On Lee Association Scheme over $mathbb{Z}_4$, Terwilliger algebras and the Assmus-Mattson Theorem (Research on finite groups, algebraic combinatorics and vertex operator algebras)

Abstract

Let C denote a linear code of length n over a finite field $Gamma$_{q} and let C^{perp} denote the corresponding dual. The Assmus-Mattson theorem states that combinatorial designs can be obtained from the supports of codewords of C with fixed weight type whenever the Hamming weight enumerators of C and C^{perp} satisfy certain conditions. This famous result has been strengthened and extended to many different settings including the Assmus-Mattson type theorems for mathbb{Z}_{4}-linear codes due to Tanabe (2003), and due to Shin, Kumar and Helleseth (2004). In this paper, we discuss an Assmus-Mattson type theorem for block codes where the alphabet is the vertex set of some commutative association scheme. This particular theorem generalizes the Assmus-Mattson type theorems mentioned above as well as the original. In proving our results, we invoke several techniques from multivariable polynomial interpolation and from the representation theory of Terwilliger algebras. This is based on a joint work with Hajime Tanaka.