r. Ahlewede, b.balkenhol, l.khachatrian,Some Properties of Fix-free Codes

7/27/2019 r. Ahlewede, b.balkenhol, l.khachatrian,Some Properties of Fix-free Codes

1/14

S o m e p r o p e r t i e s o f

F i x { F r e e C o d e s

R . A h l s w e d e

F a k u l t a t f u r M a t h e m a t i k ,

U n i v e r s i t a t B i e l e f e l d ,

P o s t f a c h 1 0 0 1 3 1 ,

3 3 5 0 1 B i e l e f e l d ,

G e r m a n y

B . B a l k e n h o l



P o s t f a c h 1 0 0 1 3 1 ,

3 3 5 0 1 B i e l e f e l d ,

G e r m a n y

L . K h a c h a t r i a n

y



P o s t f a c h 1 0 0 1 3 1 ,

3 3 5 0 1 B i e l e f e l d ,

G e r m a n y

A b s t r a c t

A ( v a r i a b l e l e n g t h ) c o d e i s x { f r e e c o d e i f n o c o d e w o r d i s a p r e x

o r a s u x o f a n y o t h e r . A d a t a b a s e c o n s t r u c t e d b y a x { f r e e c o d e i s

i n s t a n t a n e o u s l y d e c o d e a b l e f r o m b o t h s i d e s . W e d i s c u s s t h e e x i s t e n c e o f

x { f r e e c o d e s , r e l a t i o n s t o t h e d e B r u j i n N e t w o r k a n d s h a d o w p r o b l e m s .

P a r t i c u l a r y w e d r a w a t t e n t i o n t o a r e m a r k a b l e c o n j e c t u r e : F o r n u m b e r s

l

1

; : : : ; l

N

s a t i s f y i n g

N

P

i = 1

2

? l

i

3

4

a x { f r e e c o d e w i t h l e n g t h s l

1

; : : : l

N

e x i s t s .

I f t r u e , t h i s b o u n d i s b e s t p o s s i b l e .

e m a i l : b e r n h a r d @ m a t h e m a t i k . u n i - b i e l e f e l d . d e

y

e m a i l : l k @ m a t h e m a t i k . u n i - b i e l e f e l d . d e

1


2/14

1 B A S I C D E F I N I T I O N S 2

1 B a s i c D e n i t i o n s

F o r a n i t e s e t X = f 0 ; : : : ; a ? 1 g , c a l l e d a l p h a b e t , w e f o r m X

n

=

n

Q

1

X ,

t h e w o r d s o f l e n g t h n , w i t h l e t t e r s f r o m X a n d X

=

1

S

n = 0

X

n

, t h e s e t o f

a l l n i t e l e n g t h w o r d s i n c l u d i n g t h e e m p t y w o r d e f r o m X

0

= f e g , X

i s

e q u i p p e d w i t h a n a s s o c i a t i v e o p e r a t i o n , c a l l e d c o n c a t e n a t i o n , d e n e d b y

( x

1

; : : : ; x

n

) ( y

1

; : : : ; y

m

) = ( x

1

; : : : ; x

n

; y

1

; : : : ; y

m

) :

W e s k i p t h e b r a c k e t s w h e n e v e r t h i s r e s u l t s i n n o c o n f u s i o n , i n p a r t i c u l a r

w e w r i t e t h e l e t t e r x i n s t e a d o f ( x ) . W e a l s o w r i t e X

+

= X

r f e g f o r t h e

s e t o f n o n { e m p t y w o r d s .

T h e l e n g t h j x

n

j o f t h e w o r d x

n

= x

1

: : : x

n

i s t h e n u m b e r n o f l e t t e r s i n

x

n

.

A w o r d w

2 X

i s a f a c t o r o f a w o r d x

2 X

i f t h e r e e x i s t u ; v

2 X

s u c h

t h a t x = u w v . A f a c t o r w o f x i s p r o p e r i f w 6= x .

F o r s u b s e t s Y ; Z o f X

a n d a w o r d w 2 X

, w e d e n e

Yw =

fy w

2 X

: y

2 Y g;

Y Z=

fy z

2 X

: y

2 Y; z

2 Z g

a n d

Yw

? 1

=

fz

2 X

: z w

2 Y g:

A s e t o f w o r d s C X

i s c a l l e d a c o d e .

