Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 546015, 4 pages
doi:10.1155/2010/546015
Research Article
Alon-Babai-Suzuki’s Conjecture Related to
Binary Codes in Nonmodular Version
K.-W. Hwang,1 T. Kim,2 L. C. Jang,3 P. Kim,4 and Gyoyong Sohn5
1
Department of Mathematics, Donga-A University, Pusan 604-714, South Korea
Division of General Edu.-Math., Kwangwoon University, Seoul 139-701, South Korea
3
Department of Mathematics and Computer Science, Konkook University, Chungju 139-701, South Korea
4
Department of Mathematics, Kyungpook National University, Taegu 702-701, South Korea
5
Department of Computer Science, Chungbuk National University, Cheongju 361-763, South Korea
2
Correspondence should be addressed to K.-W. Hwang, khwang7@kookmin.ac.kr
Received 23 August 2009; Accepted 22 January 2010
Academic Editor: Ram N. Mohapatra
Copyright q 2010 K.-W. Hwang et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Let K {k1 , k2 , . . . , kr } and L {l1 , l2 , . . . , ls } be sets of nonnegative integers. Let F {F1 , F2 , . . . , Fm } be a family of subsets of n with Fi ∈ K for each i and |Fi ∩ Fj | ∈ L for any
i/
j. Every subset Fe of n can be represented by a binary code a a1 , a2 , . . . , an such that ai 1
∈ Fe . Alon et al. made a conjecture in 1991 in modular version. We prove
if i ∈ Fe and ai 0 if i /
Alon-Babai-Sukuki’s
in nonmodular version. For any K and L with n ≥ s max ki ,
Conjecture
|F| ≤
n−1
s
n−1
s−1
··· n−1
s−2r1
.
1. Introduction
In this paper, F stands for a family of subsets of n {1, 2, . . . , n}, K {k1 , . . . , kr }, and
L {l1 , . . . , ls }, where |Fi | ∈ K for all Fi ∈ F, |Fi ∩ Fj | ∈ L for all Fi , Fj ∈ F, i / j. The variable
x will stand as a shorthand for the n-dimensional vector variable x1 , x2 , . . . , xn . Also, since
these variables will take the values only 0 and 1, all the polynomials we will work with will be
reduced modulo the relation xi2 xi . We define the characteristic vector vi vi1 , vi2 , . . . , vin ∈ Fi . We will present some results in this paper
of Fi such that vij 1 if j ∈ Fi and vij 0 if j /
that give upper bounds on the size of F under various conditions. Below is a list of related
results by others.
Theorem 1.1 Ray-Chaudhuri and
1. If K {k}, and L is any set of nonnegative
nWilson
integers with k > max lj , then |F| ≤ s .
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Journal of Inequalities and Applications
Theorem 1.2 Alon
net
al.2.
If K and
Lnaretwo sets of nonnegative integers with ki > s − r, for
n
every i, then |F| ≤ s s−1 · · · s−r1 .
Theorem 1.3 Snevily 3. If K and L are any sets such that min ki > max lj , then |F| ≤
n−1 n−1 ··· .
s−1
n−1 s
0
Theorem 1.4 Snevily 4. Let K and L be sets of nonnegative integers such that min ki > max lj .
n−1 n−1 n−1 ··· .
Then, |F| ≤
s
s−1
s−2r1
Conjecture 1.5 Snevily 5. For any K and L with min ki > max lj , |F| ≤
n
s
.
In the same paper in which he stated the above conjecture, Snevily mentions that it
seems hard to prove the above bound and states the following weaker conjecture.
n−1 n−1 Conjecture 1.6 Snevily 5. For any K and L with min ki > max lj , |F| ≤
···
s
s−1
n−1 .
s−r
Hwang and Sheikh 6 proved the bound of Conjecture 1.6 when K is a consecutive
set. The second theorem we prove is a special case of Conjecture 1.6 with the extra condition
that m
/ ∅. These two theorems are stated hereunder.
i1 Fi Theorem 1.7 Hwang and Sheikh 6. Let K {k1 , k2 , . . . , kr } where ki k1 i − 1, k1 > s − r,
. . . ,Fm } be such that
∈ L, and
and L {l1 , l2 , . . . , ls }. Let F {F
1 , F2 ,
n−1|F
i | ∈ K for each i, |Fi | /
n−1
n−1
|Fi ∩ Fj | ∈ L for any i /
··· .
j. Then |F| ≤
s
s−1
s−r
Theorem 1.8 Hwang and Sheikh 6. Let K {k1 , k2 , . . . , kr }, L {l1 , l2 , . . . , ls }, and F j, and ki > s − r. If
{F1 , F2 , . . . , Fm } be such that |Fi | ∈ K for each i, |Fi ∩ Fj | ∈ L for any i /
n−1 n−1 n−1 m
··· .
