An Assmus-Mattson theorem for codes over commutative association schemes

作者：Morales John Vincent S; Tanaka Hajime^*

来源：Designs, Codes and Cryptography, 2018, 86(5): 1039-1062.

DOI：10.1007/s10623-017-0376-y

• Assmus-Mattson theorem
• Code
• Design
• Association scheme
• Terwilliger algebra
• Multivariable polynomial interpolation

We prove an Assmus-Mattson-type theorem for block codes where the alphabet is the vertex set of a commutative association scheme (say, with s classes). This in particular generalizes the Assmus-Mattson-type theorems for -linear codes due to Tanabe (Des Codes Cryptogr 30:169-185, 2003) and Shin et al. (Des Codes Cryptogr 31:75-92, 2004), as well as the original theorem by Assmus and Mattson (J Comb Theory 6:122-151, 1969). The weights of a code are s-tuples of non-negative integers in this case, and the conditions in our theorem for obtaining t-designs from the code involve concepts from polynomial interpolation in s variables. The Terwilliger algebra is the main tool to establish our results.

• 出版日期2018-5