Infinite families of $ 3 $-designs from o-polynomials

Cunsheng Ding; Chunming Tang

doi:10.3934/amc.2020082

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2021, Volume 15, Issue 4: 557-573. Doi: 10.3934/amc.2020082

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Infinite families of $ 3 $-designs from o-polynomials

Cunsheng Ding^1, and
Chunming Tang^2, ,

1.
Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
2.
School of Mathematics and Information, China West Normal University, Sichuan Nanchong, 637002, China

^* Corresponding author: Chunming Tang
^* Corresponding author: Chunming Tang

Received: October 31, 2019

Revised: January 31, 2020

Early access: April 2020

Published: November 2021

C. Ding's research was supported by the Hong Kong Research Grants Council, Proj. No. 16300415. C. Tang was supported by National Natural Science Foundation of China (Grant No. 11871058) and China West Normal University (14E013, CXTD2014-4 and the Meritocracy Research Funds)

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Abstract

A classical approach to constructing combinatorial designs is the group action of a $ t $-transitive or $ t $-homogeneous permutation group on a base block, which yields a $ t $-design in general. It is open how to use a $ t $-transitive or $ t $-homogeneous permutation group to construct a $ (t+1) $-design in general. It is known that the general affine group $ {\mathrm{GA}}_1( {\mathrm{GF}}(q)) $ is doubly transitive on $ {\mathrm{GF}}(q) $. The classical theorem says that the group action by $ {\mathrm{GA}}_1( {\mathrm{GF}}(q)) $ yields $ 2 $-designs in general. The main objective of this paper is to construct $ 3 $-designs with $ {\mathrm{GA}}_1( {\mathrm{GF}}(q)) $ and o-polynomials. O-polynomials (equivalently, hyperovals) were used to construct only $ 2 $-designs in the literature. This paper presents for the first time infinite families of $ 3 $-designs from o-polynomials (equivalently, hyperovals).

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Mathematics Subject Classification: Primary: 51E21, 05B05, 12E10.

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