Download Dowling, T.A.; (1971)Codes, Packings and the Critical Problem."

*
The researeh in this report was partiaUy supported by the Air Foree
Office of Scientific Research Contract AFOSR-68-1415.
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COM BIN A TOR I A L
MATHEMATICS
YEA R
2
9
7
8
February 1969 - June 1970
CoDES, PACKmGS AND THE CRITICAL PROBLEM*
by
T. A. Om'll i ng
Department of Statistics
University of North Carolina at C'aapeZ HiU
Institute of Statistics Mimeo Series No. 600.34
February, 1971
Conss, PACKINGS AND
TP:!:: CRITICAL PROBLEM'"
T. A. Dowling
University or No"pth CaroUna at Chapel, Hi t:L
1.
INTROIJLCTI~
A fundamental problem of finite projective geometry is to determine the
maximum cardinality of a set of points such that no three points of the set are
collinear. or more generally. the maximum number of points in a set. no
which lie in a subspace of dimension
t-2.
t
of
A set satisfying this latter prop-
erty we call t-independent. and a maximal su.ch set a t-packing.
The problem
has received considerable attention in recent years. not only from geometers.
but from applied mathematicians and engineers as well. for "large" t-independent
sets have important applications in coding theory. factorial designs in statistics. information retrieval systems, and a number of other areas.
Our purpose will be to establish the connection between the packing problem
and -a general combinatorial problem. called the critical. problem [11], which 1ncludes. among others. the classical problems of determining the chromatic number
of a graph or the minimum flow in a network.
The critical problem for a set of points in a projective geometry is to detennine the minimum number of hyperplanes which "distinguish" the points of the
set.
Its solution depends only on the lattice of closed sets in the subgeometry.
The characteristic polynomial of the lattice; a generalization of the concept of
the chromatic polynomial of a graph. plays a fundamental role.
The required
minimal number of hyperplanes. called the aritical exponent of the set, is the
smallest power of the order of the field which is not a root of the polynomial.
The research in this report
141M
partiaHy supported by the Air Force
Office of Scienti-fia Research ContY"act AFOSR-68-1415.
2
For the coding problem, the critical exponent of a certain subgeometry of
projective geometry represents the minimal redundancy of a linear code with prescribed minimum distance.
Enumerative formulae in terms of evaluations of the
characteristic polynomial are given for the number of t-independent sets and
t-independent spanning sets of any given cardinality.
2.
CoM3INATORIAL GEO~1ETRY
We summarize in this section some basic definitions and results of combi-
natorial geometry.
A detailed treatment is given in [11].
p~geomotry
A (finite)
a
~
AUb,
A,
~
a
is independent if
imal subset
~
b
B of
then
b
B-b
£
Aua.
for all
A such that
imal independent subset of
any set
consists of a finite set
A" A satisfying the exchange property:
closure operator
and
G(X)
A.
b
A k-flat
X.
A is atosed if
B.
A basis of a set
€
B· A;
A basis of G(X)
is a closed set
A of rank
If
G(X)
G(S)
of a pregeometry
G(X)
X with closure operator A ~
are those subsets of
G(S)
is the restriction of the rank function of
contraotion. G(X) /S
with closure operator
A S X-S
of
S independent in
An
G(S)
subsets
AS X
A set
B
A S X is a min-
X.
A is a maxAll bases of
The rank of G(X)
A.
k.
In particular, a
is of rank
r,
an
A set 1s dependent if not
A circuit is a minimal dependent set.
A restriction
S of
X,
E:
A = A.
is a basis of
(r-l)-flat is a copoint, an (r-2)-flat a coUne, etc.
subset
a,b
equivalently, a basis of
one-flat is a point, a two-flat a line, etc.
independent.
If
A set
A have the same cardinality, called the rank of
is the rank of
X together with a
G(X}
-
S.
a pregeometry defined on a
The independent sets in
G(X),
G(X)
and the rank function of
to subsets of
is a pregeometry defined on a subset
A + AuS - S.
such that
is
The independent sets of G(X)/S
A u B is independent in
G(X)
for some
S.
X-S
A
of
X
are those
3
(equivalently, for every) basis
try defined on a subset of
tractions.
