Infinite families of 3‐designs from APN functions

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 $\left(t+1\right)$‐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 almost perfect nonlinear 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.