R e c a l l t h a t a c o d e i s c a l l e d p r e x { f r e e ( r e s p . s u x { f r e e ) , i f n o c o d e w o r d

i s b e g i n n i n g ( r e s p . e n d i n g ) o f a n o t h e r o n e .

D e n i t i o n 1 A c o d e , w h i c h i s s i m u l t a n e o u s l y p r e x { f r e e a n d s u x { f r e e ,

i s c a l l e d b i p r e x o r x { f r e e . T h i s c a n b e e x p r e s s e d b y t h e e q u a t i o n s

C X

+

\ C = a n d X

+

C \ C = :

D e n i t i o n 2 A c o d e C = f c

1

; : : : ; c

N

g o v e r a n a { l e t t e r a l p h a b e t X i s s a i d

t o b e c o m p l e t e i f i t s a t i s e s e q u a l i t y i n K r a f t ' s i n e q u a l i t y , i . e . f o r

i

= j c

i

j ,

N

X

i = 1

a

?

i

= 1 :

D e n i t i o n 3 A x { f r e e c o d e C i s c a l l e d s a t u r a t e d , i f i t i s n o t p o s s i b l e t o

n d a x { f r e e c o d e C

0

c o n t a i n i n g C p r o p e r l y , t h a t i s , j C

0

j > j C j .


3/14

2 T H E E X I S T E N C E 3

2 T h e E x i s t e n c e

L e m m a 1 A n i t e x { f r e e c o d e C = f c

1

; : : : ; c

N

g o v e r X = f 0 ; : : : ; a ? 1 g

i s s a t u r a t e d i C i s c o m p l e t e .

P r o o f :

L e t

i

= j c

i

j f o r a l l 1 i N .

1 . I f

N

P

i = 1

a

?

i

= 1 , t h e n C i s s a t u r a t e d , b e c a u s e o t h e r w i s e w e g e t a c o n -

t r a d i c t i o n t o K r a f t ' s i n e q u a l i t y .

2 . N o w w e s h o w t h a t i n c a s e

N

P

i = 1

a

?

i


4/14


F o r e a c h c o d e w o r d c

i

o f l e n g t h

i

w e t h u s c o u n t a

N

?

i

l e a v e s , w h i c h h a v e

c

i

a s a p r e x a n d a l s o a

N

?

i

l e a v e s , w h i c h h a v e c

i

a s u x . T h e s e s e t s n e e d

n o t b e d i s t i n c t . H o w e v e r , t h e i r t o t a l n u m b e r d o e s n o t e x c e e d 2

N ? 1

P

i = 1

a

N

?

i

.

B y o u r a s s u m p t i o n t h i s i s s m a l l e r t h a n a

N

a n d t h e r e i s a l e a f o n t h e

N

' s

l e v e l , w h i c h w a s n o t c o u n t e d . T h e c o r r e s p o n d i n g w o r d c a n s e r v e a s o u r

N { t h c o d e w o r d .

W e d e n e n o w a s t h e l a r g e s t c o n s t a n t s u c h t h a t f o r e v e r y i n t e g r a l t u p l e

(

1

;

2

; : : : ; `

N

)

N

P

i = 1

2

?

i

< i m p l i e s t h e e x i s t e n c e o f a b i n a r y x { f r e e c o d e

w i t h l e n g t h s

1

;

2

; : : : ; `

N

.

L e m m a 3

3

4

.

P r o o f : F o r a n y =

3

4

+ " ; " > 0 , c h o o s e k s u c h t h a t 2

? k

< " . F o r t h e

v e c t o r (

1

; : : : ; `

N

) = ( 1 ; k ; : : : ; k ) w i t h N = 2

k ? 2

+ 2 w e h a v e

N

X

i = 1

2

?

i

=

1

2

+ 2

? k

( 2

k ? 2

+ 1 ) =

3

4

+ 2

? k


5/14


a n d b y i n d u c t i o n h y p o t h e s i s w e h a v e a x { f r e e c o d e C

0

w i t h t h e l e n g t h s

1

; : : : ; `

M

. W e e s t i m a t e n o w t h e s h a d o w

N

( C

0

) . A c t u a l l y , b y 2 . 1 w e g e t

a n e x a c t f o r m u l a :

j

N

( C

0

) j = 2

M

X

i = 1

2

N

?

i

?

