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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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### Keywords

- Characteristic Vector
- Binary Code
- Small Index
- Multilinear Polynomial
- Conjecture Relate