o n s t r u c t a b l e on a p a r t i c u l a r plane. n Consider t h e desarguesian p r o j e c t i v e plane o f o r d e r q , P = PG(2,qn),

q a

Affine geometries obtained from projective planes

> 2.

prime power, q

The p o i n t s i n P-a,

365

We use a method i n Hughes and P i p e r [9] f o r c o o r d i n a t i z a t i o n . n a r e assigned c o o r d i n a t e f r o m { ( x , y ) ; x,y E GF(q ) 1. The p o i n t s

on P- a r e c o o r d i n a t i z e d as R,

=

{(m)

3

U

{ ( a ) ; a E GF(qn) 1

such t h a t l i n e (O,a)(l,O)

meets ( a ) . 2n The p o i n t s i n P-R, can be t h o u g h t o f as elements o f GF(q ) p r o v i d e d a s u i t n a b l e p r i m i t i v e i r r e d u c i b l e p o l y n o m i a l o v e r GF(q ) i s s e l e c t e d . L e t x be a p r i m i 2n 2n t i v e element o f GF*(q ) = GF(q ) - t o } , Then, GF"(q

2n

) = {xi;

meets ( a ) and t h e l i n e (0,0)(0,1)

2n

i = 0,l ,...,q

-21

L e t !i0 be R

(-3 u {xj;

=

0

= I(0,a);

where

m

=

j

n O(mod q t l ) 1

a E GF(qn)l,

(0,O).

Po i s i n c i d e n t w i t h 11,

at

We d e f i n e an a f f i n e geometry G

(m).

o f dimension n and o r d e r q by: ( i ) a p o i n t o f G i s a p o i n t o f .to, ( i i ) a line

-

of G p a s s i n g t h r o u g h x and y, x,y E P ~ ,i s d e f i n e d by (1-A)y;

XY = { A X t

A E GF(q)

.

Consider two types o f c o l l i n e a t i o n s a,B on P-P, d e f i n e d by a: X + X

B:,

x

.+

(q"]), x

t

where a E R 0 1 O

C1 = { a

,a

,...,

x E p-I,

,

a, x E P-P-,

. n-

aq

'}

and C2 = { B ~ ; a E R 1 f o r m c o l l i n e a t i o n groups on P-E-. 0

Any j E C1 f i x e s l i n e s o f P-P, p a s s i n g t h r o u g h passing through f i x e s R, a

E R

0)

m

and any B E C2 maps a l i n e P a Therefore, any c o l l i n e a t i o n o f t h e s e two t y p e s

t o i t s coset.

p o i n t w i s e , and f i x e s P f i x e s t h e p o i n t s on R,

0'

my

These c o l l i n e a t i o n s a l s o a c t on G.

pointwise.

B,

f o r any

L e t a p o i n t x E P-am be on a l i n e R o f

P - R ~i n c i d e n t t o p o i n t z = .ton im,t h e n IB,(x);

a E P 1 i s t h e s e t o f p o i n t s on 0

P. L e t p o i n t x E P-R- be on a l i n e m o f P-a, p a s s i n g t h r o u g h m y t h e n i n So, f o r an a r b i t r a r y { a ( x ) ; i = O,l,...,q - 2 } i s t h e s e t o f p o i n t s o f m-Im}.

p o i n t x E P-(am u lo), {a

i B,(x);

i

a Ba E C1

x

C2}

366

R . Fuji-Hara and S.A. Vanstone

i s the s e t of points of P-(am u Let B I ( x ) = I,

to).

,...,)

be a block of type I o f 0.

Then,

u ( B I ( x ) ) i s a l s o a block of type I o f D;

? ( B I ( x ) ) = B I ( l ( x ) ) = i,, ...,
Let B I I ( e , i ) u(B I 1

= I ,, ...,l q-1 ( t , i ) ) i s a l s o a block o f type 11.

'A(BII(e,i)) = B

I1

q-1

be a block o f type I1 o f D, then

(a(9,),i)

...

i=%(a o ) ,i>,, ,6( aq-l ) ,i > l

=

),q-l>l.