M

X

i = 1

2

N

? 2

i

? 2

X

1 i < j M

2

N

? (

i

+

j

)

: ( 2 . 2 )

A c o d e w i t h l e n g t h s

1

; : : : ; `

N

i s c o n s t r u c t a b l e e x a c t l y i f

j

N

( C

0

) j 2

N

? ( N ? M ) : ( 2 . 3 )

W r i t i n g K = N ? M a n d =

M

P

i = 1

2

?

i

w e g e t a f t e r d i v i s i o n b y 2

N

f r o m

( 2 . 2 ) a n d ( 2 . 3 ) t h a t s u c i e n t f o r c o n s t r u c t a b i l i t y i s

2 ?

2

1 ?

K

2

N

:

W i t h t h e a b b r e v i a t i o n s =

N

P

i = 1

2

?

i

= +

K

2

N

a n d =

K

2

N

w e g e t t h e

e q u i v a l e n t i n e q u a l i t y

1 + ?

p

:

T h i s i s s a t i s e d f o r

3

4

, b e c a u s e 1 + ?

p

h a s t h e m i n i m a l v a l u e

3

4

( a t

=

1

4

) .

2 . 1 M i n i m a l A v e r a g e C o d e w o r d L e n g t h s

T h e a i m o f d a t a c o m p r e s s i o n i n N o i s e l e s s C o d i n g T h e o r y i s t o m i n i m i z e

t h e a v e r a g e l e n g t h o f t h e c o d e w o r d s ( s e e 2 , 5 ] ) .

T h e o r e m 1 F o r e a c h p r o b a b i l i t y d i s t r i b u t i o n P = ( P ( 1 ) ; : : : ; P ( N ) ) t h e r e

e x i s t s a b i n a r y x { f r e e c o d e C w h e r e t h e a v e r a g e l e n g t h o f t h e c o d e w o r d s

s a t i s e s

H ( P ) L ( C ) < H ( P ) + 2 :

P r o o f : T h e l e f t { h a n d s i d e o f t h e t h e o r e m i s c l e a r l y t r u e , b e c a u s e e a c h x

{ f r e e c o d e i s a p r e x c o d e a n d f o r e a c h p r e x c o d e t h e l e f t { h a n d s i d e o f

t h e t h e o r e m f o l l o w s f r o m t h e N o i s e l e s s C o d i n g T h e o r e m . I t i s a l s o c l e a r ,

t h a t t h i s l o w e r b o u n d i s r e a c h e d f o r N = 2

m

( m 2 N ) a n d P ( i ) = 2

? m

f o r

a l l 1 i 2

m

.

T h e p r o o f o f t h e r i g h t { h a n d s i d e o f t h e T h e o r e m i s t h e s a m e a s t h e p r o o f

f o r a l p h a b e t i c c o d e s , w h i c h c a n b e f o u n d i n 1 ] :


6/14

3 O N C O M P L E T E F I X { F R E E { C O D E S 6

W e d e n e

i

, d ? l o g ( P ( i ) ) e + 1 . I t f o l l o w s t h a t

N

X

i = 1

2

?

i

1

2

N

X

i = 1

2

l o g ( P ( i ) )

=

1

2

N

X

i = 1

P ( i ) =

1

2

:

B y L e m m a 2 t h e r e e x i s t s a x { f r e e c o d e C w i t h t h e c o d e w o r d l e n g t h s

1

; : : : ; `

N

.

T h e a v e r a g e l e n g t h o f t h i s c o d e i s

L ( C ) =

N

P

i = 1

P ( i )

i


7/14


( i i i ) I f t h e l e n g t h o f t h e s h o r t e s t c o d e w o r d i s d , t h e n t h e l e n g t h o f e v e r y

c o d e w o r d i s d a s w e l l .

