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A modification of the Zinoviev lower bound for constant weight codes
Discrete Applied Mathematics 11 (1985) 307 310 North-Holland
307
NOTE
A M O D I F I C A T I O N OF T I l E Z I N O V I E V L O W E R B O U N D FOR C O N S T A N T W E I G H T C O D E S liro H O N K A L A Mathematic.~ Department, Universio, o f Turl,'u, Furku, Finland
Heikki H A M A L A I N E N Kexh'ipaloh'k~t, kTnland
Markku KAIKKONEN flelsinki, t;Tttlan(l Received 12 April 1984 Re\ised 3 December 1984 IIl [31 Zinoviev presented a new method to get lower bounds for conslanl ,,'.'eight codes. In this note wc shox~ that a simple modification of the Zinoviev inethod gives further improvemel~tS.
Let A(n, 26, w) denote the maximum number of codewords in any binary code of length n, constant weight w and H a m m i n g distance at least 2i5 and let C be a code which attains the bound A ( n , 2 & w ) . Let k be a k-element subset of the set E = {1,2 . . . . . n}, O<_k
t[ceC,
w(Q)=i}.
Denote the set Uj~o D(L, i) by C(L). The code C(L) has length n k and constant weight w g. If a and b are any two codewords of the original code C with w(a I ) = i and w(bL) =j, O<_i<_g, O
23 - ( i + j ) - (g - i) - (g - j ) = 2a - 2g. Let N(L) be the number of codewords in C(L). Evaluate the sum
N(L). 1 CI: I
0166-218X/85 '$3.3(I
1985, Elsevier Science Publishers B.V. (North-tlolland)
ftonkala el al.
308 For each c e C
there are
,o
\k-iJ
k-element subsets L o f E with the property that w(cL)<_g, a n d for each c e C and for each choice o f L one codeword was formed to the code C(L). Thus
i o i
l.cF_
k- i
I11=~ a n d since N ( L ) < _ A ( n - k , 2 d - 2 g , w - g ) for all L we get the following theorem.
Theorem 1. I f O<_g
A ( n - k , 2 6 - 2 g , w-g)>_ s=o
then
\k-i/_A(n,
2&w)"
Using this f o r m u l a we f o u n d several new lower b o u n d s which are given in Table 1 at the end of this note. If we assume that k - g < f i (instead of g < 6 ) a n d form the set D ( L , i ) , where g<_i<_k, by c h a n g i n g i - g O's to l ' s in each element of the set
{cE_ L I c e C , w(cL)=i} a n d denote U~=g D(L, i) by C(L), then the code C(L) has length n - k , weight w - g a n d m i n i m u m distance at least
constant
26 - ( k - i) - (k - j ) - (i - g) - (j - g) = 2(6 - k + g). In the same way as T h e o r e m 1 we o b t a i n T h e o r e m 1'.
Theorem 1'. I f 0 <_g <_w, 0 <_k < n and k - g < 6, then
A ( n - k, 2 ( 6 - k + g), w-g)>_ '=g
\k i/
A(n, 26, w).
T h e o r e m s 1 a n d 1' have been proved i n d e p e n d e n t l y also by Zinoviev [41 a n d van Pul [21. The coefficient
7)
A modification of the Zinoviev lower bound
309
in T h e o r e m 1 is always smaller t h a n or equal to 1. T h e o r e m 2 presents a case in which the coefficient can be r e p l a c e d by 1. T h e o r e m 2. l f O< w
A(n-2, d-2, w
1)>_A(n,d,w).
