Singly even self-dual codes of length 24k + 10 and minimum weight 4k + 2

Masaaki Harada

doi:10.1007/s12095-018-0303-8

Singly even self-dual codes of length 24k + 10 and minimum weight 4k + 2

Masaaki Harada

INF - System Information Sciences

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

Abstract

Currently, the existence of an extremal singly even self-dual code of length 24k + 10 is unknown for all nonnegative integers k. In this note, we study singly even self-dual [24k + 10, 12k + 5, 4k + 2] codes. We give some restrictions on the possible weight enumerators of singly even self-dual [24k + 10, 12k + 5, 4k + 2] codes with shadows of minimum weight at least 5 for k = 2, 3, 4, 5. We discuss a method for constructing singly even self-dual codes with minimal shadow. As an example, a singly even self-dual [82, 41, 14] code with minimal shadow is constructed for the first time. In addition, as neighbors of the code, we construct singly even self-dual [82, 41, 14] codes with weight enumerator for which no singly even self-dual code was previously known to exist.

Original language	English
Pages (from-to)	597-608
Number of pages	12
Journal	Cryptography and Communications
Volume	11
Issue number	4
DOIs	https://doi.org/10.1007/s12095-018-0303-8
Publication status	Published - 2019 Jul 15

Keywords

Self-dual code
Shadow
Weight enumerator

ASJC Scopus subject areas

Computer Networks and Communications
Computational Theory and Mathematics
Applied Mathematics

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10.1007/s12095-018-0303-8

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@article{5a2c684f7cfe42908f32e21c2213b76c,

title = "Singly even self-dual codes of length 24k + 10 and minimum weight 4k + 2",

abstract = "Currently, the existence of an extremal singly even self-dual code of length 24k + 10 is unknown for all nonnegative integers k. In this note, we study singly even self-dual [24k + 10, 12k + 5, 4k + 2] codes. We give some restrictions on the possible weight enumerators of singly even self-dual [24k + 10, 12k + 5, 4k + 2] codes with shadows of minimum weight at least 5 for k = 2, 3, 4, 5. We discuss a method for constructing singly even self-dual codes with minimal shadow. As an example, a singly even self-dual [82, 41, 14] code with minimal shadow is constructed for the first time. In addition, as neighbors of the code, we construct singly even self-dual [82, 41, 14] codes with weight enumerator for which no singly even self-dual code was previously known to exist.",

keywords = "Self-dual code, Shadow, Weight enumerator",

author = "Masaaki Harada",

note = "Funding Information: Acknowledgments This work was supported by JSPS KAKENHI Grant Number 15H03633. The author would like to thank the anonymous referee for useful comments.",

year = "2019",

month = jul,

day = "15",

doi = "10.1007/s12095-018-0303-8",

language = "English",

volume = "11",

pages = "597--608",

journal = "Cryptography and Communications",

issn = "1936-2447",

publisher = "Springer Publishing Company",

number = "4",

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AU - Harada, Masaaki

N1 - Funding Information: Acknowledgments This work was supported by JSPS KAKENHI Grant Number 15H03633. The author would like to thank the anonymous referee for useful comments.

PY - 2019/7/15

Y1 - 2019/7/15

N2 - Currently, the existence of an extremal singly even self-dual code of length 24k + 10 is unknown for all nonnegative integers k. In this note, we study singly even self-dual [24k + 10, 12k + 5, 4k + 2] codes. We give some restrictions on the possible weight enumerators of singly even self-dual [24k + 10, 12k + 5, 4k + 2] codes with shadows of minimum weight at least 5 for k = 2, 3, 4, 5. We discuss a method for constructing singly even self-dual codes with minimal shadow. As an example, a singly even self-dual [82, 41, 14] code with minimal shadow is constructed for the first time. In addition, as neighbors of the code, we construct singly even self-dual [82, 41, 14] codes with weight enumerator for which no singly even self-dual code was previously known to exist.

AB - Currently, the existence of an extremal singly even self-dual code of length 24k + 10 is unknown for all nonnegative integers k. In this note, we study singly even self-dual [24k + 10, 12k + 5, 4k + 2] codes. We give some restrictions on the possible weight enumerators of singly even self-dual [24k + 10, 12k + 5, 4k + 2] codes with shadows of minimum weight at least 5 for k = 2, 3, 4, 5. We discuss a method for constructing singly even self-dual codes with minimal shadow. As an example, a singly even self-dual [82, 41, 14] code with minimal shadow is constructed for the first time. In addition, as neighbors of the code, we construct singly even self-dual [82, 41, 14] codes with weight enumerator for which no singly even self-dual code was previously known to exist.

KW - Self-dual code

KW - Shadow

KW - Weight enumerator

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