THE GEOMETRY OF TWO-WEIGHT CODES
R. CALDERBANK
AND
W. M. KANTOR
ABSTRACT
We
survey the relationships between two-weight linear
[n,
k]
codes over
GF(q),
projective
(n,
k,
/»„
/»,)
sets
in
PG(£—
\,q), and certain strongly regular
graphs.
We also describe and tabulate essentially all the
known
examples.
1.
Introduction
This paper surveys relationships between subsets of finite projective spaces,
strongly regular graphs, and linear codes. Each subject is interesting in itself and has
attracted the attention of finite geometers, combinatorists or coding theorists. What
is
remarkable
is
that results from one area can immediately be translated into the other
two.
We have sought to explain the relationships involved in this translation and to
describe and tabulate all the known
examples.
Our goal
is
to stimulate further research
by making specialists in each area aware of a wider variety of techniques.
Delsarte [14,15,16,17] was the first to investigate the relationships between the
projective sets, graphs, and linear codes that are considered here. The relationships
are a special case of
his
more general theory of association schemes arising in coding
theory. Much of this survey merely describes his results, though we provide different
proofs. However, new results have since appeared and additional examples have been
noticed, many of them geometric.
Section 2 contains basic definitions arranged by subject. The definitions are
intended to make subsequent sections more accessible to specialists in
finite
projective
geometry, combinatorics, or coding theory. The reader need not be familiar with all
three subjects in order to understand this survey.
In Section 3 we prove the equivalence of two-weight codes, projective
(«,
k,
h
1}
h
2
)
sets and certain strongly regular graphs. Section 4 describes a theorem of Goethals
and van Tilborg that characterizes a two-weight code C in terms of
the
dual code C\
In Section 5 we describe the dual of a projective
(n,
k, h
u
h
2
)
set, and the projective
dual of a two-weight code. Section 6 shows how to construct new two-weight codes
from a given two-weight code by changing the underlying field.
In the second half of this survey we describe essentially all the known examples
of two-weight codes. The most visible examples of two-weight
[n,
k]
codes arise from
subspaces of GF(q)
k
, and these are described in Section 7. There are examples that
occur when the dimension
A;
is
3
or
4
and that do not generalize to higher dimensions.
These are discussed in Section 8. In Section 9 we describe the cyclotomic examples.
These are two-weight
[n,
k]
codes constructed from subgroups of
GF(q
k
)*.
Section 10
contains examples arising from groups of collineations of
PG(k—
\,q) with exactly
two point orbits. Examples that do not fit into any of
the
above sections are collected
in Section 11.
Received
30 April 1982; revised 27 September 1985.
1980
Mathematics Subject Classification
05B25,
51E20,
94B05.
Bull. London
Math.
Soc. 18 (1986) 97-122
BLM 18
98
R.
CALDERBANK AND
W. M.
KANTOR
Section 12 contains theorems characterizing projective {n,k,h
x
,h
2
) sets subject
to
some additional geometric constraint. Theorem 12.9 is
a
new result. We conclude this
survey
by
tabulating the parameters of essentially all
the
known two-weight codes
and
the corresponding strongly regular graphs.
2.
Definitions
2A:
Two-weight
codes.
Let
q
=
p
m
,
wherep
is
prime.
If u
=
(u
t
),
v
=
(y
c
) are vectors
in GF(q)
n
then
the dot
product u
•
v
is
given
by
u
•
v
=
LjLi«»iv
An
[n,
A;]
code
C
over
GF(^)
is a
A:-dimensional subspace of
GF(q)
n
.
Vectors
in C
are called
codewords.
The
dual code
C
1
=
{veGF(q)
n
|
v
C
=
0};
it is an
[n,n-k] code.
The
weight
wt(x)
of a
vector
x
in
GF(^)"
is the
number of non-zero entries.
The
weight enumerator
W
c
(x,
y)
of
C is the
polynomial
y
i
y
i-o
where
A
i
is
the
number
of
codewords
of
weight
/.
