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BOUNDS FOR k-cAPS
m PG(r
I
q) USEFUL
m
THE THEORY OF ERROR CORRECTIOO. CODES
by
A. Barlotti*
University of North Carolina
Institute of Statistics Mimeo Series No. 484.2
August 1966
This research was supported by the Air Force Office of
Scientific Research Contract No. AF-AFOSR-'76o-65 and also
parti~ supported by Contract No. DA-31-124-AROD-254 / u.s.
A:rrrJy Research Offlce-Durham.
imJPARTMEET OF STATISTICS
UNIVERSITY OF NORTH CAROLmA
Chapel Hill, N. C.
*
On leave from the University of Florence •
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The connection between information theory and finite geometries has been
established in various papers by R. C. Boxe (see [4] for references).
A com-
p1ete introduction to the theory of these geometries can be found in the volume
of B. Segre [10].
Consider a finite r-dimensional projective space PG(r,q), based on the
Ga10is field GF(q), where q = ph, p prime.
points no three of which
are
collinear.
A k-cap in PG(r,q) is a set of k
If we denote by ms(r+l,q) the maximum
number of points in PG(r,q) such that no s of them are dependent, m,(r+l,q) is
the nUJilber of points belonging to a k-cap of PG(r,q) for which k is maximal.
The usefulness of !mowing the values of m (r+1,q) in coding theory has been
s
emphasized by R. C. Bose and I. M. Chakravarti in this volume (see [5] and [7]).
The !mown values of m,(r+l,q) are given below (see [3] and [8]).
=q
(q odd)
+1
(q even) ,
-q+2
2
~(4,q) == q + 1
(any q),
We shall mention some bounds on ~(r+l,q) in the cases r
which the values of ~(r+l,q) are unlmown.
> 3, q > 2 in
The first upper bound was obtained
by R. C. Bose [,]:
Theorem 1.
For m, (r+l,q) the fo1lowing inequality ho1ds:
<
r
1
m,(r+1,q) == 1 + ~ : 1
1
•
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If q is odd, then
m,(r+1,q)
~ qr-1 + 1.
A first improvement of this was found by Ta111n1 [9]:
Theorem 2.
If r > 3, q> 2, then
m, (r+1, q)
< qr-l + 1.
Further improvements on the upper bound on m,(r+1,q) were obtained by A.
Barlotti [2], B. Segre [9] and Bose and Srivastava [6].
We. shall give these
results with some improvements for the cases in which they are the best known.
Theorem 3.
In pG(4,q), q 2: 7 odd, we have
m,(5,q)
In geometric language:
r? - q2 + 8q -
14.
_I
if a k-cap eXists in PG( 4, q) q 2: 7 odd, then
k :::
Proof:
:::
c? - q2 + 8q -
14.
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For convenience of the reader we shall give here the whole proof
although, aPart from some initial change, it is the same as in [9],
Suppose there eXists a k-cap K in pG(4,q) with k 2: ;
§ 24.
- q2 + 8q - 13.
We
will prove that this leads to a contradiction.
A plane which cuts K in a (q+1)-arc will be called a plane of the first
kind.
2
A hj'perplane which cuts K in at least q
a hyperplane of the first kind.
- q + 7 points will be called
The following three lemmas are required for
the proof.
Lemma 1.
> 2
In PG (3,q), q2: 7 odd, any k-cap With k -= q - q + 7 belongs to
an elliptic quadric.
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The proof is g1 ven in [1].
Lemma 2.
Through any secant of K there pass at least 9q-14 planes of the
through a secant, we get
k
.:5
2 + (9q-15) (q-l) + [(q2+q+l )-(9q-15) ](q-2) •
• c? - q2 + 8q Lemma 3.
15
Through any plane of the first kind there pass at least five
hyperplanes of the first kind.
It not, then by counting the points on the hyperplanes passing through"
a plane of the first kind, we have
•
3
2
.
q - q + 8q - 14
We now proceed to the proof of Theorem 3.
Let
1C
be a plane of the first
kind and let S' and f!' be hyperplanes of the first kind passing through
Denote the conic in which
1C
cuts K by C, and by Q' and Q" the elliptic quadrics
containing the points of K in S' and f!', respectively.
1C
1C.
ObViously Q' and QtI cut
in C.
Let A be a point on C and P a point of K not contained in either S' or
S~ (The existence of P is easily shown by counting points).
There are at least
(9q-14)-(q+l) • Sq-15 planes of the first kind through the secant AP which do
I-
3
I
· not cut n in a line.
passing
~ugh
holds forQ,".
