Finite geometry and coding theory

60 Pages·0.43 MB·English

by  Peter J. Cameron

Finite geometry and
oding theory
Peter J. Cameron
S
hool of Mathemati
al S
ien
es
Queen Mary and West(cid:12)eld College
London E1 4NS
So
rates Intensive Programme
Finite Geometies and Their Automorphisms
Potenza, Italy
June 1999
Abstra
t
In these notes I willdis
usssome re
ent developments at the inter-
fa
e between (cid:12)nite geometry and
oding theory. These developments,
are all based on the theory of quadrati
forms over GF(2), and I have
in
luded an introdu
tion to this material. The parti
ular topi
s are
bent fun
tions and di(cid:11)eren
e sets, multiply-resolved designs,
odes
over Z4, and quantum error-
orre
ting
odes.
Some of the material here was dis
ussed in the Combinatori
s
Study Group at Queen Mary and West(cid:12)eld College, London, over
the past year. This a

ount is quite brief; a more detailed version will
be published elsewhere. Many of the parti
ipants of the study group

ontributed to this presentation; to them I express my gratitude, but
espe
ially to Harriet Pollatsek and Keldon Drudge.
1 Codes
There are many good a

ounts of
oding theory, so this se
tion will be brief.
See Ma
Williams and Sloane [18℄ for more details.
Let A be an alphabet of q symbols. A word of length n over A is simply
n
an n-tuple of elements of A (an element of A ). A
ode of length n over A
1
n
is a set of words (a subset of A ). A
ode of length n
ontaining M words is
referred to as an (n;M)
ode.
The Hamming distan
e d(v;w) between two words v and w is the number
of
oordinates in whi
h v and w di(cid:11)er:
d(v;w) = jfi : 1 (cid:20) i (cid:20) n;vi 6= wigj:
n
It satis(cid:12)es the standard axioms for a metri
on A . The minimum distan
e
of a
ode is the smallest distan
e between two distin
t words in C. A
ode
of length n having M
odewords and minimum distan
e d is referred to as
an (n;M;d)
ode.
The basi
idea of
oding theory is that, in a
ommuni
ation system,
messages are transmitted in the form of words over a (cid:12)xed alphabet (in
pra
ti
e, usually the binary alphabet f0;1g). During transmission, some
errors willo

ur, that is, some entries in the word willbe
hanged by random
noise. The number of errors o

urring is the Hamming distan
e between the
transmitted and re
eived words.
Suppose that we
an be reasonably
on(cid:12)dent that no more than e errors
o

ur during transmission. Then we use a
ode whose minimum distan
e d
satis(cid:12)es d (cid:21) 2e+1. Now we transmit only words from the
ode C. Suppose
0
that u is transmitted and v re
eived. By assumption, d(u;v) (cid:20) e. If u is
0
another
odeword, then d(u;u) (cid:21) d (cid:21) 2e + 1. By the triangle inequality,
0
d(u;v) (cid:21) e+1. Thus, we
an re
ognise the transmitted word u, as the
ode-
word nearest to the re
eived word. For this reason, a
ode whose minimum
distan
e d satis(cid:12)es d (cid:21) 2e+1 is
alled an e-error-
orre
ting
ode.
Thus, good error
orre
tion means largeminimumdistan
e. On the other
hand, fast transmission rate means many
odewords. In
reasing one of these
parameters tends to de
rease the other. This tension is at the basis of
oding
theory.
Usually, it is the
ase that the alphabet has the stru
ture of a (cid:12)nite
(cid:12)eld GF(q). In this
ase, the set of words is the n-dimensional ve
tor spa
e
n n
GF(q) , and we often require that the
ode C is a ve
tor subspa
e of GF(q) .
Su
h a
ode is
alled a linear
ode. A linear
ode of length n and dimension
k
k over GF(q) is referred to as a [n;k℄
ode; it has q
odewords. If its
minimum distan
e is d, it is referred to as an [n;k;d℄
ode. Almost always,
we will
onsider only linear
odes.
The weight wt(v) of a word v is the number of non-zero
oordinates of
v. The minimum weight of a linear
ode is the smallest weight of a nonzero
2
odeword. It is easy to see that
d(v;w) = wt(v (cid:0)w);
and hen
e that the minimum distan
e and minimum weight of a linear
ode
are equal.
0 0
Two linear
odes C and C are said to be equivalent if C is obtained from
C by a
ombination of the two operations:
(a) multiply the
oordinates by non-zero s
alars (not ne
essarily all equal);
and
(b) permute the
oordinates.
Equivalent
odes have the same length, dimension, minimum weight, and
so on. Note that, over GF(2), operation (a) is trivial, and we only need
operation (b).
A linear [n;k℄
ode C
an be spe
i(cid:12)ed in either of two ways:
(cid:15) A generator matrix G is a k(cid:2)n matrix whose row spa
e is C. Thus,
k
C = fxG : x 2 GF(q) g;
and every
odeword has a unique representation in the form xG. This
isuseful for en
oding: if the messages to be transmitted are allk-tuples
overthe(cid:12)eldGF(q),then we
anen
odethemessagexasthe
odeword
xG.
(cid:15) A parity
he
k matrix H is a (n(cid:0)k)(cid:2)n matrix whose null spa
e is C:
more pre
isely,
n >
C = fv 2 GF(q) : vH = 0g:
This is useful for de
oding, spe
i(cid:12)
ally for syndrome de
oding. The
n >
syndrome of w 2 GF(q) is the (n(cid:0)k)-tuple wH . Now, if C
orre
ts
e errors, and w has Hamming distan
e at most e from a
odeword v, it