B of
A minor of
X-S.
is any pregeome-
G(X)
X obtained by a sequence of restrictions and con-
Any minor may be represented in the form G(B)/A
. that. is, as a contraction of a restriction of
G(X).
equivalently the restriction of the contraction
for
The minor
G(X)!A
A S B S X,
G(B)/A
to the set
B-A S X-A.
A combinatoI'iaZ geometry (briefly, a geometry) is a pregeometry
for which the empty set and all elements of
X are closed.
pregeometry is a geometry iff every two-element subset of
Every pregeometry
G(X)
has an associated geometry
X-~
equivalence classes of
A restriction of a geometry
is
G(X)
Equivalently, a
X is independent.
G(X )
O
whose points are the
modulo the equivalence relation
a· b.
iff
a - b
is a geometry, called a subgeometry of
G(X)
G(X).
A contraction of a geometry is not necessarily a geometry.
A (finite) geometJic
then
y
covers
x
(that is
point (element covering
The set
L(X)
'Latt1~ae
0,
x
<
is a finite lattice
z
~
or atom)
y
implies
p
~
G(X),
"lattice of G{X).
tice of the associated geometry.
x
A
o
to
x,
geometry
x
E
L(X)
x v p • y.
It is isomorphic to the lat-
The
L is
rank rex) of
is well-defined as the length of all maximal chains from
G(X).
B
of
G(X),
and subset
L(X),
isomorphic to the lattice of the minor
G(S)
p • 0,
Conversely, every geometric lattice
respectively in the lattice
If
L,
and is equal to the rank of the corresponding closed set in the (pre-)
For any flat
A, B
E:
ordered by inclusion, is
the lattice of a geometry defined on the set of its points.
an element
x,y
z· y) iff there exists a
L such that
of closed sets of a pregeometry
a geometric lattice, called the
L such that if
G(S)
Ac
B,
then the interval
which have a basis in the set
S.
L(5)
denote the flats
[x.y]
of L(X)
G(B)/A.
is a restriction of a (pre-) geometry
is isomorphic to the lattice
if x, y
G(X) ,
the lattice of
consisting of those flats of
The function
p: L(X) .. L(5)
G(X)
which sends
is
4
each flat
x
to the largest flat contained in it with a basis in
the retl'ac:t from
Th~
L(X)
to
of a (locally finite) partially ordered set
the integer-valued function defined on
v(x,x) • 1,
lJ(x,y)· -lx~z<y lJ(x.z)
[20], if f,
g
then
L,
f(x)·
by
pxp
if
0 if x
~(x.y)·
is
P
Y.
~
By the MObius inversion fo~la
x < y.
are real-valued functions on
y~x
is called
L(S)."
lJ
!>fobius fu.nc::ticn
S
P such that
g(x). Ly~x
fey),
v(x,y)g(y).
The Whitnay numbers of a geometric lattice
r
==
WL(k)
are of two kinds,
L
6 (r(x) ,k),
(Second kind)
x€L
that is, the number of elements of rank
L
•
wL(k)
and
k,
v(O,x) 6(r(x),k)
(First kind) •
xli:L
The latter are the coefficients of the Ch:l1'acteristie polynomial. of
sp(A)
"(0
l.
lJ
r(l)-r(x)
•
be the n-dimensional vector space over
V is a pregeometry, with closure operator A ~ sp(A), where
n
is the subspace of V spanned by the set A c V • The closed sets are
n
- n
independent over
V
,
n
GF(q).
associated geometry of
and the independent sets are those sets
The rank of a set
V
n
n-1,
lattice of subspaces of
vn , denoted
of projective subspaces of
m· r(y)-r(x).
The
]I?
typically denoted
An
n
~litney
L
n
n
[x,y]
numbers of L
n
•
n
PG(n-l,q»
== L(V ),
interval
WL (k)
n
B
linearly
A 1s the dimension of sp(A).
is the projective geometry :lP
(- projective dimension
m
V
Then
thus the subspaces of
L,
,x)
xeL
vn = Vn (q)
To illustrate, let
GF(q).