Let P be one of these planes.
Of the q+l hyperplanes
PI there is at most one tangent to Q' at A.
The same statement
By Leuma 3 1 there are at least three hyperplanes of the first
kind passing through P which are not tangent to either Q,' or Q," at A.
Let 8
be one of these hyperplanes I let Q, be the elliptic quadric which contains the
points of K in 8 1 and-denote by n' and n" the intersections of S With S' and
S" I respectively.
S.
n' and n" are different from n since n does not belong to
For if nE8 1 then n would intersect p in a line.
The plane n' is not tangent to Q,', and therefore intersects Q' in an
irreducible conic C'. There are at mst (q2+l )_(q2_q+7 ) • q-6 points in Q'
which do not belong to K.
Thus there are at least (q+l)-(q-6) • 7 points of
But these 7 points belong to Q, and therefore Q, contains C'.
K on C'.
ilarly Q, contains (f', the intersection of K and
Sim-
n". Since n' and n" are
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different from n, C' is different from d'.
There is exactly one hyperquadric V in PG( 4, q) which contains Q', Q", and
P.
Q, cuts V in C' I C", and P,and so Q belongs to V and is the intersection of
V with S.
Since Q, Q'I and Q" are non-degenerate quadrics l V is either non-
degenerate or else a cone having a single point as vertex.
since
Q
In the latter case,
is an elliptic quadric, V is a cone haVing a point as vertex and whiCh
projects an unruled quadric.
We will show that V contains all points of K.
All the points of K belonging to S, 8', Ef lie in V.
of K not belonging to
SI
Let P* be any point
S' I or S" •. We want to prove that P* belongs to V.
Let p* be a plane of the first kind passing through the secant PP* and which
does not intersect n in a line.
Clearly p* does not belong to either 8'· or Ef.
There are at least five .hyperplanes of the first kind passing through p*.
these I two can be tangent to Q' and two others tangent to Q".
There is at
least one hyperplane S* which is tangent to neither of these quadrics.
4
Of
S* can-
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not contain
1f
since P*, which is in S*, does not cut
1f
in a line.
Let Q* be the elliptic quadric which contains the points of K in S*, and
let ft and :r' be the intersections of S* with Qt and Q", respectively.
Then,
as we saw before, ft and :r' must be different irreducible conics which belong
to Q*.
But Q* intersects V in ft, f ", and P, and so Q* belongs to V.
belongs to Q*, and thus P* belongs to
implies(l)k ~ 2(q2+l ).
V~
It follows tha.t V contains le.
But p*
This
We have
q3 _ q2 + 8q-13
.:s k .s 2(q2+l )
r? - 3q2 + 8q - 15 S 0
The last inequality is iIlIJ?ossible for' q
~
3.
This contradiction completes the
proof.
Theorem 4.
If a k-cap exists in PG{r, q) withr > 4, q ~ 7 odd, then
I
r-4
k < qr-l _ {-2 + 8 qr-3 _ 6
qi - 8.
i=O
Alternatively
r-4
h
m., (r+l, q)
\' i
8
< qr-l·- qr-2+8qr-3 - 6 L
q - •
i=O
Proof:
The proof is the same as that of Theorem 3 in [9], 24:
We have two cases.
Case 1:
There is a '-flat which contains at least
rf-q points of le.
Then by
counting the number of points of le which lie on 4-flats passing through this
3-flat and using Theorem 3 we have
(1).
See [9]
,
§ 20
.
5
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k < (q2 -q) + ( qr-4 + ••• + q +1) [(q, + 8q-14) - (q2 - q)]
-
(i)
r-4
= qr-1Case 2:
r..g, + 8 r-' - 6 \
i
8
q
q
~ q i=O
No '-flat contains 9,2_9, points of lC.
The number of pairs of points
of K and '-f'lats passing through these Points is
k •
(See B. Segre [10] §167).
(r
9, -1 ) ( qr-1-1 ) ( 9,r-2 -1J
(q'-l) (q2_1 ) (q-1)
•
On the other hand, since every '-flat contains
less than q2_q points of K, the number of pairs is not greater than
--
( r+1
r-2 2
(q2_q ) . q
4,-1 )( qr -1 )(~r-1-1 )( q"'l
(q -l)(~-l)(q -1)(q-1)
•
Combining these results, we have
q
(ii)
r+1
4
q
-
1
.
- 1
.
An upper bound for k in the general case is given by the larger of the
expressions on the right-hand sides of (i) and (ii).
It is easily shown that
(i) is larger than (ii).
Theorem 5.
