A note on the Assmus–Mattson theorem for some binary codes

The article delves into the Assmus–Mattson theorem, a pivotal result in coding and design theory, and presents a strengthened version of the theorem. It explores the implications of this strengthening, particularly in the context of binary codes and their support designs. The authors discuss the relevance of the Assmus–Mattson theorem to other areas such as lattice theory and vertex operator algebra, highlighting the Venkov and Höhn theorems. The research is inspired by Lehmer's conjecture and aims to determine the conditions under which certain codes exhibit specific design properties. The article is organized into sections that cover background material, the main theorems, proofs, and concluding remarks, making it a comprehensive resource for specialists in the field.