∅, then |F| ≤
i1 Fi /
s
s−1
s−r
Theorem 1.9 Alon et al. 2. Let K and L be subsets of {0, 1, ..., p − 1} such that K ∩ L ∅,
where p is a prime and F {F1 , F2 , . . . , Fm } a family of subsets of n such that |Fi |mod p ∈ K
j. If rs − r 1 ≤ p − 1, and n ≥ s max ki , then
for all Fi ∈ F and |Fi ∩ Fj |mod p ∈ L for i /
n n n |F| ≤ s s−1 · · · s−r1 .
Conjecture 1.10 Alon et al. 2. Let K and L be subsets of {0, 1, . . . , p − 1} such that K ∩ L ∅,
where p is a prime and F {F1 , F2 , . . . , Fm } a family of subsets of n such
for
|Fni |modp
∈ K
n that
n
j. If n ≥ smax ki , then |F| ≤ s s−1 · · · s−r1 .
all Fi ∈ F and |Fi ∩Fj |mod p ∈ L for i /
In 2, Alon et al. proved their conjectured bound under the extra conditions that rs −
r 1 ≤ p − 1 and n ≥ s max ki . Qian and Ray-Chaudhuri 7 proved that if n > 2s − r instead
of n ≥ s max ki , then the above bound holds.
We prove an Alon-Babai-Suzuki’s conjecture in non-modular version.
Theorem 1.11. Let K {k1 , k2 , . . . , kr }, L {l1 , l2 , . . . , ls } be two sets of nonnegative integers and
j, and n ≥ smaxi |Fi |.
let F {F1 , F2 , . . . , Fm } be such that |Fi | ∈ K for each i, |Fi ∩Fj | ∈ L for any i /
n n n then |F| ≤ s s−1 · · · s−r1 .
Journal of Inequalities and Applications
3
2. Proof of Theorem
Proof of Theorem 1.11. For each Fi ∈ F, consider the polynomial
fi x vi · x − ki − lj ,
2.1
j
lj <|Fi |
where vi is the characteristic vector of Fi and vi∗ is the characteristic vector of Fi∗ Fi − {1}.
Let vi the characteristic vector of Fic , and vi ∗ be the characteristic vector of Fic ∗ .
We order {Fi } by size of Fi , that is, |Fj | ≤ |Fk | if j < k. We substitute the characteristic
0 for 1 ≤ i ≤ m and fi vj 0 for
vector vi of Fic by order of size of Fi . Clearly, fi vi /
1 ≤ j < i ≤ m. Assume that
αi fi x 0.
2.2
i
We prove that {fi x} is linearly independent. Assume that this is false. Let i0 be the smallest
0. We substitute vi0 into the above equation. Then we get αi0 fi0 vi0 0.
index such that αi0 /
We get a contradiction. So {fi x} is linearly independent. Let E {E1 , . . . , Ee } be the family
of subsets of n with size at most s − r, which is ordered by size, that is, |Ei | ≤ |Ej | if i < j,
n
where e s−r
i0 i . Let ui denote the characteristic vector of Ei . We define the multilinear
polynomial gi in n variables for each Ei :
gi x r
n
xt − n − kl t1
l1
j∈Ei
xj .
2.3
We prove that {gi x} is linearly independent. Assume that
βi gi x 0.
i
2.4
Choose the smallest size of Ei . Let ui be the characteristic vector of Ei . We substitute ui into
0 and gj ui 0 for any i < j. Since n ≥ s max ki ,
the above equation. We know that gi ui /
we get βi 0. If we follow the same process, then the family {gi x} is linearly independent.
Next, we prove that {fi x, gi x} is linearly independent. Now, assume that
αi fi x βi gi x 0.
i
i
2.5
Let F1 be the smallest size of Fi . We substitute the characteristic vector v1 of F1c into the above
equation. Since |Fic | n − kl , gi v1 0 for all i. We only get α1 f1 v1 0. So α1 0. By the
same way, choose the smallest size from {Fi } after deleting F1 . We do the same process. We
also can get α2 0. By the same process, we prove that all αi 0. We prove that {fi x, gi x}
is linearly independent.
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Journal of Inequalities and Applications
Any polynomial in the set {fi x, gi x} can be represented by a linear combination
of multilinear monomials
of degree ≤ s.
The space of such multilinear polynomials has
s−r n s n dimension i0 i . We found |F| i0 i linearly independent polynomials with degree
n n n n s n at most s. So |F| s−r
≤ i0 i . Thus |F| ≤ s s−1 · · · s−r1 .
i0 i
Acknowledgments
The authors thank Zoltán Füredi for encouragement to write this paper. The present research
has been conducted by the research grant of the Kwangwoon University in 2009.
References
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