.

Similarly, any c o l l i n e a t i o n 8, o f C2 a l s o a c t s on D.

Therefore, we can

s t a t e the following r e s u l t s :

LEMMA 3.5: If

(2

coliineaGion u on P which fixes impointwise and preserves the

,zfj%ne gecmetrg G defined sn to, then a induces an autonorphism of 0.

Let x be a point on P-(em u

to) and

l e t m be a l i n e o f G.

A line

m meets k m a t u

z, 1

Let D b e a BIBD constructed by the previous method with i' U = (uoyul, u Then any block of type I , B I ( y ) , where y E P-(am u e ) , can q-1 i be generated from BI(x) by some a 8, E C1 x C 2 , ai

E

...,

B+Y)

=

>.

BI(a

caw.

i

This implies, f o r every point x

E

P - ( a o u em), c(x) i s a l i n e o f G.

Therefore, we

can s t a t e the following r e s u l t . THEOREM 3.3: If t k e m is a set of collineation C on P of which any collineation of C s'c7t;;fie; Lama A,,,

and for any two points x,y E P-(a

cuck t m t y = a ( x ) , then the BIBD D constructed on

whai: n

0

U

em) there is an

a E C

P is isomorphic to AG(n+l,q),

> 2 , q > 2.

4. SKEW RESOLUTIONS OF AG(3,q)

Let D be an (r.1)-design having two r e s o l u t i o n s R and R ' . orthogonal t o R ' provided

/ R n SI

4

1

We say t h a t R i s

Affine geometries obtained from projective planes f o r each r e s o l u t i o n c l a s s R E R and each r e s o l u t i o n c l a s s S

E

367

R'.

Orthogonal r e s -

o l u t i o n s i n b l o c k designs have r e c e n t l y been o f i n t e r e s t t o v a r i o u s a u t h o r s .

I f we can c o n s t r u c t a skew r e s o l u t i o n o f t h e l i n e s i n

f o r example [ 7 ] , [ l l l ) . AG(3,q)

(see

t h e n t h i s r e s o l u t i o n and t h e n a t u r a l r e s o l u t i o n f r o m p a r a l l e l i s m p r o v i d e a I n t h e remainder of t h i s s e c t i o n , we c o n s t r u c t a

p a i r o f orthogonal resolutions.

skew r e s o l u t i o n o f t h e l i n e s i n AG(3,q).

2 Vanstone [ 4 ] , o r t h o g o n a l l y r e s o l v a b l e ( q t q + 1 , l ) 2 -designs D a r e c o n s t r u c t e d on a f i n i t e p r o j e c t i v e p l a n e P o f o r d e r q where P-am 2 i s c o o r d i n a t i z e d by two dimensional v e c t o r s o v e r GF(q ). I n t h e plane, we can I n R. F u j i - H a r a and S.A.

choose q p o i n t s U on imsuch t h a t t h e d e s i g n D i s i s o m o r p h i c t o AG(3,q).

B u t any

two o f t h e r e s o l u t i o n s c o n s t r u c t e d by t h e method o f [ 4 ] do n o t n e c e s s a r i l y f o r m a I n t h i s s e c t i o n , we show a c o n s t r u c t i o n f o r a skew r e s o l u t i o n o f

skew r e s o l u t i o n .

i n S e c t i o n 3.

AG(3,q) o b t a i n e d f r o m PG(2,q')

Tuo blocks o f type I , B1(xl)

LEMMA 4.1:

-

and B ( X ), o f D are p a r a l l e l if and onZy 1

2

if l i n e xlx2 is i n c i d e n t a t the p o i n t Z = imn

Lo.

PROOF: L e t B1(xl)

G

q-1

{, ,

=

0

,q-l>l.