The
Mac Williams Identities
[43,
Chapter
5]
relate
the
weight enumerator
of
C to
that
of
C
1
as
follows:
W
c
^x,y)
= |^| W
c
{xHq~
\)y>
x-y). (2.1)
A two-weight code
is
a
code
C for
which
|
{/1
/
^
0 and
A
t
^
0}
|
=
2.
The distance d(x,y) between two vectors
x
and y in GF(#)
n
is the number of entries
where
x
and
y
differ. Thus d(x,y)
=
wt(x—y)
and the
minimum distance between
two
codewords
is the
minimum weight among
all
non-zero codewords.
A
code
C is
said
to
be an
[n,
k,
d]
code
if d
is
the
minimum non-zero weight
in C. If
e
=
[\(d—
1)]
then
C
is
also said
to be
e-error-correcting.
If
C is an
[n,
k]
code over
GF(^)
then there exist linear functionals
f
t
:GF{qf
>GF(q),
i=
\,...,n
such that
C
=
{(f
1
(x),...,f
n
(x))\x€GF(qn (2.2)
Let B(x,y)
be any
non-singular bilinear form
on
GF(^)*. Then there exist
y
lt
...,y
n
in GF(#)* such that/^x)
=
B(x,y
t
)
for i
=
1,
...,n.
If
B(x,y)
=
x-y
is the dot
product
then
C
=
{(x>
yi
,...,x-y
n
)\xeGF(qn (2.3)
Since dim(C)
= k
the vectors
j>
l5
...,y
n
span GF(?)
fc
. If no two of the vectors^, ...,y
n
are dependent then
the
code
C
is
said
to
be projective. Thus C
is
projective
if
and only
if the minimum weight
in the
dual code
C
1
is
at
least
3.
An
n x
n monomial matrix
M
is
a
matrix
of
the form
M =
DP,
where
D is an n
x
n
diagonal matrix
and P
is
an n x
n permutation matrix.
Two
[n,
k]
codes
C and
C
over
GF(^f)
are
said
to be
equivalent
if
there exists annxn monomial matrix
M
such that
MC
=
C".
Note that monomial transformations
are
precisely those linear transfor-
mations that preserve
the
metric
d
on
GF(^)
n
.
2B:
Projective
(n,
k, h
lt
h
2
)
sets.
A
projective
(n,
k,
h
lt
h
2
) set
O is
a proper, non-empty
set
of n
points
of
the
projective space PG(fc—
\,q)
with
the
property that every
THE GEOMETRY OF TWO-WEIGHT CODES
99
hyperplane meets
0
in h
x
points or in
h
2
points. The complement of 0 in PG(Ar—
1,
q)
isaprojective
Let
0 =
{<>
;
i)h
=
!>•••>«}
and O'
= {{y
/
i
}\i=
l,...,«}
be
two projective
(n,k,
h
v
h
2
) sets. Then
0 and 0' are
said
to be
equivalent
if
there exists AeGL(n,q)
such that A(O)
= 0'. If 0
and
0'
span
PG(k- \,q)
then
0
and
0'
determine [«,fc]
codes C and
C,
respectively, via (2.3). We remark that
0
and
0'
are equivalent
as
projective sets
if
and only
if
C and
C
are equivalent as codes. (Of course this is true
not
only
for
two-weight codes
but for any
projective codes.)
It is
possible
to
generalize
the definitions
of
equivalence
of
codes
and of
projective sets
by
allowing auto-
morphisms of
GF(<7).
This corresponds to the observation that C and
C
are
essentially
the same
for
each ae Aut GF(q).
2C:
Strongly
regular
graphs.
A connected graph on N vertices
is
said to be
strongly
regular
with parameters
(N, K,
X, ji)
if
it is regular with valency
K
and
if
the number
of vertices joined
to
two given vertices is
X
or
pi
according as the two given vertices
are adjacent
or
non-adjacent; we shall always exclude the null and complete graphs.
We label the vertices v
x
,...,v
N
, and we define
an Nx N
integral matrix
A =
(a
ti
)
by
setting
1,
if
v
t
and v
}
are adjacent
0, otherwise.
The matrix
A is
the
adjacency
matrix
of
the graph.
