Alon-Babai-Suzuki's Conjecture Related to Binary Codes in Nonmodular Version
© K.-W. Hwang et al. 2010
Received: 23 August 2009
Accepted: 22 January 2010
Published: 31 January 2010
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 , .
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 ).
Theorem (Alon et al. ).
Theorem (Snevily ).
Theorem (Snevily ).
Conjecture 1.5 (Snevily ).
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 ).
Hwang and Sheikh  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 ).
Theorem (Hwang and Sheikh ).
Theorem 1.9 (Alon et al. ).
Conjecture 1.10 (Alon et al. ).
We prove an Alon-Babai-Suzuki's conjecture in non-modular version.
2. Proof of Theorem
Proof of Theorem 1.11.
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 .
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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