...,6q-1 > ) a n d

BI(x2)

= {GO,O>,G1,l~,...,

Two d i s j o i n t l i n e s a r e p a r a l l e l i n an a f f i n e geometry i f and o n l y i f t h e y

a r e d i s j o i n t and t h e r e e x i s t s a u n i q u e p l a n e c o n t a i n i n g t h e two l i n e s . 3.1,

-

and B1(x2) a r e p a r a l l e l i f and o n l y i f l i n e s aibi

B1(xl)

G a r e p a r a l l e l o r t h e same l i n e .

-

a . can be r e p r e s e n t e d as a = a + (di j i J which ( a c o s e t o f i o )meets ( i ) = and ( j ) m ,

-

7

-

r e p r e s e n t e d as b = bi + (d! j J p a r a l l e l o r equal t o a . b J j'

-

d!). J

Then di

-

-

-.

di

-

-

-

d! = d . 1 J -

Therefore a b

j j

.

-

d! = 0. J

= a . b . t (di 11-

p a r a l l e l o r equal t o aibi.

0

dj).

-

and a.b i # j, i n J j'

-

-

Since (di

From Lemma

d . ) where di and d . a r e p o i n t s a t J J r e s p e c t i v e l y . b . a l s o can be J d. = d' d! i f and o n l y i f aibi i s J i J

d . = d! - d ' J 1 j d! = d d' d i i j j di d! and d d! a r e p o i n t s o f (i)and ( j ) = . -J J J So (j)m i s di

-

-

-

The common p o i n t o f ( i ) = and

-

d . ) i s a p o i n t on loo' a.b. i s J J J

R . Fuji-Hara and S.A. Vanstone

368

if we choose a point c from I ~ ~ - { uz }U) and construct resolution classes o f blocks of type I f o r the s e t of lines passing through c , then any of these resolu-

B u t , from the definition of Ti in section 2 ,

tion classes are skew classes of D.

every Ti obviously contains parallel lines. Let Mc be a s e t of lines of P-em passing through c i n a,-U. x

for m

E m),

E

I M c , i s a resolution c l a s s , every element of V appears precisely

once i n Rm f o r each m

u

X

E

Mc,

B (x) = V = . u I~ I E

E

(x);

Since R,={B

U

V(i)

H(i) i s an affine geometry isomorphic t o G

=

(eo-{z}; L ) , having V(i) as i t s point

s e t and { B I I ( ~ , i ) ; E L ] as lines. Define a 1

m

-

1 mapping

y

m

i j '. V(i)

V ( j ) by

-+

.(G,i>) = a , j >

1J

i f and only i f there e x i s t a block containing and 4,j> i n Rm. LEMMA 4.2: For any i , j

6

m

U and m E Mc,

7..

1J

i s an isomorphism from H(i) to H ( j ) .

PROOF: Let m, be a line o f Mc which passes through "m

= Ixj; j

3

m.

mm can be represented by

n k(mod q + 1 ) 1 U {,I,

where x i s a primitive element of GF"(q

2n

), k i s some integer.

i0 i s

defined in

section 3 as 1.

0

Let Using

'z,

=

IxJ; j

'J

= O(mod.

n q +l)l

{-I.

be the collineation on P-a,

defined in section 3;

: y

n

--f

y-xq

+1

.

elements of V ( i ) , f o r any i E U, can represented as ( a ) , i>; j = O , I

v(i) =

,...q n- 2 1 u I<-,i>)

where a i s a generator i n t o ( a r b i t r a r y point, not m). If a l i n e o f V(i) meets 0 et+k and mm a t x nad x , respectively, where t = q n+ 1 , e some integer. Then 0

O

~ ( )x = x ,x

t

and a(xet+k)

E

=

xet+k.xt

E

m_ are also on a l i n e of V(i).

i s , a line t of V(i) meets k0 a t y i f and only i f the l i n e Similarly, a line

II

of V ( j ) meet

y.xft+k, f some integer.

Let a l i n e

tively, and a line e ' of V ( j ) meets

II

meets mm a t yx

ko

That et+k

.

a t y i f and only i f the l i n e a. meets ma a t II to

of V(i) meet ko and m, a t a and b respecand m, a t d and b , respectively.

