An Assmus-Mattson theorem for codes over commutative association schemes

TL;DR

This paper extends the Assmus-Mattson theorem to codes over commutative association schemes, using Terwilliger algebra and polynomial interpolation to identify when codes form combinatorial designs.

Contribution

It generalizes the Assmus-Mattson theorem to codes over commutative association schemes, broadening its applicability and introducing new algebraic techniques.

Findings

Established a new Assmus-Mattson-type theorem for codes over association schemes.

Connected code weights to polynomial interpolation in multiple variables.

Utilized Terwilliger algebra to prove the main results.

Abstract

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 $\mathbb{Z}_4$-linear codes due to Tanabe (2003) and Shin, Kumar, and Helleseth (2004), as well as the original theorem by Assmus and Mattson (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.