If' a k-cap exists in PG(r, q) with r > 4, q = 7, then
Alternatively
m, (r+l, 7) s r-
1
r~
-
6
k
f
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••
-
for r > 4.
Proof:
Qase I:
There are three cases to consider.
There is a secant to X such that no plene through this secant cuts X
in a (g,+I>.:.Es..
By counting the points of X on the planes thrOUgh this secant
we·have
(i)
case 2:
a plane ex which cuts X in, a
There is
t~ough ex cuts X in a ('+l)~.
through ex, we have
(ii)
(g,+l)-arc but .SUCh that no 3-flat
By counting points of Ie on these 3-flats
r-3
k .:s qr-l_
I
qi
i=l
case 3:
There is at least one '-flat
S
'Which cuts
X
in a (q2+1)_cap. By counting
points of X on the 4-flats through S we obtain, using Theorem 3 ,
2
3 2
2
k.:s q + 1 + [q -q +8q-14-(q +l)J
(iii)
r-4 i
E q
1=0
r-4··
.~
= qr-l - qr-2 + 7qr-3 - 8 E qi + q .. 14 •
i=l
Comparing (i), (ii), and (iii), we see that the upper bound given by (ii)
is largest.
Hence in the general case this is an upper bound for k.
Theorem 6.
If a k-eap exists in PG(4, q), q =
k
~~
-1
Alternatively
m, (5, 5) oS J24~
(The proof is given in [2]) •
7
5, then
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Theorem 7.
If a k-cap exists in FG(r, q), r
> 4, q
_.
= 5, then
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Alternatively
for r > 4.
(The proof is given in [2]).
TheoremS,
If q > 2, r .::: 3, then the maximum possible number of points
in any k-cap in FG(r, q) cannot exceed the positive root of the quadratic
I.
I'
equation
f(x) .. (q2_ q_l ) x2 _ [q2_2q_l +(q_2)
r
-
i=O
The proof is given in [6].
r
Z qi] x-2 I: qi =
o.
i=O
Using this theorem we obtain improvements
over previous results in the following special cases ([6] § 6):
m, (5, q) oS r? - 1
q > 2 even,
2
r
i
q -2q-l+(q-2) I: q
i=O
~(r+1,q) <
2
+ 1, q > 2, even.
q
- q - 1
This last bound is improved by the following theorem:
Theorem 9,.
In mer, q), r
> 4, q > 2 even,
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m, (r+l,q)
~
r-l
q
- 2
-1.
The proof is similar to that of Theorem 5, using the bound m,(5,q)
given by the previous theorem.
S ~-l
We must consider the additional case in which
a plane cuts the It-cap in a (q+e)-arc, however.
9
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REFERENCES
[1].
A. Bar1otti,
Un'osservazione sul1e k-ca1otte degli spazi lineari
finiti di dimensione tree . Boll. Un. Mat. Ital. (;3) 11 (1956), 248-252.
[2].
A. Bar1otti.
Una 1imitazione superiore per i1
nwnero di punti appartenenti
a una k-calotta C(k,O) di uno spazio 1ineare finito.
Boll. Un. Mat.
Ital. (3) 12 (1957), 67-70.
[3] •. R.C.Bose,
Mathematical theory of the symmetrical factorial design
Sankh;ya, ~. (1947), 107-166.
[4].
R.C. Bose,
On some connections between the
information theory.
38 part
(5].
design of experiments and
Bull. of the International Statistical Institute,
rv (1961).
R.C. Bose, Error correcting, error detecting and error locating codes.
Mtmeo Series No. 426, University of North Carolina. (1965).
(6].
R.C. Bose and J.N. Srivastava, On a bound useful in the theory of factorial
designs and error correcting codes •. Ann. Math. Statist. 35 (1964),408-414.
(7].
I.M. Chakravarti, Bounds on error correcting codes (non-random).
Hi.meo
Series No. 451, University of North Carolina, (1965).
[8].
B. QVist., Some remarks concerning curves of the second degree in a
finite plane.
[9].
Ann. Acad. Sci. Fenn. Sere A. :1;. 134 (1952), pp. 1-27.
B. Segre, Le geometrie di Galois.
Cremnese, Rome, 1961.
[11]. G. Tallini, Bulle .k-ca1otte di uno spazio 1ineare finito.
Pura AWl. (4) 42 (1956), 119-164.
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Ann. Mat. Pura. AjPI>1. (4) 48(1959),1-97.
(10]. B. Segre, Lectures on M:>dern Geometry, with an appendix by Lucio LombardoRadice.
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