The

3 69

Affine geometries obtained from projective planes

b = a-x d = a.x

ettk

(ettk)

d = a.x (e-f)t

to

=

d.x f t t k

-

( f t t k ) = a.x(e-f)t

i s a c o l l i n e a t i o n i n C1.

So, yyj' i s an isomorphism which c a r r i e s

,

~ (a) -j>. ~

G

L e t m be a l i n e o f Mc which does n o t pass through

in i s an a d d i t i v e coset

m.

L e t a l i n e R o f V ( i ) meet R ~ m , and mm a t a,b and d, r e s p e c t i v e l y .

o f ma.

l i n e R ' o f V ( j ) meet g o y m and ma a t g, b and h, r e s p e c t i v e l y . b = d t P . = h t P 1

jy

-

Let a

Then

-

where Pi and Pk a r e p o i n t s a t which m meets ( i ) m and ( j ) m , r e s p e c t i v e l y . h = d t (Pi

-

Pj)

We know t h a t d = a-xettk g = (a-x

ettk

+ (Pi - P j ) ) x

= a.x ( e - f ) t t

P.

1

-

and g = h - x-(fttk).

(Pi

-

Therefore,

-(ft+k)

P.)x - ( f t t k )

J

P . i s a p o i n t i n mm, hence (Pi - P . ) x - ( ~ ~ ' ~i )s a p o i n t i n R 0' J J m e-f So y;;: - i s an isomorphism from H ( i ) t o H ( j ) , where

z . . = (P 1J

'J

i

-

P.)x-ft+k). J

L e t E = Ix1,x2,...,x

1 C m y F1(E) = I B , ( X ~ ) , B ~ ( X ~ ) , . . . , B ( x ) 1 i s a s e t o f 9 1 q skew b l o c k s such t h a t F2(E) = { f ( E , i ) ; i E U} i s a s e t o f skew blocks o f t y p e 11, where f ( E , i ) LEMMA 4.3:

J

2 Suppose D is the design obtained from PG(2,q ) and c

For any bZock

i # j, i,j

then we say t h a t E induces a skew n e t .

I E B ( x . ) ; j = 1,2,...q},

=

E

B of t y p e

11 in H(i),

B and

m

y.

U.

'J

i s a b l o c k o f type I1 i n H(k).

blocks i n D.

I f f(E,i)

B,(xI), i,j E

x,x'

E

E, x #

U, i # j, f ( E , i )

d i v i d e d by (qn-l)/(q-1).

and f ( E , j ) , XI,

0

{z}).

=

B.

Now, {Bl(x),

From Lemma 4.2, x

E

C'

f o r any

E l i s a s e t o f skew

i # j, a r e p a r a l l e l i n D, then Bk(x) and

must be p a r a l l e l .

and f ( E , j )

U

. ( B ) E H ( j ) are skew f o r any m E M

PROOF: E i s a s e t o f q p o i n t s on in such t h a t f ( E , i ) k E U, f(E,k)

E i0-(U

This i s a c o n t r a d i c t i o n .

So, f o r any

a r e skew; t h a t i s , e - f i n Lemma 4.2 cannot be

R. Fuji-Hara and S.A. Vanstone

370 LEMMA 4.4:

Suppose D i s the design obtained from PG(2,q2) and c E em-(U

E izduces u ~ e t chcn , R m

-

F1(E)

RIn-F1(E) does n o t meet any o f F2(E).

and any block o f

D i s j o i n t blocks o f type I and type I 1 are

skew, since any p a r a l l e l b l o c k o f a type I 1 b l o c k i s a l s o type 11.

-

F1(E)

+

{z}) and

F (E) i s also a skew cZass of D for any m E Mc. 2

t

PROOF: F2(E) i s a s e t o f skew blocks o f D from Lemma 4.3,

Rm

U

F2(E) i s a skew c l a s s o f D.

Therefore,

0

Next, we show a c o n s t r u c t i o n o f skew r e s o l u t i o n s o f AG(3,q), q > 2 . L e t 2 be a p r o j e c t i v e plane o f order q c o o r d i n a t i z e d by t h e method o f s e c t i o n 3. L e t 3 be a ( q ,q,l)-BIBD isomorphic t o AG(3,q) and constructed on P by t h e method o f

D

s e c t i o n 3.

