The maximal size set of weight 5 vectors in V(10,2) with minimum distance 4, is unique

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The maximal size set of weight 5 vectors in V(10,2) with

minimum distance 4, is unique

Citation for published version (APA):

Tilborg, van, H. C. A. (1976). The maximal size set of weight 5 vectors in V(10,2) with minimum distance 4, is unique. (EUT report. WSK, Dept. of Mathematics and Computing Science; Vol. 76-WSK-02). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1976

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TECHNISCHE HOGESCHOOL EINDHOVEN NEDERLAND

ONDERAFDELING DER WISKUNDE

TECHNOLOGICAL UNIVERSITY EINDHOVEN THE NETHERLANDS

DEPARTMENT OF MATHEMATICS

The maximal size set of weight 5 vectors in V(10t2)

with minimum distance 4, is unique

by

H.C.A. van Tilborg

T.H.-Report 76-WSK-02 March 1976

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Abstract

In this paper it is shown that a maximal S1ze set of weight 5 vectors of

length 10 and with mutual distance at least 4, is unique (up to an

isomor-phsim) and has 36 elements.

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- 1

-Let V(n,2) be a n-dimensional vectors pace over GF(2). The ~eight w(~) of a

vector x E V(n,2) is defined as the number of its nonzero coordinates. The

distance

d(~'l) of two vectors ~ and

X.

in V(n,2) is the number of coordinates,

where ~ and

X.

differ, Le. d(~,lo) = w(~ - lo)'

By ~'lo we mean the coordinate wise product of ~ and lo'

In this paper we shall need some results of the theory of designs.

Definition. A t - (v,k,A)

design

is an ordered pair

(X,B),

where X is a set

and

B

c

P(X),

with the following proper~ies

1)

2)

3)

Ixl

= v ,

The elements of

B

are often called

blocks

J the elements of X are often called

points.

It is well known and easy to prove that a t - (v,k,A) design is simultaneous-ly an i - (v,k,A.) design with

1 4) A.

=

A (v-i)(v-i-l) ••• 1 (k - i)(k - i - I ) (v-t+ I) (k - t + 1) , Since 1: ~ all the Clear A_O

=

I

B

I.

Let B be a specif~c block in B and let n~ be defined as the number of blocks

different from0:., "intersecting it in ~ points. The numbers n~ are called the

intersection numbers

of block B.

~

(i)n~ counts the number of i-tuples occuring on the points of B, in blocks different from B, we have

5) (.)(L -k 1),

1 1

o

:s:; i :s:; t .

A very common way of describing a t-design is by its v x A

O

incidence matrix

A, defined by

6) A . • = {··Ol

1,J

if block j contains point i

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2

-Two t-designs are called

isomorphic

iff the incidence matrix of one of the designs can be obtained from the incidence matrix of the other by a row andl or column permutation.

We are now ab le to s tate a theorem of Johnson (1962), of which we omit the proof.

Theorem. Let

e

be a collection of vectors of weight k in V(n,2) with mutual distance at least 2e.

Then 7) and

l

ei

=.!!.

n - I k k - 1 implies that

e

is a n - k + e + len - k + e

_J

e + e (k - e) ( k [n - k + e

_{J ).}

- n" e design.

Let A be a collection of vectors of weight 5 in V(10,2) with mutual distance at least 4. Application of the theorem to this case yields that IAI ~ 36 and that IAI = 36 implies that A is a 3- (10,5,3) design. We shall now formulate the main theorem of this paper.

Theorem. Let A be a maximal size set of weight 5 vectors ~n V(10,2) with mu-tual distance at least 4. Then IAI = 36 and A is unique (up to an isomorphism).

Proof. In figure 5 the reader may find an example of 36 vectors (the columns) of weight 5 in V(10,2) with mutual distance at least 4. We shall show that this example is unique. So let A have all the required properties and let

IAI = 36. We know already that A is a 3- (10,5,3) design. Moreover by 4) 8)

Since any two blocks have distance at least 4, i.e. mutual intersection at most 3, we have apart from the 4 equations in n

O,nl , ...,n4 in 5), that n4=0.

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3

-block, this solution being

9)

Let A be the incidence matrix of A. We number the rows of A by a,b,c, ••• ,j. The corresponding row vectors will be denoted by ~,~,•••

,i.

Similarly the columns of A will be numbered from 1 to 36 and the corresponding column vec-tors are denoted by .!..,~,

...

,~.

