Finite geometric configurations

Finite geometric configurations

Citation for published version (APA):

Seidel, J. J. (1976). Finite geometric configurations. In P. Scherk (Ed.), Foundations of Geometry (Selected

papers of a conference, Toronto, Canada, 1974) (pp. 215-250). University of Toronto Press.

Document status and date:

Published: 01/01/1976

Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be

important differences between the submitted version and the official published version of record. People

interested in the research are advised to contact the author for the final version of the publication, or visit the

DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page

numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne Take down policy

If you believe that this document breaches copyright please contact us at: openaccess@tue.nl

providing details and we will investigate your claim.

Proc. Conference

11

Foundations of Geometry

11

ed. P. Sherk,

Univ

. of Toronto Press (1976),

p.

2

1

5 - 250.

FINITE GEOMETRIC CONFIGURATIONS1

J.J. Seidel

l. INTRODUCTION

We present five elementary examples in 3- and 4-dimensional

spaces over Galois fields GF (q) , over the reals lR, and

over the complexes

a:.

They serve as an introduction for the subsequent sections. These sections treat discrete

geometry, that is a mixture (with a geometric flavour) of

discrete mathematics, combinatorics, finite groups,

geo-metry which has certain applications in coding theory,

statistical designs, network theory, and graph theory.

1.1. Hadamard matrices

Given a cube in

m

3 (with side 2, centre at the origin, sides parallel to the coordinate axes), we wish to construct a

regular tetrahe?ron by selecting 4 among the 8 vertices of the cube. The answer is provided by the points

(l, l, l)

[11

11]

(l,-1,-l) _{The matrix} 1 l -1 -1

H4

(-1,1,-l) l -1 1 -1 =:

(-1 ,-1 t 1) l -1 -1 1

1. Notes taken by J.G. Sunday .

c-is a Hadamar4 matrix of order 4.

DEFINITION. A Hadamard matrix Hr of the order r is a square matrix of order r, with elements ±1, satisfying

H u1' = ri .

r r r

Necessary conditions for the existence of Hr are

r

=

2, r

=

0 (mod 4).

It has been conjectured that these conditions are suffi-cient. This conjecture has been verified for r < 188 and for several infinite series. For the state of affairs we

refer toM. Hall {6), J.H. van Lint [18), J.S. Wallis [20].

1.2. A perfect ternary code

In the vector space V(4, 3) of dimension 4 over GF(3) con-sider the plane spanned by

f :; (1, 0, 1, 2) 1 g = (0, 1, 1, 1) o

There are 9 vectors in this plane, all but the origin having 3 non-zero coordinates. Hence each pair of the 9 vectors has 3 coordinates in which they differ (has Ham-ming distance 3). Calling words the 81 vectors in V(4, 3), code words the 9 vectors in the plane, we have found a 1-error-correcting code. This code (the plane) is linear and perfect. This means the following. Around each code word we draw a sphere of (Hamming) radius 1, containing 8 words. The 9 spheres are mutually disjoint, and exhaust V(4, 3) by the count

9 (1 + 8) = 81.

216

1.3. The binary projective 3-space PG(3, 2)

Consider the vector space V(4, 2) of dimension 4 over GF(2) provided with the non-degenerate alternating bilinear form

B(x, y)

=

~1~2 + ~2~1 + ~3~~ + ~~~3' for X = ( ~ ₁ I ~

2 I ~ l I ~ 4) I Y :; ( ~ l I ~ 2 I ~ l I ~ ~) • We make the following observations:

1. The symplectic group Sp(4, 2) acts transitively on V ( 4 , 2 ) \ { 0} •

2. The 16 x 16 matrix (B(x, y)] yields (28- J],

x,yEV

which is a Hadamard matrix of order 16 (here J is the all one matrix of order 16).

3. Th~ columns of the matrix

(B J - B]

constitute a binary (32, 16, 8)-code.

4. Consider the graph whose vertices are the 15 vee-tors of V\{0}, in which x andy are adjacent if and only if B(x, y)

=

0. This symplectic graph is strongly regular, and even rank 3, under the action of Sp(4, 2). Obviously, Sp(4, 2) acts not 2-transitively on the graph!

5. The extension of Sp(4, 2) by the translations of V(4, 2) does act 2-transitively on the vectors of V(4, 2). It maps the set of the triples {x, y,

z

}

with

B(x, y) + B(y, z) + B(x, z) 0

onto itself. This set defines a regular two-graph. 6. Since Sp(4, 2) ~ Symm(6) the graph under 4 is easily drawn.

1.4. Equiangular lines

In a set of equiangular lines the angle ~ between each pair of distinct lines is the same. We consider two examples in IR3_{,where we have chosen a unit vector p. along each}

l_

line (in either of the 2 directions), and their matrix C defined by

p = (<p., p .>),

l_ ]

1

C =cos ~(P-I].

The examples are the 4 diagonals of the cube and the 6 dia-gonals of the icosahedron:

1 3 _l 1 I I 0

,..0.-r:

+ + + + +l

fo

+

:j

0 + +

l:

0 + 0 + +

c4

_{+ 0}

~j'

c6

_l:

+ 0 + + + + 0 + + 0 We observe that (C ₄ - I) (C ~ + 31) 0,

c

2

= 6 51, $ 1 cos~ 1 cos ₌

_3'

=

-/5

With these examples, we associate the graphs

'

l.

;. ~

·:

:

•' :or~ ':~ ' .t ,J 'f -~ 0 0 1 1 3 't 0 0 5

If, in the second example, we take 6' instead of 6, then the graph is switched into

1

t,'

5

The effect on the matrix C₆ of this new choice of 6' is C'

=

DCD

where D is a diagonal matrix, whose diagonal elements are

) 1, 1, 1, 1, 1, -1. This exemplifies the relations between sets of equiangular lines, classes of graphs, and two-graphs. Indeed, the triples of lines are of two kinds: those with (ph, p.) (p., p.) (p., ph) > 0 such as (h, i, j)

l_ l_ ] J

=

(1, 4, 6), and those with (ph' p.)(p., p.)(p., ph) < 0

1. l_ ] ]

such as (h, i, j)

=

(1, 2, 3). In the second example, the automorphism group of the set of equiangular lines is the icosahedral group A

5• It contains as subgroups the

dihe-dral groups 010 and 06, the automorphism groups of the two

graphs mentioned above, which are related by switching with

respect to vertex 6.

