The Assmus-Mattson theorem [1] provides a direct link between linear codes and designs. This paper describes new approaches to this theorem.

Tanaka proves three versions of the theorem and two corollaries. Tanaka uses the Terwilliger algebra--in fact, only basic properties of the irreducible modules of the Terwilliger algebra--to give an alternative proof of the Assmus-Mattson theorem and to state three versions of the theorem. The second version in the paper coincides with Delsarte’s version of the Assmus-Mattson theorem [2]. The author also presents two corollaries of the new approach. A whole section is devoted to comparing the presented versions of the Assmus-Mattson theorem. Besides this comparison and numerous examples, a new proof for the minimum distance bound for s-regular codes is also given.

Apart from contributing to this field of research, the paper also contains a lot of interesting background material, making it accessible to the nonspecialized reader.