Construction of extremal self-dual codes over Z8 and Z16

Boran Kim; Yoonjin Lee

doi:10.1007/s10623-015-0137-8

Construction of extremal self-dual codes over Z₈ and Z₁₆

Boran Kim, Yoonjin Lee

Department of Mathematics

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

4 Scopus citations

Abstract

We present a method of constructing free self-dual codes over Z₈ and Z₁₆ which are extremal or optimal with respect to the Hamming weight. We first prove that every (extremal or optimal) free self-dual code over Z2m can be found from a binary (extremal or optimal) Type II code for any positive integer m≥ 2. We find explicit algorithms for construction of self-dual codes over Z₈ and Z₁₆. Our construction method is basically a lifting method. Furthermore, we find an upper bound of minimum Hamming weights of free self-dual codes over Z2m. By using our explicit algorithms, we construct extremal free self-dual codes over Z₈ and Z₁₆ up to lengths 40.

Original language	English
Pages (from-to)	239-257
Number of pages	19
Journal	Designs, Codes, and Cryptography
Volume	81
Issue number	2
DOIs	https://doi.org/10.1007/s10623-015-0137-8
State	Published - 1 Nov 2016

Bibliographical note

Funding Information:
Due to the referee’s helpful comments, we added Lemma and Theorem . We express our gratitude to the referee. Yoonjin Lee is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827) and also by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (2014-002731).

Publisher Copyright:
© 2015, Springer Science+Business Media New York.

Keywords

Code over a ring
Extremal self-dual code
Free self-dual code
Optimal self-dual code
Self-dual code

Access to Document

10.1007/s10623-015-0137-8

Cite this

@article{ed73d1fee2d940eeada3d685710ba3b1,

title = "Construction of extremal self-dual codes over Z8 and Z16 ",

abstract = "We present a method of constructing free self-dual codes over Z8 and Z16 which are extremal or optimal with respect to the Hamming weight. We first prove that every (extremal or optimal) free self-dual code over Z2m can be found from a binary (extremal or optimal) Type II code for any positive integer m≥ 2. We find explicit algorithms for construction of self-dual codes over Z8 and Z16. Our construction method is basically a lifting method. Furthermore, we find an upper bound of minimum Hamming weights of free self-dual codes over Z2m. By using our explicit algorithms, we construct extremal free self-dual codes over Z8 and Z16 up to lengths 40.",

keywords = "Code over a ring, Extremal self-dual code, Free self-dual code, Optimal self-dual code, Self-dual code",

author = "Boran Kim and Yoonjin Lee",

note = "Funding Information: Due to the referee{\textquoteright}s helpful comments, we added Lemma and Theorem . We express our gratitude to the referee. Yoonjin Lee is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827) and also by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (2014-002731). Publisher Copyright: {\textcopyright} 2015, Springer Science+Business Media New York.",

year = "2016",

month = nov,

day = "1",

doi = "10.1007/s10623-015-0137-8",

language = "English",

volume = "81",

pages = "239--257",

journal = "Designs, Codes, and Cryptography",

issn = "0925-1022",

publisher = "Springer Netherlands",

number = "2",

}

TY - JOUR

T1 - Construction of extremal self-dual codes over Z8 and Z16

AU - Kim, Boran

AU - Lee, Yoonjin

N1 - Funding Information: Due to the referee’s helpful comments, we added Lemma and Theorem . We express our gratitude to the referee. Yoonjin Lee is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827) and also by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (2014-002731). Publisher Copyright: © 2015, Springer Science+Business Media New York.

PY - 2016/11/1

Y1 - 2016/11/1

N2 - We present a method of constructing free self-dual codes over Z8 and Z16 which are extremal or optimal with respect to the Hamming weight. We first prove that every (extremal or optimal) free self-dual code over Z2m can be found from a binary (extremal or optimal) Type II code for any positive integer m≥ 2. We find explicit algorithms for construction of self-dual codes over Z8 and Z16. Our construction method is basically a lifting method. Furthermore, we find an upper bound of minimum Hamming weights of free self-dual codes over Z2m. By using our explicit algorithms, we construct extremal free self-dual codes over Z8 and Z16 up to lengths 40.

AB - We present a method of constructing free self-dual codes over Z8 and Z16 which are extremal or optimal with respect to the Hamming weight. We first prove that every (extremal or optimal) free self-dual code over Z2m can be found from a binary (extremal or optimal) Type II code for any positive integer m≥ 2. We find explicit algorithms for construction of self-dual codes over Z8 and Z16. Our construction method is basically a lifting method. Furthermore, we find an upper bound of minimum Hamming weights of free self-dual codes over Z2m. By using our explicit algorithms, we construct extremal free self-dual codes over Z8 and Z16 up to lengths 40.

KW - Code over a ring

KW - Extremal self-dual code

KW - Free self-dual code

KW - Optimal self-dual code

KW - Self-dual code

UR - http://www.scopus.com/inward/record.url?scp=84944599351&partnerID=8YFLogxK

U2 - 10.1007/s10623-015-0137-8

DO - 10.1007/s10623-015-0137-8

M3 - Article

AN - SCOPUS:84944599351

SN - 0925-1022

VL - 81

SP - 239

EP - 257

JO - Designs, Codes, and Cryptography

JF - Designs, Codes, and Cryptography

IS - 2

ER -