1.5. Complex equiangular lines

we rearrange the Hadamard matrix H~ of example 1.1 into

+ +

+ _{I + C.}

+ +

+

The skew matrix C, multiplied by the complex number i , is

a hermitian matrix satisfying

(iC) 2

= 3I. j

Hence iC

+

II) is hermitian positive. semidefinite of rank

2, and may be considered as the matrix of the hermitian

products of the vectors (

11,

0) , ( i,

/2) , (

i, w/2) ,

(i, w2

12)

where w is a primitive cube root of unity. Hence we have 4 equiangular lines in the complex 2-dimensional

plane

a:

2

• In real space

m

~ these correspond to 4

equi-isoclinic planes.

2. CODES

2.1. Definitions

Using a finite set S of cardinality s as an alphabet, we

may form sn distinct words of length n; that is, each

el-ement of the set

Sn = { x = x ₁ x ₂ • • • xn I xi f S, 1 s i s n}

is considered a word. The degree of a "spelling error"

220

might be measured by the number of positions in which a

given word differs from the intended word, so a useful

de-vice is the Hamming distance d(~,

l l

between two words in Sn:

the number of positions in which x and

¥

differ.

n It is elementary to verify that d is a metric on S , and

we may occasionally refer to the closed ball

B[5:;

rl

= {~ ~: sn

I

d(5:, ~)

s rl.

For instance, if S

=

{a, b}, then the two words x

=

aabbb and

l

=

abaab in S5 _{have Hamming distance 3, and B[aabbb;}

1]

consists of the words aabbb, babbb, abbbb, aaabb, aabab, and aabba.

n

Often, we want only a subset of S to form a

vocabu-lary of meaningful words; such

a

subset is said to be a code, and its elements are called code words.

A

code is called an (M, n, d)-code if the code consists of M code

words, each of length n, such that the Hamming distance

between any two code words is at least d. In

if c is a code word, then any element of

a[fi

this case, [d;l)] is

closer to c than

ing from c in at

to any other code word, so a word differ-d-1

most

(--2-] places might reasonably be as-sumed to be

f•

up to a spelling error; we say that the

d . ld-1) .

of mutually dtsjoint closed balls covers all of sn, we say that the code is perfect.

2.2. Binary codes

For a binary code, one may easily take the alphabet

s

to be

{1

, -1}.

Abbreviating

1

and

-1

to+ and- respectively, we may group corresponding positions for two words ~·

l

in Sn as follows

~ + + + + + + + +

-

- -

-y + + + + + + + +

No. of positions PI p2 p3

(where, of course, PI + p2 + p3 + p4 n) , so that Hamming distance is given by

d(~, y)

=

p2 + p3

and the usual inner product < , > is given by

(2.A) <~,

l>

= P₁ - P

2 - P3 + P4 = n - 2·d(~,

y

)

.

We may use these notions to establish

p4

the

THE PLOTKIN BOUND. If the parameters of a binary (M, n, d)-code satisfy n < 2d, then M $ 2d/(2d- n).

PROOF. The M code words may be used to form the rows of an M x n matrix P. According to (2.A), the Hamming distance

between distinct code words is at least d precisely when

fOr all dist1'nct ~~ V L E Sn. But t e 1nner products are h ' given as entries in the symmetric M x M matrix PPt.

There-:; ;

~

•.

fore, for an (M, n, d)-code,

(2. B) PPt $

element\lrise

ni

+

(n - 2d) (J - I)

=

2di + (n - 2d)J.

If jM is a column vector of M l's, then

(j~

P) (Pt jM) is the square of the ordinary euclidean norm of

j~P,

so (2.B) implies that

0 < .t ppt.

- JM JM

~;

2dM - (2d - n)M

2

•

If n < 2d, this establishes the Plotkin bound.

Now, a code is not essentially changed if we multiply each entry in a. column of P by -1. Thus, i t may be asswned that the first row of P consists entirely of l 's, so we may write

p

where Q is an (M - 1) x n matrix of l 's and -l 's. In the extremal case M = 2d/(2d- n), we find that

QQt 2di - (2d - n)J,

.t Q .t

JM-1 = -Jn'

Qjn (n- 2d)jM-l'

so Q may be regarded as the (+, -)-incidence matrix of a block design.

There are several interesting examples and applications

of binary codes. For instance, in the extreme case in which the Plotkin bound is valid (i.e. 2d - n

=

1 and M

=

2d), the matrix [jM P) is found to be a Hadamard matrix of

order M. If we remove the restriction that led to the Plotkin bound, we may find codes with a larger vocabulary

(i.e. more code words). For instance, there is a class of binary (8t, 4t, 2t)-codes obtained from matrices

p

= [_: ]

where H is a Hadamard matrix of order 4t. A final example is the binary (16, 5, 2)-code whose code words consist of all vectors (x₁, x

2, x3, x~, x5) with an even number of

-l 's .

For further details, we refer to [4] and to (15].

The general algebraic theory of codes is exposed in (17].

2.3. Ternary Golay codes

For ternary codes, the alphabet S consists of the elements

0, l, -1 of the finite field GF(3). In this way, the words of length n may be regarded as forming the vector space

V(n, 3). We are particularly interested in situations

where the code words form a linear subspace (such a code is called a linear code).