L e m m a 5 F o r e a c h n i t e c o m p l e t e x { f r e e c o d e C = f c

1

; : : : ; c

N

) o v e r

X = f 0 ; : : : ; a ? 1 g , a

2

d i v i d e s t h e n u m b e r o f c o d e w o r d s o f m a x i m a l l e n g t h .

P r o o f : F r o m t h e d e n i t i o n o f c o m p l e t e x { f r e e c o d e s i t f o l l o w s t h a t w i t h

e v e r y c o d e w o r d c 2 C o f m a x i m a l l e n g t h , t h e r e a r e a l s o a

2

? 1 o t h e r c o d e -

w o r d s w h i c h d i e r f r o m c o n l y i n t h e r s t a n d / o r l a s t c o m p o n e n t s . H e n c e

t h e s e t o f c o d e w o r d s o f m a x i m a l l e n g t h i s a d i s j o i n t u n i o n o f e q u i v a l e n t

c l a s s e s e a c h o f c a r d i n a l i t y a

2

. 2

L e m m a 6 F o r e a c h b i n a r y c o m p l e t e x { f r e e c o d e C t h e r e i s a t m o s t o n e

c o d e w o r d o f l e n g t h 2 o r a l l c o d e w o r d s h a v e l e n g t h 2 .

P r o o f : B y ( i ) i n P r o p o s i t i o n 1 w e k n o w t h a t C c o n t a i n s n o c o d e w o r d

o f l e n g t h o n e . I f C c o n t a i n s a c o d e w o r d c w i t h j c j > 2 t h e n b y ( i i i ) o f

P r o p o s i t i o n 1 t h e d e g r e e o f C i s g r e a t e r t h a n 2 , a n d b y ( i ) o f P r o p o s i t i o n 1

0 0 62 C a n d 1 1 62 C . H e n c e i f w e h a v e t w o c o d e w o r d s o f l e n g t h 2 t h e n t h e s e

t w o c o d e w o r d s a r e 0 1 a n d 1 0 . H o w e v e r , t h e r e i s a c o d e w o r d o f m a x i m a l

l e n g t h s t a r t i n g w i t h 0 1 o r 1 0 ( s e e L e m m a 5 ) . 2

3 . 2 O n l y T h r e e D i e r e n t L e v e l s

L e t C b e a n i t e b i n a r y c o m p l e t e x { f r e e c o d e a n d l e t C

i

, f c 2 C : j c j = i g .

L e t b i n

? 1

( c ) b e t h e n a t u r a l n u m b e r w h i c h c o r r e s p o n d s t o t h e b i n a r y r e p -

r e s e n t a t i o n o f c ( N o t e t h a t t h e l e n g t h o f c i s n o t x e d s o t h a t b i n

? 1

( c ) =

b i n

? 1

( 0 c ) ) .

L e m m a 7 L e t C = ( c

1

; : : : ; c

N

) b e a n i t e b i n a r y c o m p l e t e x { f r e e c o d e

w i t h c o d e w o r d l e n g t h s

1

; : : : ; `

N

s a t i s f y i n g

i

2 f k ; k + 1 ; k + 2 g f o r a l l

1 i N a n d s o m e k . T h e n f o r e v e r y E C

k

j

k + 1

(

E)

j 2

j E ja n d e q u a l i t y h o l d s e x a c t l y i f

j E j= 2

k

.

P r o o f : T h e u n i o n o f t h e s e t s E 0 a n d E 1 c o n t a i n s 2 j E j e l e m e n t s . H e n c e

a l w a y s j

k + 1

( E ) j 2 j E j , i f j E j


8/14


( i ) x c y 2 C

k + 2

, x ; y 2 f 0 ; 1 g i f a n d o n l y i f c 2 C

k

a n d

( i i ) j

k + 1

( C

k

) j = 4 j C

k

j .

