Designs, Codes and Cryptography

, Volume 86, Issue 5, pp 1039–1062 | Cite as

An Assmus–Mattson theorem for codes over commutative association schemes



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 (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.


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

Mathematics Subject Classification

05E30 94B05 05B05 



The authors thank Masaaki Harada for helpful discussions. HT was supported in part by JSPS KAKENHI Grant No. 25400034.


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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Graduate School of Information SciencesTohoku UniversitySendaiJapan
  2. 2.Research Center for Pure and Applied Mathematics, Graduate School of Information SciencesTohoku UniversitySendaiJapan

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