If l and -1 are abbreviated to + and - respectively, consider the 6 x 12 matrix

0 + + + + + + 0 + + + 0

+

+ + + 0 + + + + 0 +

+

+ 0 224 + 0 0 0 0 0 0 + 0 0 0 0 + 0 0 0 0 + 0 0 0 0 0 0 0 0 0 0 + 0 0 0 0 0 0 + .I

'I

(~here C₆ is the same matrix as in §1.4). Since the column

rank is obviously 6, the rows form a basis for a subspace W of dimension 6 in V(l2, 3). Now, the Hamming distance between any pair of these basis vectors is congruent to 0

(mod 3), so any pair of distinct elements from W has Ham-ming distance 3, 6, 9, or 12. Moreover, i t can be shown that no two code words in W have Hamming distance exactly 3. It follows that the 36 words in W form a ternary (36

,

12, 6)-code, called the Extended Golay code. The automor-phism group of this code is the Mathieu group M₁₂•

The ordinary Golay code is obtained by deleting any one column of the matrix [C₆ I

6) . The shortened rows then

form a basis for a subspace U of dimension 6 in V(ll, 3). The Hamming distance between any pair of distinct words in

u

is at least 5, so the 36 _{words of length ll in U form a}

ternary (36, 11, 5)-code. Moreover, each closed ball

B(~; 2] contains exactly

l + 2·11 +

2

2

·(~

1

)

= 35 words of V(ll, 3), so the family

{B(~; 2]

I

~ E U}

of disjoint closed balls covers 36·35 words of V(ll, 3).

Since there happen to be exactly 311 _{words in V(ll, 3), we}

see that this Golay code is an example of a perfect code. A third ternary code is obtained as the complement of the Golay code. More precisely, if we take the 35 _words

code words, we obtain a ternary (35, 11, 6)-code. The code

words of this last code have Hamming distances 6 and 9 only.

REMARK. There are also binary Golay codes with words be-longing to V(n, 2): a binary (212, 24, 8)-code which is

related to the Mathieu group M₂~, and a perfect binary

(212

, 23, 7)-code which is related to M_{23 •}

3. STRONGLY REGULAR GRAPHS

3.1. Definitions

Consider a graph G; a vertex P is said to be a neighbour of the (unordered) pair of vertices P₁, P

2 if P i s adjacent

to both P₁ and P

2•

LEMMA 3.1. Let G be a graph with n vertices such that

(i) each vertex is adjacent to at most k others;

(ii) each adjacent pair of vertices has at least

A

neighbours; and

(iii) each non-adjacent pair of vertices has at least

IJ neighbours.

Then (n - k - l)IJ s k(k - 1 -

A)

.

PROOF. Choose a vertex P

0 of G. Define A(P0) to be the

set of vertices adjacent to P₀, and NA(P₀) to be the set of vertices not adjacent to P₀• By assumption, IA(P

0) 1 s k

so INA(P₀

ll

~ n - 1 - k. We now proceed to count the num-ber M of edges between A(P

0) and NA(P0) in two ways. Since

. ·~

:;j

. ::

•.'

~;.

each vertex Q in NA(P

0) determines, with P0, a non-adjacent

pair, there are at least IJ vertices adjacent to both

P

₀

and Q; thus

Similarly, each vertex R in A(P

0) is adjacent to at most

k -

l vertices other than P

0 and has at least

A

neighbours

with P

0, so there are at most k - 1 - A vertices adjacent

to R but not to P

0; hence

M s IA(P

0) j · ( k - l - A.) ~ k· (k - l - A) .

These two inequalities establish the lemma.

Examining the circumstances under which equality oc-curs in Lemma 3.1, we say that a graph is strongly regular

i f

(i) each vertex is adjacent to exactly k others;

(ii) each adjacent pair of vertices has exactly A. neigh-bours; and

(iii) each non-adjacent pair of vertices has exactly

IJ neighbours.

For instance, a pentagon is a strongly regular graph with

n

=

5, k

=

2,

A

=

0, and 1J

=

1. The Petersen graph

is a strongly regular graph with n

=

10, k

=

3, X

=

0, and

~

=

1. Moreover, if T is a finite set of cardinality t,

the

(~l

unordered pairs of distinct elements of T form the

vertices of a strongly regular graph (called a triangular 1

graph T(t)) with n =

₂

t ( t - 1), k = 2 ( t - 2), A= t - 2,

and ~ = 4; two pairs are called adjacent precisely when

they have one element in common.

3.2. Graphs and matrices

The properties of a strongly regular graph are often studied

via the corresponding adjacency matrix. In particular, we

state without proof

THEOREM 3.2. Suppose A is a symmetric matrix with entries

0 and 1 such that all diagonal entries are zero. Then A

is the adjacency matrix of a strongly regular graph with

parameters (n, k,

X

,

~) if and only if there are numbers

r and s such that

Aj = k·j I

n n (A- ri)(A- si) = ~J.

Moreover, r and s are the roots of x2 + (~ - X)x + ~ - k

=

0. In addition, k, r, and s are the eigenvalues of A,

the eigenvalue k has multiplicity 1, and, if r + s ~ -1,

the eigenvalues r and s are both integers.

The adjacency matrix As of a pentagon satisfies A~ +

J so r + s =

-1;

in this instance, both r and s

happen to be irrational. Similarly, the adjacency matrix

A₁₀ of the Petersen graph satisfies Af₀ + A₁₀ - 2I = J so

228

r + s

=

-1 again; in this case, however, r and

s

happen to

be integers.

