A note on Assmus–Mattson type theorems

Tsuyoshi Miezaki; Akihiro Munemasa; Hiroyuki Nakasora

doi:10.1007/s10623-021-00848-w

A note on Assmus–Mattson type theorems

Tsuyoshi Miezaki^*, Akihiro Munemasa, Hiroyuki Nakasora

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

9 Citations (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	843-858
Number of pages	16
Journal	Designs, Codes, and Cryptography
Volume	89
Issue number	5
DOIs	https://doi.org/10.1007/s10623-021-00848-w
Publication status	Published - 2021 May
Externally published	Yes

Keywords

Assmus–Mattson theorem
Combinatorial t-design
Harmonic weight enumerator
Self-dual code
Spherical t-design
Spherical theta series
Unimodular lattice
Venkov’s theorem

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science Applications
Discrete Mathematics and Combinatorics
Applied Mathematics

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Cite this

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title = "A note on Assmus–Mattson type theorems",

abstract = "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.",

keywords = "Assmus–Mattson theorem, Combinatorial t-design, Harmonic weight enumerator, Self-dual code, Spherical t-design, Spherical theta series, Unimodular lattice, Venkov{\textquoteright}s theorem",

author = "Tsuyoshi Miezaki and Akihiro Munemasa and Hiroyuki Nakasora",

note = "Funding Information: The authors are supported by JSPS KAKENHI (17K05155, 18K03217). Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.",

year = "2021",

month = may,

doi = "10.1007/s10623-021-00848-w",

language = "English",

volume = "89",

pages = "843--858",

journal = "Designs, Codes, and Cryptography",

issn = "0925-1022",

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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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ER -

A note on Assmus–Mattson type theorems

Abstract

Keywords

ASJC Scopus subject areas

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