Infinite families of 3-designs from APN functions
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Combinatorial t-designs have nice applications in coding theory, finite geometries and several engineering areas. A classical method for constructing t-designs is by the action of a permutation group that is t-transitive or t-homogeneous on a point set. This approach produces t-designs, but may not yield (t+1)-designs. The objective of this paper is to study how to obtain 3-designs with 2-transitive permutation groups. The incidence structure formed by the orbits of a base block under the action of the general affine groups, which are 2-transitive, is considered. A characterization of such incidence structure to be a 3-design is presented, and a sufficient condition for the stabilizer of a base block to be trivial is given. With these general results, infinite families of 3-designs are constructed by employing APN functions. Some 3-designs presented in this paper give rise to self-dual binary codes or linear codes with optimal or best parameters known. Several conjectures on 3-designs and binary codes are also presented.
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