The state of affairs for the special case in which

r + s

=

-1 while r and s are both integers is quite

inter-esting. If x and y are the multiplicities of the

eigen-values r and s respectively, we are able to derive

(2r + 1) 2 - 2(2r + 1) - l6(x- y)~(2r

+

1)

=

16~2 ₁

as a consequence of the proof of the above theorem. It

follows that 2r

+

1 16~2

- 1. One class of such graphs

consists of the Moore graphs, which are restricted to ~

=

1 and A

=

0; since 2r + 1

I

15 for this class, we find that

the parameters are limited to r -1-2 7 s

=2

-3 -8 k -3-7 56 n

---ro

50 3250

The Petersen graph yields the first possibility, and i t is at present unknown whether the third possibility is

real-ized. The second possibility is realized by the

Hoffman-Singleton graph (7], which we now describe. The 50

ver-tices of the graph are given by the 15 points and 35 lines

of PG(3, 2). To be able to define adjacency, we require

the following fact (cf. [1]): "there is a 1-1

correspond-ence between the lines of PG(3, 2) and the unordered

tri-ples from {a, b, c, d, e, f, g} in such a way that two

lines intersect or are skew according as the

correspond-ing triples have an odd or an even number of elements in

con-sidered adjacent, a point and a line are concon-sidered adjacent

only if they are incident, and two lines are considered adjacent precisely when the corresponding triples are dis-joint. It is an elementary exercise to check that this graph does have the stipulated properties.

An alternative class of these graphs, cf. [2], is produced by the restriction ~

=

2 and

A

=

1,

so that 2r

+

1

63; the parameters in this instance are limited to

r k n 1 4 9 3 14 99 4 22 243 10 112 6273 31 994 49401

At present, i t is unknown whether the second, fourth, or fifth possibilities may be realized. The ·g vertices for the first possibility may be taken to be the ordered pairs from {a, b, c}; two ordered pairs are adjacent if they agree in one position. The 243 vertices for the third pos-sibility may be taken to be the elements of V(S, 3); two vectors ~ and

¥

are called adjacent if one of the differ-ences x -

X

or ~ - x is a column of the 5 x 11 matrix of generators for the complement to the Golay code. In fact,

if

f _{1 ,} ~₂, • • • , ~

11

are the columns of this matrix, then

there are 22 vectors ±c., 220 vectors ±c. ±c., and 1 vector

-1 -1 - )

o. Since

it

was shown in §2.3 that no four columns of

this matrix are dependent, i t follows that these 243 vectors are pairwise distinct, so they exhaust V(5, 3) and may be taken as representatives of the vertices of this graph. Again, one may readily check that the graph does have the

.·.~ ·; ·--~ '.·

·'

'1 '• required properties. 3.3. Rank 3 graphs

If G is a permutation group on a set ~. we may define a relation ~ on ~ x ~ by

(a, 8) ~ (y, o) if and only if y g(a) and

o

g (8)

for some g € G.

It is an elementary exercise to show that this relation is an equivalence relation, with equivalence classes of the form

{(g(a), g(8))

I

g € G}

where a,

8

€ Q. Obviously, if 6 is an equivalence class, then so is

'il

:= {(8, a) (a, 8) € 6}.

If G is transitive on ~. the diagonal T of

n

x ~ is an equivalence class. Moreover, if G is 2-transitive, there are exactly two equivalence classes, namely T and

n

x ~\T.

The next most interesting situation, in which G is slightly

I

less than 2-transitive, may be defined as follows: "G is called a rank 3 group on ~ if G is transitive on ~. the order jGj is even, and there are exactly three equivalence classes T, 6, and

r."

Using the restriction to even order, one may then show that 6

=

6 and

r

=

r,

so each unordered pair of distinct elements of ~ belongs either to 4 or to

r,

but not both. We may use this observation to define two graphs with the elements of Q as vertices: two vertices a,

(a,

a>

t 6, anq they are adjacent in the other graph if and

only if (a,

al

t

r.

Therefore, two vertices are adjacent in one graph precisely when they are non-adjacent in the other. These two complementary graphs are called rank 3 graphs. Since G is transitive on both of the equivalence classes ~and

r,

i t is transitive on the edges and on the non-edges of each of these graphs. Furthermore, G is also transitive on the vertices, so we find that any rank 3 graph is strongly regular.

We give an example to illustrate these remarks. If T is a finite set of cardinality t (t ~ 4) and Q consists of the

(t]

l2 unordered pairs from T, i t is clear that the symmetric group ST on T is a rank 3 group on

n

,

since the three equivalence classes are

T = { (w, w)

I

w. E Q}, { (w, w')

r ,;{(w,w')

the unordered pairs w and w' have no}, element in common

the unordered pairs ~ and w' have} exactly one element 1n common · The strongly regular graph with edges taken from

r

is pre-cisely the triangular graph T(t) .

4. QUADRATIC FORMS OVER GF(2)

4.

1.

Background

There are many applications of bilinear and quadratic forms to codes and strongly regular graphs. In order to

demon-strate this, we reproduce some results about forms which may be fqund with detailed references in Dieudonne's

trea-tise La geometrie des groupes classiques.

An alternating bilinear form on V(2m, 2) is a bilinear map B: V x V ~ GF(2) such that B(~, ~) = 0 for every x E V; i t follows that B(~,

y)

B(y, ~) for all ~· y E V. If

B(~,

y)

=

0 for all y E V only when x

_2

,

the form is

called non-singular.

Since GF(2) contains only two elements 0 and 1, we may consider the entries of the square matrix

B := [B(~, y)J

~EV, ¥_EV

(of order 22m) as integers. With this interpretation, one may show that the matrix 2B - J, consisting of l 's and

{-1) 's, is a Hadamard matrix. On the other hand, we may obtain an example of a binary (22m+l, 22rn, 22m-l)-code from the columns of the 22m x 22m+l matrix [B J-B).

Using a non-singular alternating bilinear form B on V, we may define the symplectic graph: the vertices are the points of V\{9}, and two vertices~· yare called ad-jacent precisely when ~

t ¥.

and B(~, ¥_) = 0. We find that the symplectic graph is a rank 3 graph under the group

Sp(2m, 2) = {a E GL(V) B(~ a I

y )

a = B(~, ¥_)

for all ~·

¥.