an be shown that the syndrome of w uniquely determines w(cid:0)v, and
hen
e v.
If h1;:::;hn are the
olumns of the parity
he
k matrix H of a
ode C,
then a word x = (x1;:::;xn) belongstoC if andonlyifx1h1+(cid:1)(cid:1)(cid:1)+xnhn = 0,
that is, the entries in x are the
oeÆ
ients in a linear dependen
e relation
between the
olumns of H. Thus, we have:
3
Proposition 1.1 A
ode C has minimum weight d or greater if and only if
any d(cid:0)1
olumns of its parity
he
k matrix are linearly independent.
n
There is a natural inner produ
t de(cid:12)ned on GF(q) , namely the dot prod-
u
t
n
X
v (cid:1)w = viwi:
i=1
If C is an [n;k℄
ode, we de(cid:12)ne the dual
ode
? n
C = fv 2 GF(q) : (8w 2 C) v (cid:1)w = 0g;
?
it is an [n;n (cid:0) k℄
ode. Then a generator matrix for C is a parity
he
k
matrix for C, and vi
e versa.
Sometimes, in the
ase when q is a square, so that the (cid:12)eld GF(q) admits
p
(cid:27) q
an automorphism (cid:27) of order 2 given by x = x , we will use instead the
Hermitian inner produ
t
n
X
(cid:27)
v Æw = viwi :
i=1
We now give a family of examples.
A 1-error-
orre
ting
ode should have minimum weight at least 3. By
Proposition 1.1, this is equivalent to requiring that no two
olumns of its
parity
he
k matrix are linearly dependent. Thus, the
olumns should all be
k
non-zero, and should span distin
t 1-dimensional subspa
es of V = GF(q) ,
where k is the
odimension of the
ode. Multiplying
olumns by non-zero
s
alars, or permuting them, gives rise to an equivalent
ode. So linear 1-
error-
orre
ting
odes
orrespond in a natural way to sets of points in the
proje
tive spa
e PG(k(cid:0)1;q).
Many interesting
odes
an be obtained by
hoosing suitable subsets
(ovoids, unitals, et
.). But the simplest, and optimal,
hoi
e is to take all
the points of the proje
tive spa
e. The
ode thus obtained is the Hamming
k