P r o o f . Let C be a b i n a r y c o n s t a n t weight code o f length n, m i n i m u m distance at least d and weight w, a n d s u p p o s e that C attains the b o u n d A(n, d, w). D e n o t e the set
{c=clc2...eneCIc, , 1 = c , , = 1} by E. N o w we f o r m a new c o n s t a n t weight code C ' o f length n a n d c o n s t a n t weight w - 1 by c h a n g i n g each c o d e w o r d c - ct c2... c,, o f the code C in the following way: (i) If c ~ C - E , change the last 1 in c to 0. (ii) I f c ~ E , change the last 0 i n c t o 1 and change c,, j a n d c n to O's. W e show that the m i n i m u m distance o f the code C ' is at least d 2. Let a and b be a n y two d i f f e r e n t c o d e w o r d s o f the code C and a ' a n d b ' be the c o d e w o r d s o f the code C ' o b t a i n e d f r o m a a n d b using the rules (i)-(ii). If { a, b } C C - E or { a, b} C E, then clearly d(a', b') >_d - 2. S u p p o s e a ~ C - E a n d b c E, a - al a2... a,,, b = bl be... b,,. Let az, be the last 1 in the c o d e w o r d a a n d bq the last 0 in b. T h e n b,, j - b,, = 1 a n d q _ n - 2. S u p p o s e first t h a t p > q , l f p > n 2, then c l e a r l y d ( a ' , b ' ) > _ d 2. W e m a y a s s u m e that p<_n 2. By the definition o f q bp- l. W h e n we change a a n d b using the rules ( i ) - ( i i ) , a/,- 1 is c h a n g e d to 0 which increases the distance, b,l = 0 is c h a n g e d to 1 and b,, ~ a n d b,, to O's. T h e r e f o r e d(a',b')>_d-2. S e c o n d , if p q, then a v = 1 is c h a n g e d to 0, bp 0 is c h a n g e d to 1 a n d b,, 1 and b,, are again c h a n g e d to O's. A g a i n d(a', b')>_d-2. T h i r d l y , if p < q , then aq=O. N o w bq 0 is c h a n g e d to 1, which increases the distance, a n d g/p, b n 1 a n d b,, are c h a n g e d to O's. So, in all cases d(a', b')>_d-2. W e note that the last two c o o r d i n a t e s in each c o d e w o r d o f the code C ' are O's. W h e n we delete t h e m we o b t a i n a c o n s t a n t weight code o f length n 2, c o n s t a n t weight w 1 a n d m i n i m u m distance at least d - 2 . T h e r e f o r e A(n 2, d - 2 ,
w - l)>_A(n, d, w). T a b l e 1. In T a b l e 1 we give the i m p r o v e d lower b o u n d s . In the c o l u m n 'suit. L ' we give s o m e lower b o u n d s which have been f o u n d using the c o n s t r u c t i o n m e t h o d o f T h e o r e m 1 for a suitable choice o f the set L. In the five last c o l u m n s we give the values o f g, k, n, 2d a n d w to which we a p p l y T h e o r e m 1.
310
H o n k a l a el al.
Table 1. Some improved lower bounds
Ill A122,6,71 A(21,6,71 A(20,6,7) A(19,6,7) A(18,6,7) A(17,6,7) A(16,6,7) A(15,6,7) A122,6,111 A(21,6,1 I) A(20,6, I 1) A(19,6,111 A( 18,6,10) A(23,10,11 ) A122,10,11)
675 465 31(I 228 160 119 90 60 1574 1286 736 332 232 38 38
[3]
320 260 198 141 95
360
Thnl I 682 570 450 338 243 166 108 67 1960 1288 760 408 239 46
Thin 2
Suit. l.
750
109 69 2576
50 46
~
1,
/i
1
2
24
~
s
I
3
24
b
s
I
4
24
s
I
5
24
s
x
I
6
24
8
8
1
7
24
8
8
1
s
24
8
8
1
9
24
8
S
1
2
24
8
12 12
2<~
w
I
3
24
8
I
4
24
8
12
1
5
24
8
12
I
5
23
S
I1
1
I
24
12
12
Acknowledgemenl
The authors
wish to thank
the referee for useful remarks.
References
[1] R.L. Graham and N.J.A. Sloane, Lower bounds for constant weight codes, lEEK 7rans. hlfol-m. Theory 26 11980) 37 43. [21 C.L.M. van Pul, A generalization of the Johnson Bound for constan! weigtlt codes (unpublished). [3] V.A. Zinoviev, On a generalization of Johnson tipper bound, International Workshop: Convohitional Codes, Multi-User Communication, USSR, Sochi 119831 206 -208. [41 V.A. Zinoviev, On the generalization of Johnson bound for equal-weight codes, Problems Inform. Frans. 20 (3) (1984) 104 107.