:tap

1. L e t

i?

aqtl,

Q

i s t h e c o l l i n e a t i o n on P d e f i n e d i n s e c t i o n 3.

take q-subsets EmyEO,E ly...,E

On a l i n e m E M c - l m _ l ,

i s a p a r a l l e l c l a s s i n H(0).

f(Eo,0),...,f(Eq-2,0)

,...,f(Eq-,,i)

i E U, f ( E w y i ) , f ( E o , i )

such t h a t f ( E ,O), q-2 Then, o f course, f o r any

i s a p a r a l l e l class i n H(i).

MZ be a s e t o f l i n e s i n P-im passing through t h e p o i n t

z = em n to. Define Ni f o r each i E Q = I-,

2 1 t o be a s e t o f l i n e s o f

0, 1, ...,q-

MZ such t h a t any l i n e o f Ni meets m a t a p o i n t o f Ei.

Nm, No,

p a r t i t i o n o f MZ. o(Ei) some j E Q.

(That i s , any

Suppose t h a t Em c o n t a i n s a p o i n t on 8,.

Ei induces a skew net.) Z t e p 2. L e t

where

i E Q i s t h e p o i n t s e t on

...,N 9-2

is a

(m) and l i n e s o f N . f o r J But p o i n t s o f a(Em) a r e a l s o on l i n e s o f Nm. P o i n t s o f P are

= l o ( x ) ; x E Ei ),

c y c l i c a l l y represented. t h a t p o i n t s o f o(Ei),

(I

Therefore, we can assume, w i t h o u t l o s s o f g e n e r a l i t y

i E Q-I-1,

a r e on l i n e s o f

N i t 1 (mod.

q+l).

Affine geometries obtainedfrom projective planes

,

\

371

‘.

i S t e p 3. I n Iu (E .); i E Q, i = 0,1 J

following rule;

,...,q-2

i ( 1 ) one s e t from each l i n e u (m) f o r 0

<

0

1, we choose q - 1 s e t s under t h e

i G q-2,

( 2 ) one s e t f r o m each Niy i E Q, (3) There a r e no two s e t s E and E’ i n chosen q - 1 s e t s such t h a t f(E,i)

= f(E’,i)

f o r some i E U.

IfE i s a q-subset on m-ao, which induces a skew n e t , such t h a t any of

f(E,O),

do n o t c o n t a i n G,i> for any i

f(E,l),...,f(E,q-l) E, u( E )

E

U, t h e n

,..., o ~ - E)~ (

o b v i o u s l y s a t i s f i e s t h e above c o n d i t i o n s . We p r o v e t h e e x i s t e n c e o f such a q-subset when q 2 3. PROOF: L e t xo,xl,

...,xq-1

be p o i n t s i n m-n.

such t h a t



E B ( x ), and l e t

I i x = a.0 n m. We show t h a t t h e r e i s no 3-subset Y o f X = { x ,x l,...,X q-l ,x 3 such t h a t f ( Y , i ) c o n s i s t s of c o l l i n e a r elements i n H ( i ) f o r some i E U. 0

Suppose t h a t , f o r a s e t of t h r e e p o i n t s Y c X-{XI, c o l l i n e a r t h r e e p o i n t s of H(0). three points o f H(i). Then,

ly.

,



Then, f o r any i E U, f ( Y , i )

consists o f

consists o f c o l l i n e a r

We assume w i t h o u t loss o f g e n e r a l i t y t h a t Y = { x

and<.s2>

The b l o c k s BI(m)

f(Y,O)

a r e elements of BI(xo),

BI(x,)

and BI(x2),

x x 1. 1’ 2 respective-

0’

c o n t a i n s t h o s e elements, B(m) = {<~,O>,<~,l>,..,,<~,q-l>~.