~
L,
are
== lP (q)
n
over
The
of rank
GF(q).
n
The
is isomorphic to the lattice
of
L
n
is isomorphic to
5
the Gaussian coefficients [14], defined as
[n]
k
k-l
IT
•
n-i
g
i-O q
k-i
-1
,
-1
and
w (k)
L
n
•
The characteristic polynomial of
L
k
n
(_l)k q(2) {k]
•
n
is
n-1
p (v)
n
Restrictions to subsets
geometries.
•
S ~ V
n
IT
i-O
i
(v-q ).
represent more typical examples of pre-
Although we shall be concerned only with pregeometries (geometries)
of this type, it should be emphasized that not every pregeometry (geometry) is
representable as a restriction of a vector space (projective geometry).
3. THE PAO<ING PROBLEM
Let P
A set T
r
be the projective geometry of rank
of points of P
r
is t-independent if
subset of T is independent, where
2 s t
S
r.
r
IT I
The
over a finite field
~
t
and every t-element
t-packing pl'ob Zern for
is to determine the maximum cardinality of a t-independent set in 1P r.
this number by
N(r,t),
GF(q).
1P
r
We denote
and call a t-independent set of cardinality .N(r,t)
a
t-packing of P r.
Of course, the analogous problem could be posed for any combinatorial geometry
K ,
p
G.
For example, if
G is the bond geometry [20] of the complete graph
the t-packing problem amounts to determining the maximum number of edges in
a graph with' p
vertices containing no circuits of length
t
or less.
classical paper of extremal graph theory, Turan [31] proved that
[p2/4]
In a
is the
6
maximum number when
t
= 3, and that the extremal graph is unique. For t
~
4,
this problem is apparently still unsolved.
Various cases of the packing problem for P
have been investigated by a
r
number of Italian geometers, including Segra [21-27], Barlotti [1-4] and TalUni
[29-30], and by Bose [5-9], who noted its connection with coding theory and experimental design, and numerous others.
In the terminology [4J of projective geometers, a t-independent set (which
is not (t+l)-independent) of cardinality
t-r, a k-cap if r
t · 3.
~4,
k is a k-aet of kind t-l, a
k-~c
if
An ova7- is a k-arc of maximum cardinality in a
projective plane (not necessarily desarguesian); a k-cap of maximum cardinality
is an ovaloid.
The geometrical structure of ovals and ovaloids in lower dimen-
An irreducible conic in a desarguesian plane
sions 1s fairly well understood.
P3
is a (q+l)-arc [23].
conic [21].
[18].
An
For
q is odd, it is an oval, and every oval is a
If
q even a conic may be extended to an oval, with
elliptic quadric is a q2+l - cap in JP 4
[5, 28, 18].
For odd q,
q+2
[17], an ovaloid if
every ovaloid is an elliptic quadric [1].
points
q > 2
A com-
prehensive survey is given in [4].
A
t-~ndependent
set in
~r
has an obvious matrix-theoretic interpretation.
Given a (homogeneous) coordinatization of lP
let a matri:& of S be any
rxn
coordinate vector of a point
matrices of
subset of
S.
t
matrix over
alSo
and any set
r
GF(q)
S
each of whose columns is a
Clearly there are
n
different
(q-l) nl
The matrix of a t-independent set has the property that every
or fewer columns is independent over GF(q).
latter property we call
A matrix with the
t-independent as well.
A few elementary observaticns on the behavior of N(r,t)
Obviously,
of n points,
N(r,2)
a
(qr_l)/(q_l),
may be made.
since every two point subset
set of a projective geometry (or of any geometry) is independent.
of the point
Also, given
7
a t-packing of a copoint
H of lP ,
any single point not in H
r
to yield a t-independent set in lP r'
be added
Since a copoint is isomorphic to' lP r-l'
N(r-l,t) + 1.
~
N(r,t}
can
A t-independent set is clearly (t-l}-independent, so
If
T
is at-packing
(t
~
3)
N(r,t-1).
~
N(r,t)
in lP
and
r
P r la, with associated geometry P r- l'
If q. 2
(3.2)
and
T-a is a (t-1)-independent set.
Hence
s N(r-l,t-l) + 1.