E V}. Since W = {~ E V

I

B(~, ~) o} is a linear subspace of V and~ is adjacent to every element of W\{~, ~}, i t follows

that the symplectic graph is a regular graph with k

=

22m-1

-

2 edges through each vertex. If _B(~o' y ) _{_o}

=

0 and B(x I ¥1)

=

1, then

-

·

_u

_{z

0 E

v

B(~, X - 0 ) 0 B(~,

:l

0 ) }

and u { z E

v

B(~, ~ ₁₎ 0 B(~, _{¥1) }}

j

are both the intersection of two hyperplanes in V(2m, 2),

so they each contain exactly 22m-2 elements; i t follows that the symplectic graph is strongly regular, with

>..

I

U o \ { Q,

~

o ,

r:

o}

I

= 22m-2 - 3

and iJ

A quadratic form on V(2m, 2) is a map Q: V + GF(2)

such that Q(~ + yl + Q(!i + Q(¥) is a bilinear form B(~, ¥) on V; i t follows that Q(Q) = 0 and that the associated bi-linear form is alternating. If this bilinear form is also non-singular, i t is known that there is a basis ~

₁

, . . • ,~

₂

m

of V with respect to which the quadratic form Q is given

by

s1(2 + (3(~ + . . . + s2m-ls2m

or

_S

_~

_{+ ( 1 (} 2 + (

~

+ ( 3 ( 4 + · · · + t; 2m-1 t; 2m

and that the number of zeros of Q is 22m-l + 2m-l or 22m-l m-1

- 2 respectively. Let

n

represent the set of all zeros of a non-singular Q.

We may now define the orthogonal graph: the vertices are the elements of Q\{9} (alternatively, one may use the elements of V\ll), and two distinct vertices~· yare

ad-:·i jacent precisely when Q(~ + y) = 0. As before, we find

~·

that the orthogonal graph is a rank 3 graph under the or-thogonal group

{a E GL(V)

I

Q(~a) = Q(~) for all x E v}.

·· We mention that the symplectic and orthogonal graphs may

:' ~·

be characterized in a more geometric manner (cf. (10),

[13)):

SHULT'S THEOREM. For a regular graph G which is not a com~

plete graph, the following are equivalent:

(i) G is isomorphic to a symplectic or an orthogonal graph;

(ii) For each pair of adjacent vertices a, b of G,

there is a vertex c adjacent to both a and b such that

each further vertex is adjacent to an odd number of vertices of the triangle {a, b, c).

4.2. Kerdock sets

We now investigate some related concepts following Kerdock, Patterson, and Goethals. Suppose 8 is the set of all al-ternating bilinear forms on V(2m, 2), allowing for singular forms and even the zero-form. Each form B(~, y) may be regarded as a matrix product

t

~ B¥

where B is a symmetric 2m x 2m matrix with entries in GF(2),

Since there are m(2m - 1) positions in such a matrix above the main diagonal, we find there are exactly 2m(2m-l) ele-ments in B.

Consider a subset S of B such that whenever B. and

1

B. E S the form B. +B. is non-singular. If the matrices

J 1 J

representing B. and B. have the same first row, then the

1 J

matrix representing Bi + BJ. has first row 0, and B. +B.

1 J

is singular. Therefore, the first rows for the matrices of the B. in S must all be distinct, so S contains at most

}_

22m-l forms. For this maximal case, we define: a subset K 2m-1

of 8 is called a Kerdock set if

IK

I

= 2 and, whenever Bi and Bj are distinct elements of K, the form Bi + Bj is non-singular. We give the following construction of a Kerdock set. GF(2), each a + ~ where element a E Tr(a) . . 2m-1 If V(2m, 2) 1s cons1dered to be GF(2 ) $

element ~ in V may be written uniquely as ~

2m-1

a E GF(2 ) and ~ E GF(2). The trace of an

GF(22m-l) is defined by 2m-2

L

k=O and it is known that

Tr(a) € GF(2),

Tr(a +B)= Tr(a)

+

Tr(B) ,

Tr (1) 1.

For each Y E GF(22m-l) , we define an alternating bilinear

form By on V(2m, 2) by

By(a,

Bl

= Tr(a8y2) + Tr(ay)·Tr(By),

?11;

i

l : i t

j

·~ 'l ., B (a, y t;) By(i;, €;) for all a,

B

to V(2m, 2) . K

=

{B y Tr (ay) , 0,

E GF(22m-l) and t; E GF(2), extending linearly We shall show 'that

I

Y € GF(22m-1)}

is a Kerdock set. It obviously has the right number of elements, so i t only needs to be shown that By + B

0 is non-singular when y

#

6. Suppose then that BY

+

B

6 is singular. Since the kernel of By

+

B

6 must have even dimension, there 2m-1

exists a € GF(2 )\{0} which is also in the kernel; i t follows that

Tr(ay) + Tr(a6) = 0

and Tr(aByL) + Tr(a862_{) ~ Tr(ay) ·Tr(By) + Tr(a6)·Tr(86)}

for all

B

€ GF(22m-1). If Tr(ay)

=

Tr(a6) 0, then

Tr(aB(y + 6) 2)

=

_{0 for all B} € GF (22m-1) and we conclude

that y

=

6. On the other hand, i f Tr(ay)

₌

Tr(a6)

=

1,

then Tr{ 13 (y + 6)(a[y + 6) + 1)}

=

0 for all 6 € GF (22m-1)

and we again deduce that y

=

6. This establishes that K

is a Kerdock set.

4.3. Kerdock codes

Suppose that B is an alternating bilinear form on V(2m, 2)

and that Q and Q' are two quadratic forms associated with B as in §4.1; i t is then an easy exercise to show that Q

+

Q' is one of the 22m linear forms on V(2m, 2).

Th~s,

there are 22m quadratic forms associated with each

alter-237

nating bilinear form on V(2m, 2). The count of alternat-ing bilinear forms in §4.2 shows that there are exactly

2m(2_m+l)

₌

_{2~m.2m(~m-l) elements in the set Q of all}

quad-ratic forms on V(2m, 2).