373

R. Fuji-Hara and S.A. Vanstone

BI(m) meets BI(x2)

a t an element

and H ( l ) having f(Y,O)

and f(Y,1),

o f H(2).

respectively.

in d i s t i n c t p o i n t s , and meet A1 too. A

0

This means A.

both meets A.

and BI(x2)

and Al

a r e p a r a l l e l i n D.

= (A ) are skew from Lemma 4.3. Contradiction, 1 701 0 Suppose t h a t Y i s a 3-subset o f X i n c l u d i n g x. Block BI(x)

BI(xo)

s i n c e x i s on II

Q,l>,...,Q,q-l>}

c o n t a i n s +,O>

and BI(xl)

-

-- B I1 (x-,0)

3.1.

and

0'

-

and Cl = BII(xm,l).

m

But C, = )ol(Co)

l e a s t one element o f f ( X , i )

contradiction.

i s a s e t o f q t 1 element i n H ( i ) i n which

I n a H ( i ) , t h e number o f blocks which has a t

i s (q;')

t

q

+

p o i n t s s e t E on m where f ( E , i )

f o r any i E U.

0

S t e p 4. L e t K

= Ei

0

i n step 3.

Sij for 0

-$

Then = R ij 0

-

(m)

i S q, 0 G j

Then

That i s , Co and C1 a r e p a r a l l e l from Lemma

1 = ( q 2 t 3q

+

2)/2.

> 3.

t h e number o f blocks, q2 t q, i n an a f f i n e plane H ( i ) when q

-

3.

L e t Co be a block o f

and Co must be skew from Lemma 4.3,

no t h r e e elements a r e c o l l i n e a r .

exists q

forms

We assume Y = {x,xo,xl respectively.

,

Therefore, f o r any i E U, f ( X , i )

<=,i>

But

and , and l e t C1 be a block c o n t a i n i n g +,l> and .

H(0) c o n t a i n i n g 0

BI(m)

and A1 be blocks o f H(0)

m

and A

{,

C

L e t A.

0

, K1

= u(Ei

1

This i s l e s s than

So, t h e r e

i s a l i n e o f H ( i ) and does n o t c o n t a i n

),...,K

q-2

= oq-2(Ei

) be subsets chosen q-2

F,(a(Kj))+ F2(a(Kj))

< q-2 i s a skew c l a s s from Lemma 4.4.

Step 5. f(Ko,i),

f(Kl,i),...,f(Kq-2,i)

a r e q-1 p a r a l l e l blocks i n H ( i ) f o r

each i E U. PROOF: From the d e f i n i t i o n i n s t e p 2, any two blocks f(ai(E.),k) and f ( 0 i'(Ej'),k), J where k E U, i # i ' ,i,i'E Q and 0 G j, j < q-2, a r e p a r a l l e l o r t h e same block. i I f they a r e same l i n e ( j = j ' ) , then f ( o E.),k) c o n t a i n s an element o r J i i' n (E.) and o (E ) a r e on t h e same Nny n some element o f Q. The c o n d i t i o n i n J j' step 3 includes those cases. 0

There i s e x a c t l y one l i n e i n H ( i ) which i p a r a l l e l t o f ( K Denote the l i n e by gi.

sk =

O < jv< q - 2

Then

Fl(a

k

k ( K j ) ) t Ia (gi);

i s a skew c l a s s f o r 0 4 k G q.

iE U l

j'

i ) , j = Oyl,...,q-2.

3 73

Affine geometries obtained from projective planes S t e p 6 . { S1 J. . ; O ~ i < q , 0 ~ j ~ q - 2 } u { S i ; O G i 4 q } u R m m i s a skew r e s o l u t i o n o f D. PROOF: We know t h a t each c l a s s o f these i s a skew class, and no block o f type I

We prove t h a t no block of type I1 appears i n two

appears i n two o f these classes.

f ( K o i ) , f(K1,i),...,f(Kq-2,i),

gi are p a r a l l e l classes i n H ( i ) f o r each k i E U, from step 5. f ( K i)E S and g. E S Hence, a ( f ( K . i ) ) E Skj and oj 1 0' J k k The s e t f a ( f ( K i f )0;G k G q } u { a (gi); 0 G k < q } i s the s e t o f a (gi) E Sk. jy l i n e s i n H ( i ) , f o r any i E U. So, no block o f type I 1 appears i n two classes. 0 classes.

iy

This f i n i s h e s the c o n s t r u c t i o n o f a skew r e s o l u t i o n i n AG(3,q). o l u t i o n i n AG(3,q)

A skew res-

can be used t o induce a r e s o l u t i o n o f the l i n e s i n PG(3,q).