If /
is the
N
x
N
identity matrix
and
if J
is the N x
N
matrix with every entry
1
then
AJ=JA
=
KJ and A
2
-(X-fi)A-(K-/i)I
= fiJ.
(2.4)
Multiplying the second equation by
/
and comparing coefficients gives
K(K- X-1) = (N-K-l)/i. (2.5)
The eigenvalues of A are K,
p
lt
and p
2
where
and
The multiplicities
of
£,
p
x
and
p
2
are
1,
e
x
, and e
2
,
respectively, where
=
2V
^ Vd / * ^
Equation (2.7) is called the
integrality
or
rationality condition
because e
1
and e
2
must
be integers.
2D:
Rank
3
groups.
Let G be
a
permutation group acting transitively on
a
set
X.
The group G
is a
rank 3
group
if
it has exactly three orbits onJfxI Suppose
| G
\
is even and let
R
be an orbit ofGonXxXother than {(x,x)\xeX}. If (x,y)eR then
(y,
x) e
R.
The orbit R determines
a
graph on the elements of X in which two elements
x,y are
joined if and only if (x,y)eR. The group G acts transitively on ordered pairs
of adjacent vertices and on ordered pairs of non-adjacent vertices.
It
follows that the
graph is strongly regular.
2E: {A
1?
X
2
}
difference
sets.
Let
Q
be
a
proper set
of
non-zero vectors
in a
vector
4-2
100
R.
CALDERBANK
AND W. M.
KANTOR
space
V
over GF(q). Then
Q is a
{X
lt
X
2
}
difference
set
over
GF(q) if
GF(q)*Q
= Q
and
if, for
v
e V, v # 0, we
have
ix
x
,
\{(x,y)\x,yeQ
and x-y = v}\=\
\X
2
,
3.
Fundamental correspondences
In this section
we
shall describe
the
equivalence
of
two-weight codes, projective
(n,
k,
h
x
, h
2
) sets,
{A
l5
A
2
}
difference sets
and
certain strongly regular graphs.
THEOREM
3.1.
(1) If the code C defined by (2.3)
is
a projective two-weight
[n, k]
code
then
{<y
t
}
\
i =
1,..., n)
is a
projective («,
k,
n
—
w
lt
n
—
w
2
)
set
that spans PG(fc
—
1,
q).
(2) Conversely if{(yi)\i
=
1, •-.,«} is a projective (n,
k,
n
—
w
x
,
n
—
w
2
)
set
that spans
PG(k
—
\,q)
then
the
code
C
defined
by
(2.3)
is
a projective two-weight
[n, k]
code with
weights
w
1
and w
2
.
Proof. Let x be any
non-zero vector
in
GF(q)
k
.
If x
1
=
{^eGF(^)
fc
|
x
-y
= 0}
then
n —
\x
1
O{y
1
,...,y
n
}\
is the
weight
of
the codeword (x-y
1
,...,x-y
n
).
Note that
if O =
{{y
i
)\i=
1,...,«}
does
not
span PG(fc—
1,q)
then
the
points
of
O
are in a
PG(r, q) subgeometry
of
PG(A;—
\,q).
Let
V=
GF(q)
k
.
Let Q c V and
suppose that
Q = -Q and 0£Q. We
define
a
graph
G(Q)
with vertices
the
vectors
of V and
where
two
vertices
are
joined
if and
only
if
their difference
is in
12.
THEOREM 3.2.
Let O =
{<(^>
\i= 1,...,«}
be a proper non-empty
set
of points
of
VG(k-\,q),
and
letQ
= {veV\
<y>eO}.
If O
spans
PG(k- \,q)
then the following
are equivalent:
(1)
Q is a
{A
l5
X
2
}
difference
set for
some
X
lt
X
2
,
(2)
G(Q) is a
strongly regular graph,
(3)
O is
a projective (n,k,n
—
w
lf
n
—
w
2
) set for some
w
l5
w
2
.
Proof. It
follows directly from
the
definitions that
(1) and (2) are
equivalent.
Delsarte essentially proves that (2)
and
(3)
are
equivalent
in
[15].
We
outline
the
proof
because
it
contains information needed later.