N(r,t}
C3.l)
then in the contraction
afiT,
t
N(r,2s+1)
is odd, this becomes an equality:
•
NCr-l,2s),
2,
q •
which is easily proved by adding a column of zeros, then a row of ones, to a
2s-ind~pendent
matrix, thereby reversing the inequality (3.1).
We list below all known values [4] of N(r,t)
for which at least one of
r, t, q is arbitrary.
N(r ,2)
N(3,3}
•
•
r
.Cq -1) I (q-l) ,
rq+l,
q odd
tQ+2.
q even
2
N{4,3}
•
q +1,
N(r.3)
•
2
N(r, r)
•
r+1,
r-l
,
Various upper and lower bounds for N(r,t)
[3. 9, 26. 27. 29}.
Bounds for general
r. t
q > 2
q • 2
q • 2.
are known for special cases
are most easily expressed in
8
terms of a related function
R(n.t)
defined in Section 4, and will be given
there.
Note that iteration of (3.1) yields
r-t+2 1
q q-l - +
s
N(r, t)
(3.3)
A t-independent set in P r
of]P
G(T)
L(T)
(- cardinality)
k-flats of P
the set
oS
r
t-1.
A~P,
Thus
up to rank
B(T)
This property permits an enumeration of the number of
meeting T in a given j-element subset, by M'6bius inversion on
t-1
or fewer elements, for all
as follows.
let
be the number of k-flats of lP
gk(A)
the number of k-flats of lP r
k
•
can be equivalently defined as a subgeometry
P of subsets of T with
For
2
is isomorphic to the Boolean algebra
j S k S t-l,
f (A)
-
such that every (t-l)-element subset of points is closed.
r
the lattice
t
r
containing A,
intersecting TinA.
Then for
and
k S t-l,
&tt(A)
which yields, by the Mobius inversion formula,
• B2A
l
(3.4)
If
na11ty
r-i
- [k-l]'
D
~
For a set
containing
~e
then the closure of
A of cardinality j,
A is
n-j
(l-j)'
B
in 1P
is an l-flat, so
r
the number of
and since the interval
B£P of cardi-
[A,B]
~(A,B) - (_l)l-j. Thus (3.4) becomes, 1f IAI -
Boolean algebra,
(3.5)
-K.
is of cardinality i,
B~P
~(B)
~(A,B) ~ (B).
of
P
is a
j,
•
right-hand side of (3.4) 1s independent of
emphasize its dependence on n.
A.
We denote it by
+jk(n),
to
9
THEOREM
let
1.
Let T be a t-independent set in P
0 S j S k S t-l.
of cardinality
r
Then the number of k-flats intersecting
T
n,
and
in any given
j-element subset is
CoROLLARY.
T is
The number of k-flats not meeting
. r
b=o
k
'Ok(n)
Note that
~Ok(n)
is a polynomial in
n
of degree
Some obvious nec-
k.
essary conditions for the existence of a t-independent set of cardinality
n
are
(3.6)
'Ok (m-l)
for all
mS n
~
~Ok(m)
and k S t-l.
~
0
The upper bound for
observation appears to be larger than (3.3).
(4l
0k
: OSkSt-l)
N(r.t)
based solely on this
Nevertheless, the sequence
provides some information which may be useful.
Geometrical argu-
ments establishing the non-existence of order ideals in P
with .Ok(n) flats
r
of rank k would imply the non-existence of a t-independent set of cardinality
n.
To take a simple example. in P3(3).
n • 5,
and
.00(5). 1,
~Ol(5)
D
8.
t-3,
with
~02(5).
3.
(3.8) is satisfied for
Since three concurrent lines
contain 10 points, and three non-concurrent lines 9 points, a 5-arc in
~3(3)
is impossible.
4. THE CoDING PROBLEM
Let F· GF(q).
T
a • (a ,a •••• ,an ) ,
l 2
Then
w i s a norm on
In the vector space
n
F
(.... V )
-
n
let
w(a),
denote the number of non-zero coordinates
Fn,
called Hamming
~eight,
at
of
inducing the metric
a.
d
10
(Hamming distance) on Fn defined by
indices
a i '" b i • An
n
a k-dimensional subspace C of F
a
~
b.
i
The
such that
l~dundanay
code of distance
of an
d(a,b)· w(a-b),
(n,k)-tinear oodkJ of distanoe t+l
d(a,b) ~ t+l
such that
(n,k)-cod~
is its codimension r
is optimal if k
t+l
that is, the number of
for all
a
n-k.
is maximal for given n, t;
valently, if its redundancy is minimal.