Resolution o f l i n e s i n PG(3,q) was f i r s t done by Denniston [ 21.

We i n c l u d e an

a1t e r n a t e p r o o f here f o r completeness.

LEMMA 4.5:

Let R be a skew resoZution of AG(3,q).

Then, for each cZass S o f R,

there i s a unique paraZle1 class of pZanes Cs of AG(3,q) such that there i s no l i n e o f S contained in any plane of Cs.

Moreover, C

PROOF: Suppose t h a t S i s a skew c l a s s o f R.

S

=

Cs, i f and only i f S

L e t P be a plane o f AG(3,q).

= S'.

P con-

t a i n s a t most one l i n e o f S. The number o f planes o f AG(3,q) c o n t a i n i n g a l i n e o f 3 2 3 2 But t h e t o t a l number o f planes i n AG(3,q) i s q t q t q. T h e r e f o r 6 S is q t q

.

t h e r e a r e q planes which do n o t c o n t a i n any l i n e o f S. c o n t a i n i n g any l i n e o f S.

Suppose Po i s a plane n o t

Then, t h e l i n e s o f S , which meet

meet any p a r a l l e l plane t o Po a t d i s t i n c t p o i n t s .

Po a t d i s t i n c t p o i n t s ,

That i s , any p a r a l l e l plane t o

Po i s a l s o a plane o f AG(3,q) which does n o t c o n t a i n any l i n e o f S. say t h a t , i f S i s a skew c l a s s o f AG(3,q), planes i n AG(3,q)

So, we can

t h e r e i s e x a c t l y one p a r a l l e l c l a s s o f

where any plane o f the c l a s s does n o t c o n t a i n any l i n e o f S.

L e t R be a skew r e s o l u t i o n , and l e t S and S ' be d i s t i n c t classes o f R. here a r e p a r a l l e l classes o f planes Cs and C s l which do not. c o n t a i n any l i n e o f S and S ' ,

respectively.

plane o f C s ( = CsI).

R.

Suppose now t h a t Cs and C s , be the same class. Each l i n e o f

The number o f l i n e s i n

2

P must be contained i n d i s t i n c t skew classes o f

P i s q + q, b u t t h e number o f skew classes o f R

excluding S and S ' i s q2 t q l i n e s o f P, c o n t r a d i c t i o n .

L e t P be a

-

0

1.

That i s , some skew c l a s s o f R must c o n t a i n two

374

R. Fuji-Horn and S.A. Vanstone

Here we d e f i n e a packing o f PG(3,q)

t o be a r e s o l u t i o n w i t h respect t o t h e

l i n e s i n PG(3,q).

THEOREM 4.1: Any skeu resoZution i n PG(3,q) induces a packing of PG(3,q) and

conljersely, any Tacking of PG(3,q) induces a skew resoZution i n AG(3,q). PROOF: Suppose t h a t R i s a skew r e s o l u t i o n o f AG(3,q).

L e t I: be t h e p r o j e c t i v e

space obtained by a t t a c h i n g an i n f i n i t y p l a n e P , t o AG(3,q). o f AG(3,q)

are i n c i d e n t a t t h e same p o i n t on P,.

t h e r e i s a unique l i n e

eS

In

z, p a r a l l e l

lines

F o r each skew c l a s s S o f R ,

i n P, where any p o i n t on P . ~ i s n o t i n c i d e n t w i t h any

l i n e o f S, because planes o f Cs a r e i n c i d e n t w i t h a l i n e o f P, and o n l y t h e l i n e s i n those planes are i n c i d e n t w i t h the l i n e of I . ~ # i s , . So

IS u L ~ ;S

P,.