307
NOTE
A M O D I F I C A T I O N OF T I l E Z I N O V I E V L O W E R B O U N D FOR C O N S T A N T W E I G H T C O D E S liro H O N K A L A Mathematic.~ Department, Universio, o f Turl,'u, Furku, Finland
Heikki H A M A L A I N E N Kexh'ipaloh'k~t, kTnland
Markku KAIKKONEN flelsinki, t;Tttlan(l Received 12 April 1984 Re\ised 3 December 1984 IIl [31 Zinoviev presented a new method to get lower bounds for conslanl ,,'.'eight codes. In this note wc shox~ that a simple modification of the Zinoviev inethod gives further improvemel~tS.
Let A(n, 26, w) denote the maximum number of codewords in any binary code of length n, constant weight w and H a m m i n g distance at least 2i5 and let C be a code which attains the bound A ( n , 2 & w ) . Let k be a k-element subset of the set E = {1,2 . . . . . n}, O<_k
t[ceC,
w(Q)=i}.
Denote the set Uj~o D(L, i) by C(L). The code C(L) has length n k and constant weight w g. If a and b are any two codewords of the original code C with w(a I ) = i and w(bL) =j, O<_i<_g, O
23 - ( i + j ) - (g - i) - (g - j ) = 2a - 2g. Let N(L) be the number of codewords in C(L). Evaluate the sum
N(L). 1 CI: I
0166-218X/85 '$3.3(I
1985, Elsevier Science Publishers B.V. (North-tlolland)
ftonkala el al.
308 For each c e C
there are
,o
\k-iJ
k-element subsets L o f E with the property that w(cL)<_g, a n d for each c e C and for each choice o f L one codeword was formed to the code C(L). Thus
i o i
l.cF_
k- i
I11=~ a n d since N ( L ) < _ A ( n - k , 2 d - 2 g , w - g ) for all L we get the following theorem.
Theorem 1. I f O<_g
A ( n - k , 2 6 - 2 g , w-g)>_ s=o
then
\k-i/_A(n,
2&w)"
Using this f o r m u l a we f o u n d several new lower b o u n d s which are given in Table 1 at the end of this note. If we assume that k - g < f i (instead of g < 6 ) a n d form the set D ( L , i ) , where g<_i<_k, by c h a n g i n g i - g O's to l ' s in each element of the set
{cE_ L I c e C , w(cL)=i} a n d denote U~=g D(L, i) by C(L), then the code C(L) has length n - k , weight w - g a n d m i n i m u m distance at least
constant
26 - ( k - i) - (k - j ) - (i - g) - (j - g) = 2(6 - k + g). In the same way as T h e o r e m 1 we o b t a i n T h e o r e m 1'.
Theorem 1'. I f 0 <_g <_w, 0 <_k < n and k - g < 6, then
A ( n - k, 2 ( 6 - k + g), w-g)>_ '=g
\k i/
A(n, 26, w).
T h e o r e m s 1 a n d 1' have been proved i n d e p e n d e n t l y also by Zinoviev [41 a n d van Pul [21. The coefficient
7)
A modification of the Zinoviev lower bound
309
in T h e o r e m 1 is always smaller t h a n or equal to 1. T h e o r e m 2 presents a case in which the coefficient can be r e p l a c e d by 1. T h e o r e m 2. l f O< w
A(n-2, d-2, w
1)>_A(n,d,w).