We denoee'by
(16] is
RCn,t)
a,b
An
E
C,
(n,k)-
equi-
the redundancy
(Jading prob7..em is to determine R{n, t).
Suppose C is any subspace of Fn such that d{a,b) ~ t+l for all a ~
of an optimal code.
in
the
C.
Then for
a
The
~
0,
w(a)· d(a,O)
t+l.
Thus
~
wee)
~ t}.
C contains no elements of
t-ba 7.7.
Sn,t •
Conversely, if
for
~
a
~
b
in
{e:
1
C is a subspace such that
C,
for
d(a,b). w(a-b)
C n Sn, t • $,
and
atb (C
then
imply
d(a,b)
~
a-b
~
C,
t+l
so
a-b ~ Sn, t' Thus an (n,k)-linear code of distance t+l is equivalently a kn
dimensional subspace of F containing no elements of the t-ball S t'
n,
Let C be an (n,n-r)-code of distance t+l. If M is an rxn matrix
with
Ker H • C,
then
~
C n 8n ,t •
implies
Me
~
0
for all e
E
8 n, t-
By
the definition of . Sn, t' it follows that H is t-independent. Conversely,
given an r - n t-independen~ matrix H, its kernel, ,of dimension at
least n-I,
can contain no elements of 5n, t'
code of distance t+l iff there exists in ~
Thus there exists an
(n,n-r)
n-point t-independent set.
an
r
This result, first noted by Bose [5] in connection with an analogous problem, accounts for the importance of the packing problem in coding theory _ A
regarded as functions of n, r
consequence is that each of
R(n,t), N(r,t),
respectively, for fixed
determines the other completely by the relation
t,
R(n,t)
S
r
<~
N(r,t)
~
b
n.
11
Thus
~
(4.l)
R(n,t)·
min{r:
N(r,t)
(4.2)
N(r,t)
max{n:
R(n,t) ... r},
a
n},
the equality in brackets above being justified by the fact that
R(n,t)
is a
unit-increasing function of n,
R(n+l,t)
•
R(n,t) + c,
~
£
{O,U.
From (3.1) and (4.1) we obtain
(4.3)
R(n,t)
~
R(n-l,t-l) + 1,
which becomes an equality if
t
R(n,2s+1)
A
is odd and q. 2,
•
R(n,2s) + 1,
number of lower bounds for
by (3.2) and (4.1):
(q • 2).
R(n, t) are known.
The Rao-Hamming bound
[16, 19] is
R(n,2s)
~
{tog q B(n,s)},
R(n,2s+1)
~
{log (B(n,s)+(n-l)(q_l)s+l)},
s
q
where
B(n,m)
m
Q
l
n
(i) (q-1)
i
•
1...0
This, of course, gives implicitly an upper bound for
The Varshimov-Gilbert upper bound [16] for
(4.4)
R(n, t)
s
N(r,c).
R(n,t}
is of a similar form:
{log q (l+B(n-l,t-l»}.
The upper bounds given below for
t · 25
are derived from codes constructed by
12
Bose and Ray-Chaudhuri [7, 8], and Hoequenghem [15]:
u~ing
are obtained from these,
(4.5)
R(n,2s)
R(n,2s+1)
(4.6)
R{n,2s+1)
sa
2).
(2s+2){log q (n+1)} - 1.
S
These provide lower bounds for
5.
(q
2s{log q (n+l)}
s
R(n,2s)
(q .. 2)
s{!o92n} + 1
S
t · 26+1
(3.2) and (3.1):
s{lo92(n+l)}
S
The bounds for
N(r,t).
WNNECTION ~'lllli THE CRITI CAL PROBLEM
A sequence
(L l ,L 2 , ••• ,L r )
of linear functionals on
tinguish a (spanning) set S £ Vn-{O}
such that
L (a> .; O.
i
a~S
if for every
The critical. problem for the set
Crapo and Rota [11], is to determine the minimum length
of linear functionals distinguishing
The integer
S.
is said to dis-
V
n
there exists an
L
i
S,
as formulated by
C"
c{S)
C·
c(S)
of a sequence
is called the
critical. exponent of S.
