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

Open Access
Research Article

Abstract

Let Open image in new window and Open image in new window be sets of nonnegative integers. Let Open image in new window be a family of subsets of Open image in new window with Open image in new window for each Open image in new window and Open image in new window for any Open image in new window . Every subset Open image in new window of Open image in new window can be represented by a binary code a Open image in new window such that Open image in new window if Open image in new window and Open image in new window if Open image in new window . Alon et al. made a conjecture in 1991 in modular version. We prove Alon-Babai-Sukuki's Conjecture in nonmodular version. For any Open image in new window and Open image in new window with Open image in new window Open image in new window , Open image in new window .

Keywords

Characteristic Vector Binary Code Small Index Multilinear Polynomial Conjecture Relate 

1. Introduction

In this paper, Open image in new window stands for a family of subsets of Open image in new window , Open image in new window and Open image in new window , where Open image in new window for all Open image in new window , Open image in new window for all Open image in new window . The variable Open image in new window will stand as a shorthand for the Open image in new window -dimensional vector variable Open image in new window . Also, since these variables will take the values only Open image in new window and Open image in new window , all the polynomials we will work with will be reduced modulo the relation Open image in new window . We define the characteristic vector Open image in new window of Open image in new window such that Open image in new window if Open image in new window and Open image in new window if Open image in new window . We will present some results in this paper that give upper bounds on the size of Open image in new window under various conditions. Below is a list of related results by others.

Theorem 1.1 (Ray-Chaudhuri and Wilson [1]).

If Open image in new window , and Open image in new window is any set of nonnegative integers with Open image in new window , then Open image in new window .

Theorem (Alon et al. [2]).

If Open image in new window and Open image in new window are two sets of nonnegative integers with Open image in new window , for every Open image in new window , then Open image in new window .

Theorem (Snevily [3]).

If Open image in new window and Open image in new window are any sets such that Open image in new window Open image in new window , then Open image in new window .

Theorem (Snevily [4]).

Let Open image in new window and Open image in new window be sets of nonnegative integers such that Open image in new window . Then, Open image in new window .

Conjecture 1.5 (Snevily [5]).

For any Open image in new window and Open image in new window with Open image in new window Open image in new window Open image in new window , Open image in new window .

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 Open image in new window and Open image in new window with Open image in new window Open image in new window Open image in new window , Open image in new window .

Hwang and Sheikh [6] proved the bound of Conjecture 1.6 when Open image in new window is a consecutive set. The second theorem we prove is a special case of Conjecture 1.6 with the extra condition that Open image in new window . These two theorems are stated hereunder.

Theorem 1.7 (Hwang and Sheikh [6]).

Let Open image in new window where Open image in new window , Open image in new window , and Open image in new window . Let Open image in new window be such that Open image in new window for each Open image in new window , Open image in new window , and Open image in new window for any Open image in new window . Then Open image in new window .

Theorem (Hwang and Sheikh [6]).

Let Open image in new window , Open image in new window , and Open image in new window be such that Open image in new window for each Open image in new window , Open image in new window for any Open image in new window , and Open image in new window . If Open image in new window , then Open image in new window .

Theorem 1.9 (Alon et al. [2]).

Let Open image in new window and Open image in new window be subsets of Open image in new window such that Open image in new window , where Open image in new window is a prime and Open image in new window a family of subsets of Open image in new window such that Open image in new window Open image in new window for all Open image in new window and Open image in new window Open image in new window for Open image in new window . If Open image in new window , and Open image in new window Open image in new window , then Open image in new window .

Conjecture 1.10 (Alon et al. [2]).

Let Open image in new window and Open image in new window be subsets of Open image in new window such that Open image in new window , where Open image in new window is a prime and Open image in new window a family of subsets of Open image in new window such that Open image in new window Open image in new window Open image in new window for all Open image in new window and Open image in new window Open image in new window for Open image in new window . If Open image in new window Open image in new window , then Open image in new window .

In [2], Alon et al. proved their conjectured bound under the extra conditions that Open image in new window and Open image in new window Open image in new window . Qian and Ray-Chaudhuri [7] proved that if Open image in new window instead of Open image in new window Open image in new window , then the above bound holds.

We prove an Alon-Babai-Suzuki's conjecture in non-modular version.

Theorem.

Let Open image in new window , Open image in new window be two sets of nonnegative integers and let Open image in new window be such that Open image in new window for each Open image in new window , Open image in new window for any Open image in new window , and Open image in new window Open image in new window . then Open image in new window .

