TY - JOUR
T1 - A note on Assmus–Mattson type theorems
AU - Miezaki, Tsuyoshi
AU - Munemasa, Akihiro
AU - Nakasora, Hiroyuki
N1 - Funding Information:
The authors are supported by JSPS KAKENHI (17K05155, 18K03217).
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.
PY - 2021/5
Y1 - 2021/5
N2 - In the present paper, we give Assmus–Mattson type theorems for codes and lattices. We show that a binary doubly even self-dual code of length 24m with minimum weight 4m provides a combinatorial 1-design and an even unimodular lattice of rank 24m with minimum norm 2m provides a spherical 3-design. We remark that some of such codes and lattices give t-designs for higher t. As a corollary, we give some restrictions on the weight enumerators of binary doubly even self-dual codes of length 24m with minimum weight 4m. Ternary and quaternary analogues are also given.
AB - In the present paper, we give Assmus–Mattson type theorems for codes and lattices. We show that a binary doubly even self-dual code of length 24m with minimum weight 4m provides a combinatorial 1-design and an even unimodular lattice of rank 24m with minimum norm 2m provides a spherical 3-design. We remark that some of such codes and lattices give t-designs for higher t. As a corollary, we give some restrictions on the weight enumerators of binary doubly even self-dual codes of length 24m with minimum weight 4m. Ternary and quaternary analogues are also given.
KW - Assmus–Mattson theorem
KW - Combinatorial t-design
KW - Harmonic weight enumerator
KW - Self-dual code
KW - Spherical t-design
KW - Spherical theta series
KW - Unimodular lattice
KW - Venkov’s theorem
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U2 - 10.1007/s10623-021-00848-w
DO - 10.1007/s10623-021-00848-w
M3 - Article
AN - SCOPUS:85101404651
SN - 0925-1022
VL - 89
SP - 843
EP - 858
JO - Designs, Codes, and Cryptography
JF - Designs, Codes, and Cryptography
IS - 5
ER -