Therefore i f S # S ' E R then

E R ! i s a packing o f p r o j e c t i v e space

Suppose Q i s a packing o f PG(3,q).

z.

L e t Po be an a r b i t r a r y plane o f PG(3,q).

Each class A o f Q contains p r e c i s e l y one l i n e xA o f Po.

Since i f A contains two

l i n e s from a plane then those l i n e s a r e i n t e r s e c t i n g , and each c l a s s o f Q must c o n t a i n a t most one l i n e o f Po.

The l i n e s o f A - { e A 1 meet Po a t d i s t i n c t p o i n t s ,

which means l i n e s o f A - { k A ) a r e skew i n t h e a f f i n e space PG(3,q)-P

0'

0

From the p r o o f o f Theorem 4.1 i t i s easy t o extend t h e r e s u l t t o show t h a t t h e r e e x i s t s a skew r e s o l u t i o n i n AG(n,q) The converse o f t h i s may be f a l s e . r e s o l u t i o n i n AG(4,q)

i f t h e r e e x i s t s apacking i n PG(n,q).

The authors b e l i e v e t h a t t h e r e e x i s t s a skew

b u t i t i s w e l l known t h a t t h e r e i s no packing i n PG(4,q).

Several r e c u r s i v e c o n s t r u c t i o n s f o r skew r e s o l u t i o n s e x i s t ( [ 6 ] ) .

i s no skew r e s o l u t i o n i n any AG(n,q),

As yet, there

n even.

ACKNOWLEDGEMENT: The authors would l i k e t o thank Professor U.S.R.

Murty f o r

valuable comnents made d u r i n g t h e p r e p a r a t i o n o f t h i s paper.

BIBLIOGRAPHY

1.

2. 3.

P. Dembowski, F i n i t e Geometries, Springer, 1968. R.H.F. Denniston, Some Packing o f P r o j e c t i v e Spaces, Rend. Accad. Naz. Linc e i , 52 (1972). L.J. Dickey and R. Fuji-Hara, A Geometrical C o n s t r u c t i o n o f Doubly Resolva b l e (n2tn+1 ,l)-designs, Ars Cornbinatoria v o l . 8 , 1979.

Affine geometries obtained from projective planes

4. 5. 6. 7. 8. 9. 10. 11.

315

R. F u j i - H a r a and S.A. Vanstone, On Autornorphisrn of DoubZy Resolvable Designs. Proceedings o f t h e 7 t h A u s t r a l i a n Conference on Combinatorics, 1979. Springer-Verlag L e c t u r e Note 829 (1980). R. Fuji-Hara and S.A. Vanstone, Transversal Designs and Doubly Resolvable Designs. Europ. J , of Comb., 1 (1980), 219-223. R. Fuji-Hara and S.A. Vanstone, Recursive Constructions f o r Skew Resolutions i n A f f i n e Geometries, Aequationes Mathernuticue ( t o appear). R. F u j i - H a r a and S.A. Vanstone, On t h e spectrum o f doubly r e s o l v a b l e K i rkman systems , Congressus Nwnerantiwn, 28 ( 1 980), 399-407. J.W.P. H i r s c h f e l d , Projective Geometries over F i n i t e FieZds, Clarendon P r s s , Oxford , 7 979. D.R. Hughes and R.C. Piper, Projective Planes, Springer-Verlag, New York, Hei de 1berg , B e r l in H. Lenz, Zur Begriindung der a n a l y t i s c h e n Geometrie, Bayer Akad. Wiss.Math Natur. K Z , S.-B., 17-72, 1954. A. Rosa, Room squares generalised, Annals of Discrete Math., 8 (1980), 4557.

.

Department o f Combinatorics and O p t i m i z a t i o n U n i v e r s i t y o f Waterloo Waterloo, Ontario, Canada N2L 361