We may represent each Q

~

Q as a vector in V(22m, 2)

whose

2~m

coordinates 0 and 1 are the values

Q(~)

over all

~ <:

v

12m, 2). From these binary "words" of length 2 2m , we

form a Kerdock code by fixing a Kerdock set K and taking

as code words

{Q, l + Q

I

the bilinear form B(Q) E K}.

Since

~here

are 22m-l bilinear forms in K, 22m quadratic

forms for each bilinear form, and 2 code words for each

quadratic form, we have a total of 24m code words. Since

2m-1 m-1

the smallest number of l's in a combination is 2 - 2 ,

h d k d . f d b . ( 4m 2m

t e Ker oc co e 1s oun to e a b1nary 2 , 2 , 22m-1

m-1

- 2 ~)-code.

5. EQUIANGULAR LINES IN Rd

5.1. Background

If a collection of n lines through a point 0 in Rd has the

property that each pair of distinct lines determines the

same angle ~. the collection is called a set of equiangular

lines. A unit vector p. may be chosen along each line of

1

such a collection in either of two directions; i t follows that the inner product between distinct unit vectors is

given by 2.18 .,

)

···1 <p., p.> =±cos ~ -1 J

The matrix of inner products may thus be rewritten as

l ±cos ~

p I

+

C cos ~

±cos ~ l

where

c

is a symmetric n x n matrix with 0 along the main

diagonal and ±l elsewhere. In order that the n lines span

Rd, we may assume that n > d. Since there are now more

directions than dimensions, the smallest eigenvalue of P

is 0, and its multiplicity is at least n d. Because C

is obtained from P in the above manner, we find that the

smallest eigenvalue of C is -1/cos ~, and its multiplicity

is at least n - d. Conversely, suppose that C _n is a

sym-metric n x n matrix with 0 along the main diagonal and ±l

elsewhere, that y₀ is the smallest eigenvalue of en' and

that y _o has multiplicity n - d. It follows that C _n - y I _o

is a symmetric positive-semidefinite n x n matrix of rank

d, and thus that there is an n x d matrix U such that

(5 .A)

Then n rows of U may be considered vertices in Rd, and (5.A)

shows that en- y₀I is the matrix of inner products for

these vectors. Since the non-diagonal elements of Cn - Y0I

are all ±1, we see that the n lines spanned by these vectors

are equiangular.

we look for large matrices C whose smallest eigenvalue has

n

a large multiplicity. For instance, there is a 16 x 16 symmetric Hadamard matrix H_{16 which may be written as}

H₁₆

=

_{C 16 - I where} (C_{16 -} 1)2

=

161; i t follows that (C_{16 - 51) (C 16} + 31) = 0, so the eigenvalues of C₁₆ are 5 and -3 (with multiplicities 6 and 10 respectively). Proceeding in the manner described above, we discover a set of 16 equiangular lines in R16-10(=R6 ).

5.2. Graphs and equiangular lines

The matrix C associated above with a set of equiangular lines may be interpreted as the adjacency matrix A of a graph via the transition

C = J - I - 2A,

i.e. , 1 and -1 inC correspond to 0 and 1 in A respectively. For instance, the adjacency matrix A1 o of the Petersen graph was found in § 3. 2 to satisfy 2

+ _{A1 o} 21 Aj

A1 o

-

= J, = 3j

so the corresponding

_c

_{1 0 satisfies} 2

c

1 0 91. The

multi-plicities of the eigenvalues 3 and -3 of C₁₀ are both 5. Using the procedure of §S.l, we discover a set of 10 equi-angular lines in R1o-s (=Rs).

The correspondence between graphs and sets of equian-gular lines is not 1-1, due to the fact that we originally could have chosen the unit vectors along the lines in ei-ther of two ways. Examining how a choice of -pi instead of pi is reflected in the adjacency matrix

c,

we define a

.i

"switching relation" on (±!)-adjacency matrices: C and C' are related if there is a diagonal matrix D whose diagonal elements are ±1 such that C'

=

DCD. This switching relation is easily seen to be an equivalence relation. rhe switching classes of graphs are now found to be in 1-1 correspondence with the sets of equiangular lines. For graphs with 3

ver-tices, there are two switching classes, consisting of graphs with an even number of edges or graphs with an odd number of edges (the latter are called odd 3-graphs). Graphs with

4 vertices are distributed among switching classes according as the number of odd 3-subgraphs is 0, 2, or

4

.

5.3. Two-graphs

If Q(i) denotes the collection of i-subsets of the set

n,

a graph is occasionally defined as a pair (Q, f) where f:

n<

2

) + {1, -1} is an adjacency function. Similarly, we define a two-graph to be a pair (Q, g) where g:

n<

3l +

{1, -1} is a map satisfying

g(a,

a, y)

·g(a,

a, ol

·g(a,

y,

o) ·g(f3,

y, o)

= 1

for all distinct a, (3, y, o E n. If a triple {a, (3, y} is

called coherent when g(a, (3, y) = -1, the above condition (essentially the cohomology condition og

=

1) allows that, whenever we know which triples containing a particular ver -tex a are coherent, we know the coherent triples for the whole two-graph.

Given a graph (n, f), we may define a two-graph (0, g)

by g(a, 8, y)

:=.

f(a, 8) ·f((3, y) ·f(y, a); the coherent tri -ples of the two-graph are precisely the odd 3-subgraphs of the original graph. On the other hand, if we start with a two-graph (Q, g), we may associate with i t the switching class

{ w,

f)

I

f(a, f3)·f(f3, y)·f(y, a)= g(a, (3, y)} of ordinary graphs. In this way, we discover

THEOREM 5.1. There is a 1-1 correspondence between

(i) two-graphs,

(ii) switching classes of graphs, and

(iii) sets of equiangular lines.

COROLLARY. If a graph (Q, f) belongs to the switching class of the two-graph (Q, g), then Aut(Q, f) is a subgroup of Aut(i1, g).

