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Alon-Babai-Suzuki's Conjecture Related to Binary Codes in Nonmodular Version
Journal of Inequalities and Applications volume 2010, Article number: 546015 (2010)
Abstract
Let and
be sets of nonnegative integers. Let
be a family of subsets of
with
for each
and
for any
. Every subset
of
can be represented by a binary code a
such that
if
and
if
. Alon et al. made a conjecture in 1991 in modular version. We prove Alon-Babai-Sukuki's Conjecture in nonmodular version. For any
and
with
,
.
1. Introduction
In this paper, stands for a family of subsets of
,
and
, where
for all
,
for all
. The variable
will stand as a shorthand for the
-dimensional vector variable
. Also, since these variables will take the values only
and
, all the polynomials we will work with will be reduced modulo the relation
. We define the characteristic vector
of
such that
if
and
if
. We will present some results in this paper that give upper bounds on the size of
under various conditions. Below is a list of related results by others.
Theorem 1.1 (Ray-Chaudhuri and Wilson [1]).
If , and
is any set of nonnegative integers with
, then
.
Theorem (Alon et al. [2]).
If and
are two sets of nonnegative integers with
, for every
, then
.
Theorem (Snevily [3]).
If and
are any sets such that
, then
.
Theorem (Snevily [4]).
Let and
be sets of nonnegative integers such that
. Then,
.
Conjecture 1.5 (Snevily [5]).
For any and
with
,
.
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.
Conjecture 1.6 (Snevily [5]).
For any and
with
,
.
Hwang and Sheikh [6] proved the bound of Conjecture 1.6 when is a consecutive set. The second theorem we prove is a special case of Conjecture 1.6 with the extra condition that
. These two theorems are stated hereunder.
Theorem 1.7 (Hwang and Sheikh [6]).
Let where
,
, and
. Let
be such that
for each
,
, and
for any
. Then
.
Theorem (Hwang and Sheikh [6]).
Let ,
, and
be such that
for each
,
for any
, and
. If
, then
.
Theorem 1.9 (Alon et al. [2]).
Let and
be subsets of
such that
, where
is a prime and
a family of subsets of
such that
for all
and
for
. If
, and
, then
.
Conjecture 1.10 (Alon et al. [2]).
Let and
be subsets of
such that
, where
is a prime and
a family of subsets of
such that
for all
and
for
. If
, then
.
In [2], Alon et al. proved their conjectured bound under the extra conditions that and
. Qian and Ray-Chaudhuri [7] proved that if
instead of
, then the above bound holds.
We prove an Alon-Babai-Suzuki's conjecture in non-modular version.
Theorem.
Let ,
be two sets of nonnegative integers and let
be such that
for each
,
for any
, and
. then
.
2. Proof of Theorem
Proof of Theorem 1.11.
For each , consider the polynomial
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F546015/MediaObjects/13660_2009_Article_2185_Equ1_HTML.gif)
where is the characteristic vector of
and
is the characteristic vector of
. Let
the characteristic vector of
, and
be the characteristic vector of
.
We order by size of
, that is,
if
. We substitute the characteristic vector
of
by order of size of
. Clearly,
for
and
for
. Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F546015/MediaObjects/13660_2009_Article_2185_Equ2_HTML.gif)
We prove that is linearly independent. Assume that this is false. Let
be the smallest index such that
. We substitute
into the above equation. Then we get
. We get a contradiction. So
is linearly independent. Let
be the family of subsets of
with size at most
, which is ordered by size, that is,
if
, where
. Let
denote the characteristic vector of
. We define the multilinear polynomial
in
variables for each
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F546015/MediaObjects/13660_2009_Article_2185_Equ3_HTML.gif)
We prove that is linearly independent. Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F546015/MediaObjects/13660_2009_Article_2185_Equ4_HTML.gif)
Choose the smallest size of . Let
be the characteristic vector of
. We substitute
into the above equation. We know that
and
for any
. Since
, we get
. If we follow the same process, then the family
is linearly independent. Next, we prove that
is linearly independent. Now, assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F546015/MediaObjects/13660_2009_Article_2185_Equ5_HTML.gif)
Let be the smallest size of
. We substitute the characteristic vector
of
into the above equation. Since
,
for all
. We only get
. So
. By the same way, choose the smallest size from
after deleting
. We do the same process. We also can get
. By the same process, we prove that all
. We prove that
is linearly independent.
Any polynomial in the set can be represented by a linear combination of multilinear monomials of degree
. The space of such multilinear polynomials has dimension
. We found
linearly independent polynomials with degree at most
. So
. Thus
.
References
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Snevily HS: On generalizations of the de Bruijn-Erdős theorem. Journal of Combinatorial Theory. Series A 1994, 68(1):232–238. 10.1016/0097-3165(94)90103-1
Snevily HS: A sharp bound for the number of sets that pairwise intersect at positive values. Combinatorica 2003, 23(3):527–533. 10.1007/s00493-003-0031-2
Snevily HS: A generalization of the Ray-Chaudhuri-Wilson theorem. Journal of Combinatorial Designs 1995, 3(5):349–352. 10.1002/jcd.3180030505
Hwang K-W, Sheikh N: Intersection families and Snevily's conjecture. European Journal of Combinatorics 2007, 28(3):843–847. 10.1016/j.ejc.2005.11.002
Qian J, Ray-Chaudhuri DK: On mod- Alon-Babai-Suzuki inequality. Journal of Algebraic Combinatorics 2000, 12(1):85–93. 10.1023/A:1008715718935
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.
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Hwang, KW., Kim, T., Jang, L. et al. Alon-Babai-Suzuki's Conjecture Related to Binary Codes in Nonmodular Version. J Inequal Appl 2010, 546015 (2010). https://doi.org/10.1155/2010/546015
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DOI: https://doi.org/10.1155/2010/546015