ode H(k;q) of length n = (q (cid:0)1)=(q(cid:0)1) and dimension n(cid:0)k over GF(q).
To reiterate: the parity
he
k matrix of the Hamming
ode is the k(cid:2)n ma-
k
trix whose
olumns span the n one-dimensional subspa
es of GF(q) . It is a
[n;n(cid:0)k;3℄
ode.
One further
ode we will need is the famous extended binary Golay
ode.
This is a [24;12;8℄
ode over GF(2), and is the unique
ode (up to equiv-
alen
e) with this property. The o
tads, or sets of eight
oordinates whi
h
4
support words of weight 8 in the
ode) are the blo
ks of the Steiner system
S(5;8;24) (or, in other terminology, the 5-(24;8;1) design).
We dis
uss brie(cid:13)y some operations on
odes. Let C be a linear
ode.
(cid:15) Pun
turing C in a
oordinate i is the pro
ess of deleting the ith
oor-
dinate from ea
h
odeword.
(cid:15) Shortening C in a
oordinate i is the pro
ess of sele
ting those
ode-
words in C whi
h have entry 0 in the ith
oordinate, and then deleting
this
oordinate from all
odewords.
(cid:15) Extending C, by an overall parity
he
k, is the pro
ess
onsisting of
adding a new
oordinate to ea
h
odeword, the entry in this
oordinate
being minus the sum of the existing entries (so that the sum of all

oordinates in the extended
ode is zero). We denote the extension of
C by C.
(cid:15) The dire
t sum of
odes C1 and C2 is the set of all words obtained by

on
atenating a word of C1 with a word of C2.
The weight enumerator of a linear
ode is an algebrai
gadget to keep
tra
k of the weights of
odewords. If C has length n, its weight enumerator
is
n
X
n(cid:0)i i
WC(x;y) = aix y ;
i=0
where Ai is the number of words of weight i in C. Note that WC(1;0) = 1,
and WC(1;1) = jCj.
The weight enumerator has many important properties. For example,
the weight enumerator of the dire
t sum of C1 and C2 is the produ
t of the
weight enumerators of C1 and C2. For our purposes, the most important
result is Ma
Williams' Theorem:
?
Theorem 1.2 Let C be a linear
ode over GF(q), and C its dual. Then
1
WC?(x;y) = WC(x+(q (cid:0)1)y;x(cid:0)y):
jCj
We illustratethis theoremby
al
ulatingthe weight enumerators of Ham-
ming
odes. It is easier to (cid:12)nd the weight enumerators of their duals:
5
Proposition 1.3 Let C be the q-ary Hamming
ode H(k;q) of length n =
k ?
(q (cid:0)1)=(q (cid:0)1) and dimension n(cid:0)k. Then every non-zero word of C has
k(cid:0)1
weight q .
?
We say that C is a
onstant-weight
ode.
Proof Let h1;:::;hn be the
olumns of the parity
he
k matrix H of C.
?
We
laim that ea
h word of C has the form (f(h1);:::;f(hn)), where f
belongs to the dual spa
e of the k-dimensional spa
e V of
olumn ve
tors
(cid:3) ?
of length k; and every element of V gives rise to a unique word of C .
? ?
This holds be
ause H is a generator matrix of C , so the words of C are
linear
ombinations of the rows of H. Now the ith row of H has the form
(ei(h1);:::;ei(hn)), where ei is the ith dual basis ve
tor. So the
laim is
proved.
(cid:3)
Now for any non-zero f 2 V , the kernel of f has dimension k(cid:0)1, and so
k(cid:0)1

