A note on Assmus–Mattson type theorems

Tsuyoshi Miezaki*, Akihiro Munemasa, Hiroyuki Nakasora

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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 languageEnglish
Pages (from-to)843-858
Number of pages16
JournalDesigns, Codes, and Cryptography
Volume89
Issue number5
DOIs
Publication statusPublished - 2021 May
Externally publishedYes

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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