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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
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
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 
:
We prove that 
is linearly independent. Assume that
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
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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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