P r o o f . Let C be a b i n a r y c o n s t a n t weight code o f length n, m i n i m u m distance at least d and weight w, a n d s u p p o s e that C attains the b o u n d A(n, d, w). D e n o t e the set
{c=clc2...eneCIc, , 1 = c , , = 1} by E. N o w we f o r m a new c o n s t a n t weight code C ' o f length n a n d c o n s t a n t weight w - 1 by c h a n g i n g each c o d e w o r d c - ct c2... c,, o f the code C in the following way: (i) If c ~ C - E , change the last 1 in c to 0. (ii) I f c ~ E , change the last 0 i n c t o 1 and change c,, j a n d c n to O's. W e show that the m i n i m u m distance o f the code C ' is at least d 2. Let a and b be a n y two d i f f e r e n t c o d e w o r d s o f the code C and a ' a n d b ' be the c o d e w o r d s o f the code C ' o b t a i n e d f r o m a a n d b using the rules (i)-(ii). If { a, b } C C - E or { a, b} C E, then clearly d(a', b') >_d - 2. S u p p o s e a ~ C - E a n d b c E, a - al a2... a,,, b = bl be... b,,. Let az, be the last 1 in the c o d e w o r d a a n d bq the last 0 in b. T h e n b,, j - b,, = 1 a n d q _ n - 2. S u p p o s e first t h a t p > q , l f p > n 2, then c l e a r l y d ( a ' , b ' ) > _ d 2. W e m a y a s s u m e that p<_n 2. By the definition o f q bp- l. W h e n we change a a n d b using the rules ( i ) - ( i i ) , a/,- 1 is c h a n g e d to 0 which increases the distance, b,l = 0 is c h a n g e d to 1 and b,, ~ a n d b,, to O's. T h e r e f o r e d(a',b')>_d-2. S e c o n d , if p q, then a v = 1 is c h a n g e d to 0, bp 0 is c h a n g e d to 1 a n d b,, 1 and b,, are again c h a n g e d to O's. A g a i n d(a', b')>_d-2. T h i r d l y , if p < q , then aq=O. N o w bq 0 is c h a n g e d to 1, which increases the distance, a n d g/p, b n 1 a n d b,, are c h a n g e d to O's. So, in all cases d(a', b')>_d-2. W e note that the last two c o o r d i n a t e s in each c o d e w o r d o f the code C ' are O's. W h e n we delete t h e m we o b t a i n a c o n s t a n t weight code o f length n 2, c o n s t a n t weight w 1 a n d m i n i m u m distance at least d - 2 . T h e r e f o r e A(n 2, d - 2 ,
w - l)>_A(n, d, w). T a b l e 1. In T a b l e 1 we give the i m p r o v e d lower b o u n d s . In the c o l u m n 'suit. L ' we give s o m e lower b o u n d s which have been f o u n d using the c o n s t r u c t i o n m e t h o d o f T h e o r e m 1 for a suitable choice o f the set L. In the five last c o l u m n s we give the values o f g, k, n, 2d a n d w to which we a p p l y T h e o r e m 1.
310
H o n k a l a el al.
Table 1. Some improved lower bounds
Ill A122,6,71 A(21,6,71 A(20,6,7) A(19,6,7) A(18,6,7) A(17,6,7) A(16,6,7) A(15,6,7) A122,6,111 A(21,6,1 I) A(20,6, I 1) A(19,6,111 A( 18,6,10) A(23,10,11 ) A122,10,11)
675 465 31(I 228 160 119 90 60 1574 1286 736 332 232 38 38
[3]
320 260 198 141 95
360
Thnl I 682 570 450 338 243 166 108 67 1960 1288 760 408 239 46
Thin 2
Suit. l.
750
109 69 2576
50 46
~
1,
/i
1
2
24
~
s
I
3
24
b
s
I
4
24
s
I
5
24
s
x
I
6
24
8
8
1
7
24
8
8
1
s
24
8
8
1
9
24
8
S
1
2
24
8
12 12
2<~
w
I
3
24
8
I
4
24
8
12
1
5
24
8
12
I
5
23
S
I1
1
I
24
12
12
Acknowledgemenl
The authors
wish to thank
the referee for useful remarks.
References
[1] R.L. Graham and N.J.A. Sloane, Lower bounds for constant weight codes, lEEK 7rans. hlfol-m. Theory 26 11980) 37 43. [21 C.L.M. van Pul, A generalization of the Johnson Bound for constan! weigtlt codes (unpublished). [3] V.A. Zinoviev, On a generalization of Johnson tipper bound, International Workshop: Convohitional Codes, Multi-User Communication, USSR, Sochi 119831 206 -208. [41 V.A. Zinoviev, On the generalization of Johnson bound for equal-weight codes, Problems Inform. Frans. 20 (3) (1984) 104 107.