Finite geometries

Codes and Combinatorial Structures

3.4 Finite geometries

0₁ and O₂ are orbitals implies that the number of common neighbours of two adjacent vertices, or two non-adjacent vertices, is constant; and the transitivity ofG shows that

r

is regular. So

r

is a rank-3 strongly regular graph.

Sporadic simple groups are often related with strongly regular graphs. For example, there is a strongly regular graph with parameters (162,105,81) and the MacLaughlin group of order 898, 128, 000 is a subgroup of index 2 of the automorphism group of this graph. Similarly there is a strongly regular graph with parameters (416,100,96) and the Suzuki group of order 448, 345, 497, 600 is a subgroup of index 2 of the automorphism group of this graph. For a survey on strongly regular graphs and rank-3 groups the reader is encouraged to consult [14, 15, 10].

The code formed by the span of the adjacency matrix of a graph

r

is also the code of the l-(v, k, k) design obtained by taking the rows of the adjacency matrix as the incidence vectors of the blocks; the automorphism group of this design will contain the automorphism group of the graph. Thus a relation is established between graphs, designs, codes and groups. The interplay between these structures will be more clear in Chapter 4, where these relations undergo a considerable development and are explicitly defined. In Chapters 7, 8 and 9 we construct codes from the adjacency matrices of the graphs defined therein and an interplay between codes and graphs is established.

3.4.1 Projective geometries

Let V be a vector space over the field IF. The projective geometry (space) defined by V is denoted by PG(V). If the vector space V has dimension m over IF, then the projective geometry PG(V) has projective dimensionm-I;

PG(V) is also denoted by P(V)

=

PGm-l(lF). The elements of PGm-l(lF) are non-trivial subspaces ofV, and the structure of the set is given by set-theoretical containment. The projective dimension of an element U in PGm -1(IF) is denoted by pdim(U) and is defined to be one less than the dimension of U as a vector space over IF. Thus the points of PG(V) are the I-dimensional subspaces of V, the lines are the 2-dimensional subspaces of V, and the hyperplanes are the (m - I)-dimensional subspaces ofV.

If IF

=

lFq , a point of the projective geometry PGm-l(lFq ) is given in homogeneous coordinates by the non-zero vector (Xl,"" Xm ) E lF_q^m. Each point then hasq-lsuch coordinates representatives since

(XI, ... ,

x_{m )} and

A(XI, ... ,

x_{m )} yield the same I-dimensional subspace of lFqm for any non-zero vector AE lF_{q .}

A hyperplane H of the projective geometry _PGm-1(lF_{q ),} in homogeneous coordinates is determined by by the non-zero vector

(YI, ... ,Ymf

which spans Hl... A point

((XI, ... ,

x_{m ))} is on the hyperplane H if and only if

(XI, ...

,x_{m ) .}

(YI, ... ,Ymf =

O.

Grassman's identity for subspaces of V holds for subspaces of a projective space PG(V). Thus if U and Ware arbitrary elements of PG(V), then

pdim(U)

+

pdim(W) - pdim(U

n

W) = pdim(U

+

W). (3.4) IfH is a hyperplane ofPG(V) and U is an element ofPG(V) with pdim(U)

= t,

then from the identity 3.4 we get that pdim(H

n

U) = t or t - 1, and the former occurs if and only if

U

S;;;

H.

The number of subspaces ofV of dimension k, where 0

<

k :::; m, is given by

(qm _ l)(qm _ q) (qm _ qk-l)

Nm,k(q)

=

(qk _ l)(qk _ q) (qk _ qk-l) . (3.5) In particular the number of points and the number of hyperplanes of

PGm-1(IF

q ) is q;_~l.

Similarly, if

U

is an r-dimensional subspace of an m-dimensional vector space V and k is an integer with 0

:S

r

<

k

:S

m, then the number of subspaces ofV of dimension k that contain U is given by

(qm _ qr)(qm _ qr+l) (qm _ qk-l)

(qk _ qr)(qk _ qr+l) (qk _ qk-l) . (3.6) In particular, if k

=

m - 1, this gives the number of hyperplanes that contain

U

""m-r-l i

as L..-i=O q.

Given two projective spaces, an isomorphism is a bijective map that preserves incidence structure. An isomorphism between two projective spaces is called collineation, and an isomorphism from a projective space to itself is called an automorphism or collineation. The full automorphism group of

PG(V)

is given by the well-known fundamental theorem of projective geometry which follows:

Theorem 3.4.1 (Fundamental Theorem of Projective Geometry) The full automorphism group of

PG

_{m- 1}(IFq ) is prLm(q) for any q

:2:

2 and m

:2:

3.

Proof: See [8] . •

The elements of prLm(q) preserve the subspaces of V

=

IF^qm, and thus they form a permutation group on the points of

PGm-1(V).

We may construct within this group an automorphism a of order q~~l that permutes the points of the geometry in a single cycle of this length, called a Singer cycle, see [9, Theorem 6.2]. The group generated by a Singer cycle is called a Singer group.

3.4.2 Affine geometries

Definition 3.4.2 An affine geometry (space) of an m-dimensional space V over a filed IF consists of all cosets x

+

^U where U is a subspace ofV andx is an element ofV, and is denoted byAG(V) or AGm(lF).

The dimension of the geometry is the same as the dimension of the vector space and affine dimension of an element x

+

U in AG(V) is the dimension ofU as a subspace of V The points of this space are all the vectors ofV, the lines are the cosets of the I-dimensional subspaces ofV, and the hyperplanes are the cosets of the (m-I)-dimensional subspaces ofV. Ifa subspaceU has dimensionr, then any coset of U is called an r-fiat ofAG(V). Two r-fiats x

+

U and y

+

W in AG(V) are parallel if U

=

^W. The number of r-fiats in AGm(F_{q ),} where 0 ~ r _~ m, is

N _ qm-r(qm - l)(qm-l - 1) (qm-r+l - 1)

m,r(q) - (qr _ l)(qr-l - 1) (q - 1) (3.7) Theorem 3.4.3 (Fundamental Theorem of Affine Geometry) The

full automorphism group of AGm(lF_{q ),} where m

2':

2 and q

2':

2, is the affine semilinear group AfL_m(q).

Proof: See [9]

•

The affine semilinear group AfL_{m (}q) preserves the cosets ofV and thus acts as a permutation group on all the points ofAGm(lF_{q ).}

For both projective and affine geometries, the full automorphism groups are doubly transitive on the points. This will be of use later in the construction of designs from the geometries PGm(lF_{q )} and AGm(lF_q) where m

2':

2.