ontains (q (cid:0)1)=(q(cid:0)1) one-dimensional subspa
es, and so it vanishes at
?
this many of the
olumns of H. So the
orresponding word of C has weight
k k(cid:0)1 k(cid:0)1
(q (cid:0)1)=(q (cid:0)1)(cid:0)(q (cid:0)1)=(q (cid:0)1) = q :
?
It follows that the weight enumerator of C is
(qk(cid:0)1)=(q(cid:0)1) k (qk(cid:0)1(cid:0)1)=(q(cid:0)1) qk(cid:0)1
x +(q (cid:0)1)x y ;
and so the weight enumerator of C is
(cid:16) (cid:17)
1 (qk(cid:0)1)=(q(cid:0)1) k (qk(cid:0)1(cid:0)1)=(q(cid:0)1) qk(cid:0)1
(x+(q (cid:0)1)y) +(q (cid:0)1)(x+(q (cid:0)1)y) (x(cid:0)y) :
k
q
Finally on this topi
, we mention that the weight enumerator of the ex-
tended binary Golay
ode is
24 16 8 12 12 8 16 24
x +759x y +2576x y +759x y +y :
? ?
A
ode C is self-orthogonal if C (cid:18) C , and is self-dual if C = C . The
extended binary Golay
ode just mentioned is self-dual; other examples are
the extended binary Hamming
ode of length 8 (a [8;4;4℄
ode with weight
8 4 4 8
enumerator x +14x y +y ), and the binary repetition
ode of length 2 (a
2 2
[2;1;2℄
ode with weight enumerator x +y ).
6
Using Ma
Williams' Theorem, we see that the weight enumerator of a
self-dual
ode C of length n over GF(q) satis(cid:12)es
1
WC(x;y) = n=2WC(x+(q (cid:0)1)y;x(cid:0)y): (1)
q
This gives a system of equations for the
oeÆ
ients of WC, but of
ourse not
enough equations to determine it uniquely.
Gleason [13℄ found a simple des
ription of allsolutions of these equations,
using
lassi
alinvarianttheory. Wedes
ribehiste
hnique forself-dualbinary

odes.
Let G be a (cid:12)nite group of 2(cid:2)2 matri
es over C. Let f(x;y) be a poly-
nomial of degree n. We say that f is an invariant of G if
(cid:18) (cid:19)
a b
f(ax+by;
x+dy) = f(x;y) for all 2 G:

d
Sin
e the sum and produ
t of invariants is invariant, the set of G-invariants
is a subalgebra of the algebra C[x;y℄ of all polynomials in x and y over C.
G
We denote this subalgebra by C[x;y℄ .
If f(x;y) is a G-invariant, then its homogeneous
omponent of degree k
i j
(the sum of all terms aijx y with i + j = k) is also G-invariant. So the
G
algebra C[x;y℄ is graded, a

ording to the following de(cid:12)nition:
L
Let A = k(cid:21)0Ak be an algebra over C. We say that A is graded if
Ai (cid:1) Aj (cid:18) Ai+j for all i;j (cid:21) 0. If dim(Ak) is (cid:12)nite for all k (cid:21) 0, then the
Hilbert series of A is the formal power series
X
k
dim(Ak)t :
k(cid:21)0
G
Molien'sTheoremgivesanexpli
itformulafortheHilbertseriesofC[x;y℄
for any (cid:12)nite group G:
Theorem 1.4 Let G be a (cid:12)nite group of 2 (cid:2) 2 matri
es over C. Then the
G
Hilbert series of C[x;y℄ is given by
X
1
(cid:0)1
(det(I (cid:0)tA)) :
jGj
A2G
7
Now let C be a self-dual binary
ode. Sin
e all words in C have even
weight, the weight enumerator of C satis(cid:12)es
WC(x;(cid:0)y) = WC(x;y):
Also, sin
e WC(x;y) is homogeneous of degree n, we
an rewrite Equation 1
as (cid:18) (cid:19)
x+y x(cid:0)y
WC p ; p = WC(x;y):
2 2
These two equations assert that the polynomial WC is an invariant of the
group G = hA1;A2i, where
(cid:18) (cid:19) (cid:18) p p (cid:19)
1 0 1= 2 1= 2
A1 = ; A2 = p p :
0 (cid:0)1 1= 2 (cid:0)1= 2
Now it is easily
he
ked that
2 2 8
A1 = A2 = (A1A2) = I;
so G is a dihedral group of order 16. Now Molien's Theorem, and some
G

al
ulation, shows that the Hilbert series of C[x;y℄ is
1
:
2 8
(1(cid:0)t )(1(cid:0)t )
>From this we see that the dimension of the nth homogeneous
omponent is
equal to the number of ways of writing n as a sum of 2s and 8s.
We know some examples of self-dual
odes: among them, the repetition

