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

  • * Corresponding author: Chunming Tang

    * Corresponding author: Chunming Tang
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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  • 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).

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


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