Let
N - q
k
. We
order
the
vectors
v
x
,
...,v
N
of V and use
this ordering
to
define
the adjacency matrix
A = (a^) of
G(Q).
Let
/:GF(^)
+
->C*
be any
non-principal
character
of the
additive group GF(<y)
+
.
If ue V
then
the map x
u
(v) = x(uv) for all
ye
V is a
character
of
the abelian group
V. We
define
a
vector
e
v
eC
N
by
setting
(e
v
)i
=
Xv(Vi) fori=\,...,N.
(3.3)
LEMMA
3.4. The
vector
e
v
is an
eigenvector
of A
with eigenvalue
(q
— 1) (n
—
w
v
)
—
w
v
, where
{q
—
1) (n
—
w
v
)
=
|
v
1
ft Q
|. The vectors
e
v
, veV,
are
a
basis
ofC
N
.
Proof. We
have
(e
v
A)
j
=
I
1
Xv(v
i
)a
ij
= I
XviPi)
=
(
2
XviM))
=
(e
v
),(
I
Xv(u)),
ueQ
ueQ
THE GEOMETRY
OF
TWO-WEIGHT CODES
101
where
I
Xv(u)=
I
XvM+
2 ( I ,
ueil ueCl
ueCl aeGF(g)
uv-0 uv-l
= (q-\)(n-w
v
)-w
v
.
Finally
N
N
V*w = I /(w«)/(wwi) = I
i-l i-\
{N,
ifv
+ w
=
0,
[0, otherwise.
This completes
the
proof
of
the lemma.
If G(Q)
is
strongly regular then
A
has just
3
eigenvalues
and
Lemma
3.4
implies
that
(3)
holds. Conversely,
if
(3)
holds then
A
has
3
distinct eigenvalues
and
the
multiplicity
of the
valency
n(q
—
1)
of
G(Q)
is 1. The
adjacency matrix
A
satisfies
an
equation
of the
form
A
2
=
al+bA
+
cJ for
scalars
a, b, and c and
this readily yields
the strong regularity asserted
in
Theorem
3.2.
COROLLARY 3.5.
Let O =
{<^>
|
/,...,»} be
as
in
Theorem
3.2, let C
be the code
defined
by
(2.3),
and
suppose
that
conditions
(1), (2),
and (3)
of
Theorem
3.2
hold.
Then
the
eigenvalues
of
A
are
n(q—\),
n(q-
l)-^w
l5
and
n(q—
\)
—
qw
2
,
with
multiplicities
1,
A
m
,
and A
W2
respectively,
where
A
Wt
is the
number
of
codewords
of C
with weight
w
t
. Ifw
2
>w
1
then
,
A
">
=
7^—^\ {^{q
k
-V-nq
k
-\q-1)). (3.6)
\W
2
VVjJ
Proof
The
only part that
is not
immediate
is
(3.6),
but we
shall postpone
the
proof
of
(3.6) until (5.6).
COROLLARY
3.7.
If
conditions
(1), (2), and (3)
of
Theorem
3.2
hold then
the
parameters (N, K,
X,
fi)
of
G(Q) are
given
by
N=q\
K
=
n(q-\),
X
=
K
2
+ 3K— q{w
l
+ w
2
) — Kq{w^ + vv
2
) + q
2
w
1
w
2
,
and
2
H
= 1^1 =
K^
+
K-Kqiw.
+
w^
+
q^
w
2
.
Proof
By
Corollary
3.5
(y4
_
(
^_l)_^
M
;
i
)/)(^_(^_l)_^
M
,
2
)/)
= c
y (3.8)
for some constant
c.
Premultiplying both sides
by
J
we obtain
c
=
q*w
x
w
2
/q
k
.
Now
compute
X
and
fj.
by
comparing
(3.8)
with (2.4).
COROLLARY 3.9. IfG(Q)
is strongly regular
with
parameters
(N,
K,
X,
fi)
as in
(3.7)
then
H,
Wn
K*
+
K-Kq(w
1
+
w
2
)
+
(q
k
-l)-±zi
=
0,
(3.10)
and
q(w
2
-
Wl
)
=
((X-
M
y+4(k-
M
))i (3.11)