It is proved in [11] that the critical exponent of S depends only on the
lattice
L{S)
consi.sting of those subspaces of
V
with a basis in
n
S.
In
fact, more can be said:
THEOREM (Cmpo-RotaJ.
(L ,L 2 , ••• ,L r )
l
where
p(v)
CoRoLLARY.
of length
The number of sequences of linear functionals
r
on
Vn which distinguish the set
r
S is . p(q ),
is the characteristic polynomial of L{S).
The critical exponent
c
for the set
S
p(qr) .. 0,
r .. O,l, ••• ,c-l
p(qr) > 0,
r
~
c.
is determined by
13
The critical probieul embraces a number of well-known combin3torial problems.
A classical example is the problem of coloring the vertices of a graph
no two adjacent vertices receive the sarne color.
80
that
The bond geometry of the graph,
defined on the set of edges, has a representation in a vector space over any
field.
If
is a representation of the geometry in
5
graph is colorable in
A sequence
5
).l.
iff no vector of
r
Thus if
\J: V ... F •
n
then any sequence
r
This implies
~
qr
(L ,L , ••• ,I. )
l 2
r
of linear functionals clearly dis tinguishes
is in the kernel of the linear transformation
C
is a subspace of dimension
such that
(L l ,L2 ,···,L r )
\.I ..
n-k,
with equality iff
imum dimension of a subspace
critical exponent of
then the
n
n
the critical exponent of S.
5
vn * and Ker
the dual space
r ~ c(S),
colors iff
v .. V (q),
lJ ...
C.
L , L ,
l
2
k
such that
Ker lJ S C distinguishes
• • • t
L
r
It follows that if
C of
V
such that
n
S is given by
C'"
C n S ..
S.
are independent in
k .. k(S)
Cn S
~,
is the max-
is empty, then the
n-k.
The Crapo-Rota theorem and the foregoing remarks suggest that evaluations
of the characteristic polynomial at powers of
subspaces of
case.
V
n
q
can be used to enumerate the
of each rank containing no points of
S.
This is indeed the
The proof requires the following [10].
THEOREM (Crapo). If Q is the lattice of a 8ubgeometry of a geometry with
lattice L,
then
•
where
p: L ... Q
is the retract.
We shall also need the identity:
m
(5.1)
L
i-O
(i)
(_1)i q 2
em] [m-i]
i
j
•
6jm •
14
To prove (5.1), consider
in L.
~he
number
above
j
x
Clearly
m
.
By
of elements of corank
gj(x)
~6bius
m-r(x)
[,j
J
.. r
6(r(y),m-j).
y~
inversion,
.
6 (r{x) ,m-j)
In particular, if x· 0,
6
jm
..
•
THEOREM
number
a
2.
If
5
is a spanning set of
of subspaces of
n-m
V
n
•
V
n
not containing the zero, the
of dimension n-m
containing no points of
S is given by
l
f1I!::rr
n
(5.2)
J
(q m-q i
..
>Ja
n-m
l1 0
where
PROOF.
p(v)
...
m
i dq> [i
m p(q m-i ),
I (-1)
J
1"'0
is the characteristic polynomial of L(5).
The subspaces of
in the retract
V
n
p: L ... L{S).
n
S are the preimage of 0
containing no points of
Since
[x,l]:: L
m for
x
E:
L
n
of rank
n-m,
by
Crapo's theorem,
(5.3)
where
..
p(v)
n
I
a
meO
p (v),
n-m m
m-l m
(q -q )
rri-O
i
Pm (v) -
is the characteristic polynomial of
L •
m
Note that
m-i
(5.4)
[m-i]
j
Setting
.. Pj(q
)
Pj(qj)
\l.
qm-i
in (5.3) and substituting into the right-hand side of
(5.2). and using (5.4) and (5.1). we obtain
15
m
I
i-a
(i)
(_1)i q 2 [m]
i
n
L
j-O
a
n-j P j (q
m-i
)
.. nr Pj(qj) an-j l (_l)i q (i)2
j-O
i-a
m
.