P r o o f :

( i ) L e t C

0

k

=

c 2 f 0 ; 1 g

k

n C

k

: x c y 2 C

k + 2

; x ; y 2 f 0 ; 1 g

,

C

0

k + 2

=

x c y 2 C

k + 2

; x ; y 2 f 0 ; 1 g : c 2 C

0

k

a n d l e t

D =

k + 1

( C

0

k

) =

c 0 ; c 1 ; 0 c ; 1 c 2 f 0 ; 1 g

k + 1

: c 2 C

0

k

.

F r o m L e m m a 5 w e k n o w t h a t j C

0

k + 2

j = 4 j C

0

k

j . W e c o n s i d e r n e w

c o d e s

C

0

1

= (

C n C

0

k + 2

)

C

0

k

a n d

C

0

2

= (

C n C

0

k + 2

)

D. I t c a n b e

e a s e l y v e r i e d , t h a t b o t h C

0

1

a n d C

0

2

a r e x { f r e e c o d e s . M o r e o v e r , C

0

1

i s c o m p l e t e , s i n c e C i s c o m p l e t e . T h e r e f o r e w e c a n a p p l y L e m m a 7

w i t h r e s p e c t t o E = C

0

k

; j C

0

k

j 2 j C

0

k

j .

H o w e v e r t h i s l e a d s t o t h e c o n t r a d i c t i o n , b e c a u s e C

0

2

i s a x - f r e e c o d e ,

b u t

X

c 2 C

0

2

2

? j c j

=

X

c 2 ( C n C

0

k + 2

)

2

? j c j

+

X

c 2 D

2

? j c j

>

X

c 2 ( C n C

0

k + 2

)

2

? j c j

+

X

c 2 C

0

k + 2

2

? j c j

=

X

c 2 C

2

? j c j

= 1 :

( i i ) W e c o n s i d e r t e h l o w e r s h a d o w o f C

k + 2

:

?

k + 1

( C

k + 2

) ,

n

c 2 f 0 ; 1 g

k + 1

:

k + 2

( c ) \ C

k + 2

6= ;

o

:

B y ( i ) w e h a v e

?

k + 1

( C

k + 2

) =

k + 1

( C

k

) .

T h e r e f o r e C

k + 1

= f 0 ; 1 g

k + 1

n

k + 1

( C

k

) , s i n c e C i s c o m p l e t e .

N o w j

k + 1

( C

k

) j 1 . 2

3 . 3 R e l a t i o n s t o t h e d e B r u i j n N e t w o r k

T h e b i n a r y d e B r u i j n N e t w o r k o f o r d e r n i s a n u n d i r e c t e d g r a p h B

n

=

(

V

n

;

E

n

) , w h e r e

V

n

=

X

n

i s t h e s e t o f v e r t i c e s a n d ( u

n

; v

n

)

2 E

n

i s a n e d g e

i

u

n

2 f ( b ; v

1

; : : : ; v

n ? 1

) ; ( v

2

; : : : ; v

n

; b ) : b 2 f 0 ; 1 g :

T h e b i n a r y d e B r u i j n N e t w o r k B

4

i s g i v e n a s a n e x a m p l e :


9/14


u

u u

u u

u

?

?

?

?

@

@

@

@

?

?

?

?

@

@

@

@

u

u u

u u

u

?

?

?

?

@

@

@

@

?

?

?

?

@

@

@

@

u u

?

?

?

?

?

?

?

@

@

@

@

@

@

@

@

@

@

@

@

@

@

?

?

?

?

?

?

?u u

@

@

@

@

@

@

@

@

@

@

@

@

@

?

?

?

?

?

?

?

?

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( 0 ; 1 ; 0 ; 0 )( 0 ; 0 ; 1 ; 0 )

( 1 ; 0 ; 1 ; 1 ) ( 1 ; 1 ; 0 ; 1 )

( 1 ; 0 ; 0 ; 1 )

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( 1 ; 0 ; 1 ; 0 )( 0 ; 1 ; 0 ; 1 ) ( 1 ; 1 ; 0 ; 0 )( 0 ; 0 ; 1 ; 1 )

A s u b s e t A V

n

i s c a l l e d i n d e p e n d e n t , i f n o t w o v e r t i c e s o f A a r e c o n -

n e c t e d , a n d w e d e n o t e b y I ( B

n

) t h e s e t o f a l l i n d e p e n d e n t s u b s e t s o f t h e

d e B r u i j n n e t w o r k . W e n o t e , t h a t f o r a l l b 2 f 0 ; 1 g , ( b ; b ; : : : b ) 62 A 2 I ( B

n

) ,

b e c a u s e ( b ; b ; : : : b ) i s d e p e n d e n t i t s e l f . T h e i n d e p e n d e n c e n u m b e r f ( n ) o f

B

n

i s f ( n ) = m a x

A 2 I ( B

n

)

j A j .

