Linear codes and incidence structures of bent functions and their generalizations

In this paper, we consider further applications of (n, m)-functions for the construction of 2-designs. For instance, we provide a new application of the extended Assmus-Mattson theorem, by showing that linear codes of certain APN functions with the classical Walsh spectrum support 2-designs. With this result, we give several sufficient conditions for an APN function with the classical Walsh spectrum to be CCZ-inequivalent to a quadratic one. On the other hand, we use linear codes and combinatorial designs in order to study important properties of (n, m)-functions. In particular, we provide a characterization of a quadratic Boolean bent function by means of the 2-transitivity of its automorphism group. Finally, we give a new design-theoretic characterization of (n, m)-plateaued and (n, m)-bent functions and provide a coding-theoretic as well as a design-theoretic interpretation of the extendability problem for (n, m)-bent functions.