ode of length 2 and the extended Hamming
ode of length 8, with weight
enumerators respe
tively
2 2
r(x;y) = x +y ;
8 4 4 8
h(x;y) = x +14x y +y :
i j
Moreover, any polynomialof the form r h is a weight enumerator (of the
dire
t sum of i
opies of the repetition
ode and j
opies of the extended
Hamming
ode). It
an be shown that, for (cid:12)xed n, these polynomials (with
2i+8j = n) are linearly independent; thus they span the nth homogeneous

omponent of the algebra of invariants of G. This proves Gleason's Theorem:
8
Theorem 1.5 A self-dual binary
ode has even length n = 2m, and its
weight enumerator has the form
bn=8
X
2 2 (n(cid:0)8j)=2 8 4 4 8 j
aj(x +y ) (x +14x y +y )
j=0
for some aj 2 Q, j 2 f0;:::;bn=8
g.
The te
hnique has other appli
ations too. We give one of these. A self-
orthogonal binary
ode has the property that all its weights are even. Su
h
a
ode is
alled doubly even if all its weights are divisible by 4.
If C is a doubly even self-dual
ode, then the weight enumerator of C is
(cid:3) (cid:3)
invariant under the group G = hA1;A2i, where A2 is as before and
(cid:18) (cid:19)
(cid:3) 1 0
A1 = :
0 i
(cid:3)
It
an be shown that G is a group of order 192, and the Hilbert series of its
algebra of invariants is
1
:
8 24
(1(cid:0)t )(1(cid:0)t )
Now there exist doubly even self-dual
odes whi
h have lengths 8 and 24,
namely the extended Hamming
ode and the extended Golay
ode. The
weightenumeratoroftheextendedHamming
odeisgivenabove. Theweight
enumerator of the extended Golay
ode is
24 16 8 12 12 8 16 24
g(x;y) = x +759x y +2576x y +759x y +y :
Again, these two polynomialsareindependent, and we have Gleason's se
ond
theorem:
Theorem 1.6 A doubly even self-dual
ode has length n divisible by 8, say
n = 8m, and its weight enumerator has the form
bn=24
X
8 4 4 8 (n(cid:0)24j)=8
aj(x +14x y +y ) (cid:2)
j=0
24 16 8 12 12 8 16 24 j
(cid:2)(x +759x y +2576x y +759x y +y )
for some aj 2 Q, j 2 f0;:::;bn=24
g.
Several further results of the same sort are given in Sloane's survey [26℄.
9
The (cid:12)naltopi
inthis se
tion
on
erns the
overing radiusof a
ode. This
is a parameter whi
h is in a sense dual to the pa
king radius, the maximum
number of errors whi
h
an be
orre
ted.
Let C be a
ode of length n over an alphabet A. The
overing radius of
C is
maxmind(v;
):
v2An
2C
That is, it is the largest value of the distan
e from an arbitrary word to the
nearest
odeword. Said otherwise, it is the smallest integer r su
h that the
n
spheres of radius r with
entres at the
odewords
over the whole of A .
We saw that, if the number of errors is at most the pa
king radius, then
nearest-neighbour de
oding
orre
tly identi(cid:12)es the transmitted
odeword.
The
overing radius has a similar interpretation: if the number of errors
is greater than the
overing radius, then nearest-neighbour de
oding will

ertainly give the wrong
odeword.
We give one result on the
overing radius of binary
odes whi
h will be
used in Se
tion 5. We say that a
ode C has strength s if, given any s

oordinate positions, all possible s-tuples over the alphabet o

ur the same
number of times in these positions. The maximum strength is the largest
integer s for whi
h the
ode has strength s.
Theorem 1.7 Let C be a
ode of length n over an alphabet A of size q and
n
v an arbitrary word in A .
(a) If C has strength 1, then the average distan
e of v from the words of C
is n(q (cid:0)1)=q.
(b) If C has strength 2, then the varian
e of the distan
es of v from the
2
words of C is n(q (cid:0)1)=q .
Proof (a) For 1 (cid:20) i (cid:20) n, let di(
) = 0 if v and
agree in the ith
oordinate,
1 otherwise. Then
n
X
d(v;
) = di(
):
i=1
So the average distan
e from v to C is
n
XX
1
di(
):
jCj
i=1
2C
10