L e m m a 8 L e t C b e a b i n a r y c o m p l e t e x { f r e e c o d e o n t h r e e l e v e l s :

C = C

n

C

n + 1

C

n + 2

; C

i

6= ; . T h e n

( i ) C

n

2 I ( B

n

) a n d

( i i ) f o r e v e r y A 2 I ( B

n

) t h e r e e x i s t s a c o m p l e t e x { f r e e c o d e o n t h r e e

l e v e l s n ; n + 1 ; n + 2 f o r w h i c h A = C

n

, a n d t h e c o d e i s u n i q u e .

P r o o f :

( i ) I m m i d e a t e l y f o l l o w s f r o m T h e o r e m 2 ( i i ) .


10/14

3 O N C O M P L E T E F I X { F R E E { C O D E S 1 0

( i i ) F o r a n A 2 I ( B

n

) w e c o n s t r u c t a c o m p l e t e x { f r e e c o d e

C = C

n

C

n + 1

C

n + 2

a s f o l l o w s : C

n + 1

= f 0 ; 1 g

n + 1

n

n + 1

( A ) ,

C

n + 2

=

x c y 2 f 0 ; 1 g

n + 2

; x ; y 2 f 0 ; 1 g : c 2 A

. 2

W e n o t e , t h a t t h e e x a c t v a l u e o f t h e i n d e p e n d e n c e n u m b e r f ( n ) o f B

n

i n

g e n e r a l i s n o t k n o w n .

C l e a r l y f o r a n y x

n

; y

n

2 A 2 I ( B

n

) ; x

n

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n

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b i n

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n

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t h e o r e t i c a l p r o b l e m :

F o r g i v e n m 2 N , n d a s e t S = f 1 a

1

< : : : < a

s

< m g o f m a x i m a l

c a r d i n a l i t y , f o r w h i c h f a

i

; 2 a

i

; 2 a

i

+ 1 ; a

i

+ m g \ f a

j

; 2 a

j

; 2 a

j

+ 1 ; a

j

+ m g = ;

h o l d s f o r a l l 1 i < j j S j .

I n t h e c a s e m = 2

n

w e h a v e e x a c t l y t h e p r o b l e m o f n d i n g a m a x i m a l

i n d e p e n d e n t s e t w i t h c a r d i n a l i t y f ( n ) i n t h e d e B r u i j n n e t w o r k . H e n c e w e

s o l v e t h i s p r o b l e m ( f o r m = 2

n

) a s y m p t o t i c a l l y .

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l i m

n ! 1

f ( n )

2

n

=

1

2

:

P r o o f : L e t A 2 I ( B

n

) w i t h j A j = f ( n ) . C l e a r l y f ( n )


11/14

4 C O M P U T E R R E S U L T S 1 1

j f 0 ; 1 g

n

n ( S

n

0

S

n

1

) j =

n

2

X

i = 0

n

2

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=

n

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2

:

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n

0

j =

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n

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n ! 1

j S

n

0

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2

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=

1

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n

0

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n

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n

= S

n

0

. 2

4 C o m p u t e r R e s u l t s

1 . ) F o r 2 n 6 w e h a v e c a l c u l a t e d t h e i n d e p e n d e n t n u m b e r ( f ( n ) ) o f t h e

b i n a r y d e B r u i j n n e t w o r k o f o r d e r n v i a a c o m p u t e r p r o g r a m . A m a x i m a l

i n d e p e n d e n t s e t S = f 1 a

1

< : : : < a

s


12/14


N e c e s s a r y d = 1 , b e c a u s e i n c a s e d = 0 , w e h a v e e = 0 , f o r o t h e r w i s e ,

t h e c o d e w o r d 0 1 w o u l d b e s u x . H o w e v e r , 0 0 i s

e x c l u d e d , b e c a u s e o t h e r w i s e 0 0 0 o r 1 0 0 w o u l d b e s u x .

c = 0 , b e c a u s e f o r c = 1 w e g e t 1 1 0 o r 1 1 1 a s s u x .

b = 1 , b e c a u s e f o r b = 0 w e g e t 0 0 0 o r 1 0 0 a s p r e x .