5.4. 2-transitive two-graphs

Shult employs this last observation to construct a two-graph with a 2-transitive automorphism group. Suppose that

(Q, f_{0 )} is a graph, X € Q, f = {vertices adjacent to X} X

and ~x = {vertices non-adjacent to x}. We suppose further-more that Aut

W,

f) is transitive on

n,

and that there exist automorphisms h₁ and h₂ of the subgraphs fx and Ax

respect-ively such that h1 (y) is adjacent to h2(z) precisely when

Y

and z are non-adjacent. Under these conditions, i t fol-lows that there is a two-graph (11 u {w}, g) with a

2-tran-?II?

sitive automorphism group. In fact, the switching class of the graph (Q u

{w},

f) where

i f either y f (y, z) =

r

f 0(y, z) w or z

=

w, otherwise

may be shown to contain an element interchanging w and x; since the original assumptions make i t clear that the sta-bilizer of w is transitive on the remaining vertices, we see that the switching class of (Q u {w}, f) has an auto -morphism group which is 2-transitive on Q u

{w}.

Theorem 5.1 now establishes the existence of the required two-graph.

0 FIGURE A

Q

FIGURE B

For example, the pentagon in figure A has a transitive auto-morphism group. If h₁ interchanges the two vertices of

f

0

=

{1, 4}

and h 2 is chosen to be the identity on ~ 0 =

{2,

3},

then we find that our conditions are fulfilled. It follows that there is a two-graph with six vertices whose automorphism group is 2-transitive; this two-graph is as-sociated with the switching class of the graph representing the diameters of the icosahedron (figure B), and its auto-morphism group is A₅• This procedure generalizes to produce

a two-graph with q + l vertices and an automorphism group PSL(2, q) which is 2-transitive.

Another example may be obtained from the symplectic graph on V(2m, 2)\{Q} introduced in §4.1. We saw that Sp(2m, 2) was transitive on the vertices of that graph. I f _~ is a vertex, then

r

{:f ~ 0

I

B(~, y) = 0} and b.

~ ~

{~ B(~, z) = 1}. I f hi is the mapping

hi:

r

~

r

xi y 1-+ X + y ~

and h₂ is chosen to be the identity on 6 , then our

condi-~

tions are fulfilled and there exists a two-graph on 22m vertices with a 2-transitive automorphism group Sp(2m, 2)· V(2m, 2). If the vertices are the elements of V(2m, 2), we find that ~· ~· ~ are coherent exactly when B(~, ~) +

B(~, ~) + B(~, ~) = 0.

As a final example, we consider the orthogonal graph on the quadric

{~ E V(2m, 2)\{Q}

I

Q£(~) = O}

where £ ±1, as introduced in §4.1. The graph has 0£(2m,

2) as a transitive automorphism group, and a construction

as above yields a two-graph with 2m-l(2m + £) vertices and a 2-transitive automorphism group Sp(2m, 2).

For further examples such as those involving Conway's group ·3, we refer to [13].

6. REGULAR TWO-GRAPHS

6.1. Definitions and properties

We proceed with notions due to G. Higman [11] I (16]. A

two-graph (Q, g) is called regular if each pair a,

B

of

-~

-vertices is contained in exactly r coherent triples. In

terms of the ideas of §5, i t may be shown that this is equivalent to insisting that any matrix C (of size n x n) in the switching class associated with (Q, g) has exactly two eigenvalues, say p₁ > _{p 2 ,} and (C- p₁I)·(C- p₂I)

=

0.

We shall set forth several consequences of this latter

de-finition. The multiplicities of the eigenvalues p

1 and

p

2 will be denoted by ~

1

and ~

2

respectively, so

~~ + _~2 = (6 .A) n 1

-

plp2, ~lpl + _~2.p2 0. It follows that (6.B) If

P

₁ + p

2 ~ 0, then p1 and p₂ are odd integers. (6.C) If p

1 + p2

=

0, then p1

=

-p2

=

;n-=-I and

c

2

₌

(n- l)I; furthermore, i t may be shown that n

=

2 (mod 4)

and that n - 1 is the sum of two squares. Examples have

k

been constructed when n - 1

=

p

=

1 (mod 4), and also for n = 226; the first unknown possibility occurs when n = 46.

Solving (6. A) for _~2.' we find

p3

-PI p2 1 ~2

=

1

-

_pl

-

_Pz and we deduce (6.D) If p 1

=

3, then n

=

10, 16, or 28. If p1

=

5, then n = 16, 26, 36, 76, 96, 126, 176, or 276. Moreover, exam-245

ples of each possibility are known, except for

P

₁ 5 and

n

= 76 or 96.

Furthermore

(6. E)

This is a consequence of the following (cf. also (21])

THEOREM 6.1. The number of equiangular lines in Rd is at most ¥<d + 1).

PROOF. Suppose p , ... , p are unit vectors along n

equi-1 n

angular lines in Rd, so <p., p.> =±a if i

~

j. Let

l. ]

Pi: Rd + Rd,

X~

<~,

pi>pi

be the projection onto the vector pi; then Pi is a symmetric linear map (1 5 i 5 n). Moreover, if i ~ j,

Tr (P. P.) l. ] k E<P.P.ek, l. J- ~k> l:<p., ek>·<p., p.>·<p., ~k> k l. - l. ] ] <p., p.>2 = a2. l. ]

Since an inner product of two symmetric linear maps Q and R is given by tr(QR), we may deduce from this that P₁, • • • ,

p are linearly independent. Since the space of symmetric n

linear maps on Rd has dimension

~(d

+ 1), i t follows that

n 5 ict<d + 1), as required.

6.2. Unitary two-graphs

Let q be an odd prime power, PG(2, q 2 ) the projective plane

246

<j

obtained from V(3, q2) , H a non-degenerate hermitian form

on V(3, q 2), and

n :

= { < ~ > £ PG < 2 , q 2 >

1

u

< ~ , ~ > =

o }

the quadric associated with H. It is known that

1111

= q3

+ 1 and that PfU(3, q 2 ) acts 2-transitively on

n.

