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 APN functions with the classical Walsh spectrum support 2-designs. On the other hand, we use linear codes and combinatorial designs in order to study important properties of $(n,m)$-functions. In particular, 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.

翻译：本文进一步探讨了$(n,m)$-函数在2-设计构造中的应用。例如，通过证明具有经典Walsh谱的APN函数的线性码支持2-设计，我们给出了扩展Assmus-Mattson定理的新应用。另一方面，我们利用线性码和组合设计来研究$(n,m)$-函数的重要性质。特别地，我们给出了$(n,m)$-平缓函数与$(n,m)$-bent函数的新设计理论刻画，并提供了$(n,m)$-bent函数可扩展性问题的编码理论与设计理论解释。