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

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

(2.1)

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

(2.2)

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 :

(2.3)

We prove that is linearly independent. Assume that

(2.4)

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

(2.5)

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

1. 1.

Ray-Chaudhuri DK, Wilson RM: On -designs. Osaka Journal of Mathematics 1975, 12(3):737–744.

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-O

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)90103-1

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/s00493-003-0031-2

5. 5.

Snevily HS: A generalization of the Ray-Chaudhuri-Wilson theorem. Journal of Combinatorial Designs 1995, 3(5):349–352. 10.1002/jcd.3180030505

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.2005.11.002

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: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.

Author information

Correspondence to K-W Hwang.

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