Suppose 6 is the set of distinct(!), (~) , (!) inn such that

{ a square in GF(q2) if q : -1 (mod 4), H(!, x>·H(~, !)•H(!, ~)'"' . a non-square l.n GF(q2) if q - 1 (mod 4).

Then (1!, 6) is a regular two-graph, with n = q3 + 1, ~ = q2 - q

+

1,

1

(C- q2I)·(C

+

qi) = 0

~2 = g(q2- q + 1),

(it is a slightly complicated process to establish these results about the parameters). Theorem 5.1 now implies that there is a set of q3 _{+ 1 equiangular lines in Rq}2_-q+l

with cos ~

=

1/q. Thus, the maximum number of equiangular lines in Rd is at least d/d.

6.3. Equiangular lines

We now tabulate the maximum number n(d) of equiangular lines in Rd as follows: d n(d) reference 2 3 3-4 6 5 10 5.2 6 16 5.1 7-14 28 5.4, 6.1 15 36 5.4 16 40 17-18 48 ?117

d n(d) reference 19 72 20 90 21 126

u

(3, 52) 22 176 Higman-Sims 23-42 276 constructed below 43 344 U(3, 72) 6.4. The 276-two-graph

A non-trivial regular two-graph on 276 vertices may be con-structed as follows [5]. From (6.A), (6.B), and (6.C) , we deduce that the only possible eigenvalues p₁,

P

_{2 of Care}

55 and -5. Now the complementary Golay code consists of the 243 vectors, having Hamming distances 6 and 9, in a 5-dimensional subspace of V(l l, 3). Let

PR.: V(ll, 3) -+ GF(3) I

~

be the projection onto the ith coordinate (1 ~ i ~ 11); there are 33 pairs (PRi' x) where PRi is one of these 11 projections and x E GF(3). Let the 276 vertices of a graph consist of these 33 pairs and the 243 elements of the com-plementary Golay code. Adjacency is defined by:

(PR. I X) is adjacent to (PR. I y) <=> X

=

y ~ J (PR. I X) is adjacent to

!::

= (y 1 I • • • I y 1 I) ~ <=> X

=

PR. (y) 1 - yi ~ (x 1, • • • , x11) is adjacent t o y = (y1 , • • • , y11 )

<=> their Hamming distance is 9.

we find that the adjacency matrix

c

for this graph satisfies (C + 5I)·(C- 55!)= 0

so the two-graph associated with the switching class of

'lAO

\

this graph is regular. This two-graph is the only non-trivial regular two-graph on 276 vertices, since the above process may be reversed. Another way to construct 276 equi

-angular lines in R23 is to consider in the Leech lattice in R24 the 276 rhombi having a fixed diagonal at the second

smallest distance occurring in the latti~e.

REFERENCES

1. F.C. Bussemaker and J.J. Seidel. Symmetric Hadamard matrices of order 36. T.H. Report 70-WSK-02, Tech. Univ. Eindhoven, Netherlands (1970).

2. E.R. Berlekamp, J.H. van Lint, and J.J. Seidel. A strongly regular graph derived from the perfect ternary Golay code. A Survey of Combinatorial Theory 1973

(J.N. Shrivastava, editor), 25-30. North Holland Pub. Co.

3. P.J. Cameron and J.J. Seidel. Quadratic forms over GF(2). Proc. Kon. Ned. Akad. Wet. Ser. A, 76 (1973):

1-B.

4. J.M. Goethals. Some combinatorial aspects of coding theory. A Survey of Combinatorial Theory 1973 (J.N. Shrivastava, editor), 189-208. North Holland Pub. Co. 5. J.M. Goethals and J.J. Seidel. The regular two-graph

on 276 vertices. Discrete Math. 12 (1975): 143-158. 6. M. Hall. Combinatorial Theory. Waltham, Mass. (1967) . 7. A.J. Hoffman and R.R. Singleton. On Moore graphs with

diameters 2 and 3. IBM J. Research Develop. 4 (1960): 497-504. B. 9. 10.

11.

12. 13.

P.W.H. Lemmens and J.J. Seidel. Equiangular lines. J. Algebra 24 (1973): 494-512.

J.J. Seidel. Strongly regular graphs. Progress in Combinatorics 1969 (W.T. Tutte, editor), 185-197. Academic Press.

On two-graphs and Shult's characterization of symplectic and orthogonal geometries over GF(2). T. H. Report 73-WSK-02, Tech. Univ. Eindhoven, Nether-lands (1973).

A survey of two-graphs. Proc. Int. Call. Teorie Combinatorie, Ace. Naz. Lincei, Rome (to appear).

Graphs and two-graphs. Fifth Southeastern Conf. on Combinatorics, Graph Theory

&

Computing. Utilitas Math. Publ. Inc., Winnipeg (1974): 125-143.

E.E. Shult. Characterizations of certain classes of graphs. J. Comb. Theory (B), 13 (1972): 1-26.

14. The graph-extension theorem. Proc. Amer. Math. Soc. 33 (!972): 278-284.

15. N.J. Sloane and J.J. Seidel. A new family of

non-linear codes obtained from conference matrices. Int.

Conf. on Combinatorial Math. 175 (1970): 363-365.

16. D.E. 'l'aylor. Regular two-graphs. Proc. London Math.

Soc. (to appear).

17. J.H. van Lint. Coding theory. Springer Lecture

Notes 201.

18. Combinatorial theory seminar. Springer Lecture

Notes 382.

19. J.H. van Lint and J.J. Seidel. Equilateral point sets

in elliptic geometry. Proc. Kon. Ned. Akad. Wet. Ser.

A, 69 (1966): 335-348.

20. J.S. Wallis. Hadamard matrices. Springer Lecture

Notes 292.

21. Ph. Delsarte, J.M. Goethals, J.J. Seidel. Bounds for

systems of lines, and Jacobi polynomials. Philips