F i n a l l y a 6= 0 , b e c a u s e f o r a = 0 w e g e t 0 1 a s p r e x .

a n d a 6= 1 , b e c a u s e f o r a = 1 w e g e t 1 1 0 a s p r e x . 2

T h i s i s a c o n t r a d i c t i o n .

3 . ) W e p r e s e n t a n e x a m p l e o f a c o m p l e t e b i n a r y x { f r e e c o d e f o r e a c h

p o s s i b l e l e n g t h { d i s t r i b u t i o n L w i t h j L j 2 9 :

0 1

2 : 2 x 1

0 1 0 0 1 0 1 1

4 : 4 x 2

0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1

8 : 8 x 3

0 1 0 0 0 1 0 0 1 1 0 1 1 1

0 0 1 0 1 0 1 0 0 0 1 1 1 0 1 1

9 : 1 x 2 + 4 x 3 + 4 x 4

0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 0 1 1 0 1 1 1 0

0 0 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 1 0 1 1 0 1 1 1 1 1 1 1

1 6 : 1 6 x 4

0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 1 0

0 1 0 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 1 1 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1

1 0 0 1 1

1 7 : 1 x 3 + 1 2 x 4 + 4 x 5

0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 1 1

0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 1 1 0 1

0 0 0 1 1 1 0 0 1 1

1 8 : 2 x 3 + 8 x 4 + 8 x 5

0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1

1 0 1 1 0 1 1 1 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 1 0

1 1 0 0 1 0 0 1 0 0 1 1 1 1 0 0 1 1

1 9 : 2 x 3 + 9 x 4 + 4 x 5 + 4 x 6

0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 0 1 1 1 0 0 1 1 1 1 1 1 1

0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 0 0 1 1 0 1 0 1 1 1 1 0 1 1

0 1 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 1 1 1 0 0 1 1

2 0 : 3 x 3 + 5 x 4 + 8 x 5 + 4 x 6

0 0 1 0 1 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 0 1 1 0 1

1 1 1 1 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 1 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 1 0

1 0 0 1 1 0 0 0 0 1 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 0 1 1 1

2 1 : 3 x 3 + 6 x 4 + 4 x 5 + 8 x 6


13/14


0 1 0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 0 1 1 1 1 0 0 1 0 0 1 0 1 0 0

0 0 0 1 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 1 1 1 0 1 1

0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1 0 1 1

2 2 : 1 x 2 + 5 x 4 + 1 2 x 5 + 4 x 6

0 0 1 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1

1 1 1 1 0 1 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 0 0 0 1 1 0 1 1 0 0 0 1 1 1 0 0 0

0 1 0 0 1 0 0 1 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 0 1 0 0 1 1 0 0 1 1 1 1 1 0 0 1 1

2 2 : 3 x 3 + 6 x 4 + 5 x 5 + 4 x 6 + 4 x 7

0 1 0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 0 0

1 0 1 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1 1 1 0 0 1 0 1 0

1 0 1 0 1 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1

2 3 : 1 x 2 + 6 x 4 + 8 x 5 + 8 x 6

0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 1 0 1 1 1 1 1 0 1 0 0

1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 1 1 1 0 1 1 0 0 1 1 1 1 0 1 1 1

0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1 0 0 1 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1

2 4 : 1 x 2 + 6 x 4 + 9 x 5 + 4 x 6 + 4 x 7

0 0 1 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0

0 0 0 1 0 0 1 0 1 0 0 0 0 1 1 0 1 0 1 1 0 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 0 0 1 1 0 1 0

