# Alon-Babai-Suzuki's Conjecture Related to Binary Codes in Nonmodular Version

- K-W Hwang
^{1}Email author, - T Kim
^{2}, - LC Jang
^{3}, - P Kim
^{4}and - Gyoyong Sohn
^{5}

**2010**:546015

https://doi.org/10.1155/2010/546015

© K.-W. Hwang et al. 2010

**Received: **23 August 2009

**Accepted: **22 January 2010

**Published: **31 January 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
,
.

## Keywords

## 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]).

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]).

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.

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

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 .

## Declarations

### 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.

## Authors’ Affiliations

## References

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## Copyright

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