Instead of A I ' we shall write (a, 1), etc. Since the intersection numbers a,

of each block are given by 9), we have that w.l.o.g. A looks like

-:6 27 28 29 30 31 32 33 34 35 36 1 2 345 6 7 8 9 10 II 12 13 14 15 16 J7 18 19 20 21 22 23 :4 25 a I 1 I I 1 I I I I 1 I I 1 1 I 1 I 1 b I I I I I I 1 I 1 I I I 1 I I I I I c 1 I I I I I 1 1 I 1 1 1 I J I I I 1 d I 1 1 I 1 J 1 I I 1 I 1 1 I I J I I

"

I I I I I I 1 I 1 I 1 I I I I I 1 I

-f g h i j _---~ I Figure 1.

where all the empty positions in the first 5 rows should be red as 0. The blocks 32,33, .•• ,36 each have 4 ones in the rows f,g, ••• ,j. Since each of them has intersection number n

4 = 0, we have w.l.o.g. 32 33 34 35 36 f 1 1 1 1 g 1 1 1 1 h 1 1 1 1 i 1 1 1 1 ] 1 1 1 1 Figure 2.

We shall continue to number the ones ~n fig. 4 (and fig. 5) in the order of the consecutive arguments, that we will use.

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4

-Since w(22·S2) ::; 3 (n

4 =

0),

entry (22,j) must be one. Similarly w(g·12)::; 3, so (22,i) = I and also w(t2:E) ::; 3, so (23,j)= I, etc.

According to our numbering we have denoted all the ones that follow from this argument, by a 2 in figure 4.

Since w(~:

l.!.)

~ I (nO = 0 for b10ck 2)., we know tha t a t Ie as t one of the en-tries (f,2) and (g,2) equals I. Similarly at least one of entries (f,3) and (g,3) is one. However w(~:

1) ::;

3, so w.1.o. g. we have

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3

f l O g 0 I Figure 3.

We can repeat this argument for columns 4 and 5 versus column 30, etc. All the ones that follow from this argument are denoted by a 3 in figure 4.

3\ 132 \ I 2 3 4 5 6 789 10 II J2 13 14 15 16 17 18 19 20 2 I 22 23 24 25 26 27 28 29 30 33 34 35 36 a I I 1 I I I I \ I I I I I I I I I I b I I I I I I I I I I I I J I I I J I c I I I 1 I I I I I I I I I I I J 1 1 d 1 I I I I I 1 I 1 I I I I I I I I I

"

1 I I I I I I 1 I 1 I I I I I J I I f 3 3 3 3 2 2 2 2 \ I I I g 3 3 3 3 2 2 2 2 I I I I 1 3 3 3 3 2 2 2 2 I I I I i 3 3 3 3 2 2 2 2 I J I I j 3 3 3 3 2 " ₂ ₂ _I _J _J _I -Figure 4.

In figure 5 we start our numbering again with I. The reader is advised to take a sheet of squared paper and follow our arguments on it.

Since the row permutation (d,e)(f,g) together with the column permutation

(2,3) (4,6) (5,7) (8, 10) (9 , 11) ( 14, 16) (15, 17) (24,25) (27,28) (29 ,30) (35,36) leaves figure 4 invariant but permutes the entries (f,22) and (g,22) and

since a similar permutation leaves the design invariant but permutes (f,22) and (h,22), we have w.l.o.g. that entry (£,22) equals one. This gives rise to the number 2 in figure 5.

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5

-In the following we shall often have a statement like "the inequality(ies) A (and B) imply that entry (x,k) is 1, which will be denoted by the number

9. on entry (x,k) in figure 5". We will abbreviate this statement by

A (and B) ~ 9. on (x,k)

.

Using this notation we get

w(..!i'.!..~) ::; 3 and w(..!i'~) ::; 3 ~ 4 on (i, 14)

,

w (§"2) ::; 3 and w (§.'32) ::; 3 ~5 on (j ,8)

,

wc.~'

2)

::; 3 and w(i'E.) ::; 3 ~6 on (g,4) w (3.'1) ::; 3 and w(3.'~) :<::: ₃ ~7 _{on (h ,2)}

.

The row permutation (a,b) (i,j) and the column permutation

(8,14) (9, 15) (10, 16) (11, 17) (12, 18) (13, 19) (20,21) (23,26) (24,27) (25,28) (32,33)

leaves figure 5at this moment invariant but permutes positions (i,31) and

(j ,31 ), Hence w (29 '1.!..) :<::: 3 ~ 8 on (i , 31 )

,

w (20'll.) ::; _{3 an d w}

(3.2:

_l!)