n
j
Pj(q)
I
j=O
8
[ID] rm-i]
i
j
_ 6
n j jm
l
. C-i-a
11 (qm_qi) Ja •
n-m
We recall from Section 4 that an (n,k)-linear code of distance
n
k-dimensional subspace of
containing no vectors of the t-ball
F
latter consisting of all non-zero vectors with
(The set
5
V.
n
Equivalently,
such that
IE I
~
t+l
5
n, t"
e€Sn, t i f f
R(n,t)
PROPOSITION
and distance
1.
t+l
S
n, t
the
as the set
B
Vn
E of
The coding problem is to determine the
o f a s ub space of
is the redundancy
'rhe redundancy
Vn
or there exists a circuit
n
F
containing no vector of
Thus the coding problem is the critical, prob'tem for 5
tical exponent of
5n, t'
or fewer elements of a given basis
eEB
e€E s.Bue.)
and
minimum codimension
t
is a
or fewer non-zero coordinates.
could be defined independently of coordinates in
n,t
of all vectors linearly dependent on
of
t
t+l
R(n, t)
R(n,t)
n, t'
and the eri-
of an optimal code.
of an optimal code of length
n
is given by
r .. O,l, ••• ,R(n,t) - 1
r ::!: R(n,t),
where
Pn,t(v)
is the characteristic polynomial of the geometric lattice
n
of subspaces of F
(F" GF(q»
with a basis in
S
o,t
..
{e:
1
~
wee)
~
tl.
Lu • t
16
CoROU.ARY
The maximum cardinality of a t-independent set in
N(r,t)
m
maX{n:
P
r
n, t(q) >
Recall that any coordinatization of V
n
dual space
n
by premultiplication by an
2.
is
defines a coordinatization of the
lJ: V .... F
r
is represented
n
matrix, whose rows are the coordinates of the
v ." defining
n
linear functionals of
PRoPOSITION
rxn
r
a}.
such that each linear transformation
V"
~
The number of
lJ.
rxn
We can thus state
t-independent matrices over GF(q)
The number of n-point t-independent sets in P
r
is
1s
r I (q-l) nnl.
Pn, t(q)
Set
in (5.2). and note that ITi:'~l (qr_ri )
m· r
full linear group
PROPOSITION
GF(q)
3.
GL (q).
r
is the order of the
Hence
The number of t-independent
rxn
matrices of rank
rover
is
WROLl.MY.
The number of n··points t-independent sets spanning
~
r
is
Of course, for these results to be of any use in attacking the coding or
packing problem, the characteristic polynomials
Pn,t(v)
must be determined.
Without underestimating the difficulties encountered in evaluating the polynomiala, it seems likely that an investigation of the structure of the geometric
lattices
L
n, t
might provide considerable insIght into the problem.
At present,
17
we can offer a general expression for the polynomials
Pn.t(v)
only for
t . 2.
a somewhat trivial case for the packing problem.
The elements of Ln, 2
Bpac~s
of column monomial matrices.
generalization to
GF(q)
which it reduces when
yo which are the null
[13J are those eubspaces of
The order relation is representable as a
of the refinement order of the partition lattice. to
q. 2.
The characteristic polynomial or L
n.2
n-l
• Tf (v-l-i(q-l».
is
i"'O
This yields the critical exponent
R(n,t)
•
[logq«n-l)(q-l)+l)] + 1
and hence,
r
N{r,2)
• s....:l
q-l
•
The lattice
rank n-t,
Ln, t has the property that it is isomorphic to Labove
n
as implied by
THEOREM
3.
The retract
corank
t-l or less.
PROOF.
Let x
p; L
n
+L
be an element of Ln, t
n.t
preserves all elements
of corank k
t-l.
S
XE:L
n
of
Without loss of
x as the kernel of a matrix of the form M· (lk,A),
T
T
lk is the identity of order k. Then if B a (-A ,In_k)' MB • 0, so
'generality, we can take
where
the rows of B are a basis of x,
non-zero elements, each row of
Thus
x has a basis in Sn, t'
and since each row of
B has at most k+l s
so
p(x)· x.
t
AT
has at most k
non-zero elements.
18
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