Post on 14-Apr-2018
7/27/2019 r. Ahlewede, b.balkenhol, l.khachatrian,Some Properties of Fix-free Codes
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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
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
L . K h a c h a t r i a n
y
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
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
7/27/2019 r. Ahlewede, b.balkenhol, l.khachatrian,Some Properties of Fix-free Codes
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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 .
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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
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2 T H E E X I S T E N C E 4
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
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2 T H E E X I S T E N C E 5
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 ] :
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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
7/27/2019 r. Ahlewede, b.balkenhol, l.khachatrian,Some Properties of Fix-free Codes
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3 O N C O M P L E T E F I X { F R E E { C O D E S 7
( 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
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3 O N C O M P L E T E F I X { F R E E { C O D E S 8
( 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 :
7/27/2019 r. Ahlewede, b.balkenhol, l.khachatrian,Some Properties of Fix-free Codes
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3 O N C O M P L E T E F I X { F R E E { C O D E S 9
u
u u
u u
u
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h
h
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h
h
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h
h
(
(
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(
(
& %
' $
& %
' $
( 0 ; 0 ; 0 ; 0 )
( 1 ; 1 ; 1 ; 1 )
( 0 ; 0 ; 0 ; 1 ) ( 1 ; 0 ; 0 ; 0 )
( 0 ; 1 ; 1 ; 1 ) ( 1 ; 1 ; 1 ; 0 )
( 0 ; 1 ; 0 ; 0 )( 0 ; 0 ; 1 ; 0 )
( 1 ; 0 ; 1 ; 1 ) ( 1 ; 1 ; 0 ; 1 )
( 1 ; 0 ; 0 ; 1 )
( 0 ; 1 ; 1 ; 0 )
( 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 ) .
7/27/2019 r. Ahlewede, b.balkenhol, l.khachatrian,Some Properties of Fix-free Codes
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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
6= y
n
:
b i n
? 1
( x
n
) 6= 2 b i n
? 1
( y
n
) ; b i n
? 1
( x
n
) 6= 2 b i n
? 1
( y
n
) + 1 ;
b i n
? 1
( x
n
) 6= b i n
? 1
( y
n
) + 2
n ? 1
b i n
? 1
( y
n
) 6= 2 b i n
? 1
( x
n
) ;
b i n
? 1
( y
n
) 6= 2 b i n
? 1
( x
n
) + 1 ; b i n
? 1
( y
n
) 6= b i n
? 1
( x
n
) + 2
n ? 1
H e n c e , t h e d e t e r m i n a t i o n o f f ( n ) i s a s p e c i a l c a s e o f t h e f o l l o w i n g n u m b e r {
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 .
T h e o r e m 3
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 )
7/27/2019 r. Ahlewede, b.balkenhol, l.khachatrian,Some Properties of Fix-free Codes
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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
i
2
=
n
n
2
:
H e n c e j S
n
0
j =
2
n
?
(
n
n
2
)
2
, a n d l i m
n ! 1
j S
n
0
j
2
n
=
1
2
.
I t i s e a s e l y s e e n t h a t S
n
0
2 I ( B
n
) a n d w e s e t A
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
7/27/2019 r. Ahlewede, b.balkenhol, l.khachatrian,Some Properties of Fix-free Codes
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4 C O M P U T E R R E S U L T S 1 2
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
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4 C O M P U T E R R E S U L T S 1 3
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
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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 .