1 1 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 1 0 0 1 0 0 1 1 0 0 1 1 1 1 1 0 0 1 1

2 4 : 4 x 3 + 3 x 4 + 5 x 5 + 8 x 6 + 4 x 7

0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 1 1 1

1 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 0

0 0 0 1 1 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 0 1 0 1 0 0 0 0 1 0 1 1

1 0 0 1 0 1 1

2 5 : 1 x 2 + 7 x 4 + 5 x 5 + 8 x 6 + 4 x 7

0 1 1 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 1 1 0 1 0

0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 0 0 1 0 1 0 0 0

1 1 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 1 1 0 1 0 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0

0 0 1 0 0 1 1 1 0 1 0 0 1 1

2 6 : 1 x 2 + 1 x 3 + 3 x 4 + 9 x 5 + 8 x 6 + 4 x 7

1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 1 1 1

1 1 0 0 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 1 0 0 1 0 0 0 0 0 1 0 0 1

0 1 0 1 0 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1 0 1 0 0 0 0 1 0 1 0 0 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 0 1 1 0 1 0 0 0

1 1 1 0 1 0 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 0 1

2 7 : 1 x 2 + 7 x 4 + 6 x 5 + 5 x 6 + 4 x 7 + 4 x 8

1 0 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 1 1 1 1 0 1 0 0 0 1 1 0 0 0

0 1 1 0 0 1 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 1

0 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 0 1 0 0 1 1 1 0 1 0 0

0 1 0 0 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1

2 8 : 1 x 2 + 1 x 3 + 4 x 4 + 6 x 5 + 8 x 6 + 8 x 7


14/14

R E F E R E N C E S 1 4

1 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 0 0 1 1 0 1 1 1

1 1 1 1 0 0 1 0 1 0 1 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 1 0 0 1

1 1 1 0 0 1 0 1 0 1 0 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1 0 1 0 0 0 0 1 0 1 0 0 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1

0 1 1 0 1 0 0 0 1 1 1 0 1 0 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 0 1

2 8 : 1 x 2 + 8 x 4 + 2 x 5 + 9 x 6 + 4 x 7 + 4 x 8

1 0 0 0 1 0 0 0 0 1 1 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 0 1

1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0

1 1 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 1 1 0 0 1 1

1 1 1 0 0 1 1 0 1 1 0 0 1 0 0 1 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 1

2 9 : 1 x 2 + 1 x 3 + 4 x 4 + 6 x 5 + 9 x 6 + 4 x 7 + 4 x 8

R e f e r e n c e s

1 ] R . A h l s w e d e a n d I . W e g e n e r , S u c h p r o b l e m e ,

T e u b n e r , S t u t t g a r t , 1 9 7 9 .

2 ] R . B . A s h , I n f o r m a t i o n t h e o r y ,

I n t e r s c i e n c e T r a c t s i n P u r e a n d A p p l i e d M a t h e m a t i c s 1 9 ,

I n t e r s c i e n c e , N e w Y o r k , 1 9 6 5 .

3 ] J e a n B e r s t e l a n d D o m i n i q u e P e r r i n ,

T h e o r y o f c o d e s , P u r e a n d A p p l i e d M a t h e m a t i c s , 1 9 8 5 .

4 ] D a v i d G i l l m a n a n d R o n a l d L . R i v e s t ,

C o m p l e t e v a r i a b l e { l e n g t h x { f r e e { c o d e s ,

D e s i g n s , C o d e s a n d C r y p t o g r a p h y , 5 , 1 0 9 { 1 1 4 , 1 9 9 5 .

1 9 9 5 K l u w e r A c a d e m i c P u b l i s h e r s , B o s t o n .

M a n u f a c t u r e d i n T h e N e t h e r l a n d s .

5 ] C . E . S h a n n o n , P r e d i c t i o n a n d e n t r o p y o f p r i n t e d E n g l i s h ,

B e l l S y s t e m s T e c h n i c a l J o u r n a l 3 0 , 5 0 { 6 4 , 1 9 5 1 .

r. Ahlewede, b.balkenhol, l.khachatrian,Some Properties of Fix-free Codes

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