:<::: 3 ~9 on (h, 20)

,

w(30'20) ::; _{3 and w(30'32,.)} ::; 3 ~ 10 on (j , 30 )

,

w(ll.'

3.Q.)

::; _{3 an d w}

C.U:

_1Q.)

::; ₃ _~ _{lIon (g, 2 1)}

,

w(£'22) ::; _{3 and w(£' 30)} ::; ₃ ~ 12 on (g,25)

,

w(28' E.) ::; _{3 and w (28'l!.)} :<::: 3 ~ 13 on (h,28)

.

Since w(2..:D ::; 3 implies that entry (h,6) = 0, w(.!2'11) ::; 3 ~mplies that en-try (h,13) =

a

and w(3.'2) = 5 implies that (h,25) =0 one has by w(:::'.~....!:~) = 3 that exactly one of the entries (h,lO) and (h,ll) equals 1. Byw(:::,.~..~) one gets now that entry (h,9) = O. From w(§.'2) :<::: 3 we deduce that entry (g,9) = I, which gives rise to the number 14 in figure 5. Now

w(24'V :<::: 3 _{and w}<.~.:~:

32,)

::; 3 ~ 15 on (h ,24)

,

w (31:§.) :<::: 3 and w(23'~) ::; 3 ~ 16 on (i,23)

,

w(~'20) ::; _{3 and w(26'28)} :<::: 3 ~ 17 on (g,26)

,

w(3.Z:.!..~) ::; _{3 and w (32.' 26)} ::; ₃ _~ _{18 on (j,27)}

,

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6 -w

(1·f!)

~ 3 and w

(1

·24) ~ ₃ "? 20 on (i ,5) w(~·2) ~ 3 and w(~.25) ~ 3 "? 2I on (i,6) w(2·~) ~ 3 and w(2 ·28) ~ 3 "? 22 on (j,7) w

(.!.Q.•

.!..!.)

~ 3 an d w

(.!.Q.•

£)

~ 3 "? 23 on (h, 10) w

(.!.!.•

'!'Q)

~ _{3 and w(1.J:23)} ~ 3 "? 24 on (f, 11) w<"!1:11) ~ 3 and w

(..!1..

24) ~ 3 "? 25 on (f, 12) w

(12:..!1.)

~ _{3 and w}

(11:

2.!.)

~ 3 "? 26 on (j,13) w(J1.~) ~ 3 and w

(.!2..•.

27) ~ 3 "? 27 on (h, 15) w(..!2..·lD ~ _{3 and w}_(1~:,~,§) ~ 3 "? 28 on (f, 16)

w(!2•

..!i)

~ _{3 and w}_<"!2:~) ~ 3 "? 29 on (i,17)

w(~.12.) ~ 3 and w(~.~) ~ 3 "? 30 on (g,18) w(12..j!) ~ 3 and w(12..~) ~ 3 "?31 on (f, 19)

.

-"--_._---I 2 3 4 5 6 7 8 <) 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 a I I I I I I I I I I 1 1 I I I 1 1 I b I I I ! I 1 1 1 I I I I 1 I I J 1 I c I I 1 I 1 I 1 I I I I I I I I I I J d I I I I 1 I I 1 I I I I I I 1 I I 1 e I I 1 I I 1 I 1 I I 1 I I I I I 1 I ---~--- -f 1 I I _4'J.' _~5 _I ₂₈ ₃₁ ₂ _I _I ₃ _I _{1 I} I 1 I I g 1 6 I 14 1 1 30 II I 12 17 I I 1 I I I I h 7 I I 23 I 27 I <) I 15 I J3 I I 1 1 I 1 i ~o 21 1 I I 4 29 I 1 16 I I I 8 I I I I

l.

19 ', ₅ ₂₆ J I I I I I I 1 18 10 I I 1 1 ----~-~..._.__.~._---~._---_. __.•._.._----~- _.---

--Figure 5.

It is a matter of straight foreward checking that we have indeed obtained a

3- (l0,5,3) design, with all blocks at mutual distance at least 4. D

In the previous proof we started with a fixed block and none of the automor-phisms involved this block, which means that the automorphism group of the design in figure 5 is transitive on blocks. Consequently (see Dembowski [1968J) it acts transitively on the points too.

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- 7

-References

[IJ P. Dembowski, 1968, Finite geometries, Springer-Verlag, Berlin, etc.

[2J S.M. Johnson, 1962, A new upper bound for error correcting codes,