2. Proof of Theorem

Proof of Theorem 1.11.

For each Open image in new window , consider the polynomial

where Open image in new window is the characteristic vector of Open image in new window and Open image in new window is the characteristic vector of Open image in new window . Let Open image in new window the characteristic vector of Open image in new window , and Open image in new window be the characteristic vector of Open image in new window .

We order Open image in new window by size of Open image in new window , that is, Open image in new window if Open image in new window . We substitute the characteristic vector Open image in new window of Open image in new window by order of size of Open image in new window . Clearly, Open image in new window for Open image in new window and Open image in new window for Open image in new window . Assume that

We prove that Open image in new window is linearly independent. Assume that this is false. Let Open image in new window be the smallest index such that Open image in new window . We substitute Open image in new window into the above equation. Then we get Open image in new window . We get a contradiction. So Open image in new window is linearly independent. Let Open image in new window be the family of subsets of Open image in new window with size at most Open image in new window , which is ordered by size, that is, Open image in new window if Open image in new window , where Open image in new window . Let Open image in new window denote the characteristic vector of Open image in new window . We define the multilinear polynomial Open image in new window in Open image in new window variables for each Open image in new window :
We prove that Open image in new window is linearly independent. Assume that
Choose the smallest size of Open image in new window . Let Open image in new window be the characteristic vector of Open image in new window . We substitute Open image in new window into the above equation. We know that Open image in new window and Open image in new window for any Open image in new window . Since Open image in new window Open image in new window , we get Open image in new window . If we follow the same process, then the family Open image in new window is linearly independent. Next, we prove that Open image in new window is linearly independent. Now, assume that

Let Open image in new window be the smallest size of Open image in new window . We substitute the characteristic vector Open image in new window of Open image in new window into the above equation. Since Open image in new window , Open image in new window for all Open image in new window . We only get Open image in new window . So Open image in new window . By the same way, choose the smallest size from Open image in new window after deleting Open image in new window . We do the same process. We also can get Open image in new window . By the same process, we prove that all Open image in new window . We prove that Open image in new window is linearly independent.

Any polynomial in the set Open image in new window can be represented by a linear combination of multilinear monomials of degree Open image in new window . The space of such multilinear polynomials has dimension Open image in new window . We found Open image in new window linearly independent polynomials with degree at most Open image in new window . So Open image in new window . Thus Open image in new window .

Notes

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

  1. 1.
    Ray-Chaudhuri DK, Wilson RM: On -designs. Osaka Journal of Mathematics 1975, 12(3):737–744.MathSciNetMATHGoogle Scholar
  2. 2.
    Alon N, Babai L, Suzuki H: Multilinear polynomials and Frankl-Ray-Chaudhuri–Wilson type intersection theorems. Journal of Combinatorial Theory. Series A 1991, 58(2):165–180. 10.1016/0097-3165(91)90058-OMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    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)90103-1MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Snevily HS: A sharp bound for the number of sets that pairwise intersect at positive values. Combinatorica 2003, 23(3):527–533. 10.1007/s00493-003-0031-2MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Snevily HS: A generalization of the Ray-Chaudhuri-Wilson theorem. Journal of Combinatorial Designs 1995, 3(5):349–352. 10.1002/jcd.3180030505MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Hwang K-W, Sheikh N: Intersection families and Snevily's conjecture. European Journal of Combinatorics 2007, 28(3):843–847. 10.1016/j.ejc.2005.11.002MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Qian J, Ray-Chaudhuri DK: On mod- Alon-Babai-Suzuki inequality. Journal of Algebraic Combinatorics 2000, 12(1):85–93. 10.1023/A:1008715718935MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© K.-W. Hwang et al. 2010

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  • K-W Hwang
    • 1
  • T Kim
    • 2
  • LC Jang
    • 3
  • P Kim
    • 4
  • Gyoyong Sohn
    • 5
  1. 1.Department of MathematicsDonga-A UniversityPusanSouth Korea
  2. 2.Division of General Edu.-Math.,Kwangwoon UniversitySeoulSouth Korea
  3. 3.Department of Mathematics and Computer ScienceKonkook UniversityChungjuSouth Korea
  4. 4.Department of MathematicsKyungpook National UniversityTaeguSouth Korea
  5. 5.Department of Computer ScienceChungbuk National UniversityCheongjuSouth Korea

Personalised recommendations