An Assmus-Mattson theorem for codes over commutative association schemes

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.

24 pages

References in corpus (2)

• A Lie theoretic interpretation of multivariate hypergeometric polynomials
• The Rahman polynomials and the Lie algebra sl_3(C)