TY - JOUR

T1 - An Assmus–Mattson theorem for codes over commutative association schemes

AU - Morales, John Vincent S.

AU - Tanaka, Hajime

N1 - Funding Information:
Acknowledgements The authors thank Masaaki Harada for helpful discussions. HT was supported in part by JSPS KAKENHI Grant No. 25400034.

PY - 2018/5/1

Y1 - 2018/5/1

N2 - 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 Z4-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.

AB - 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 Z4-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.

KW - Assmus–Mattson theorem

KW - Association scheme

KW - Code

KW - Design

KW - Multivariable polynomial interpolation

KW - Terwilliger algebra

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U2 - 10.1007/s10623-017-0376-y

DO - 10.1007/s10623-017-0376-y

M3 - Article

AN - SCOPUS:85021207713

VL - 86

SP - 1039

EP - 1062

JO - Designs, Codes, and Cryptography

JF - Designs, Codes, and Cryptography

SN - 0925-1022

IS - 5

ER -