Construction of MDS self-dual codes over Galois rings

Jon Lark Kim; Yoonjin Lee

doi:10.1007/s10623-007-9117-y

Construction of MDS self-dual codes over Galois rings

Jon Lark Kim, Yoonjin Lee

Department of Mathematics

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

25 Scopus citations

Abstract

The purpose of this paper is to construct nontrivial MDS self-dual codes over Galois rings. We consider a building-up construction of self-dual codes over Galois rings as a GF(q)-analogue of (Kim and Lee, J Combin Theory ser A, 105:79-95). We give a necessary and sufficient condition on which the building-up construction holds. We construct MDS self-dual codes of lengths up to 8 over GR(3²,2), GR(3³,2) and GR(3⁴,2), and near-MDS self-dual codes of length 10 over these rings. In a similar manner, over GR(5²,2), GR(5³,2) and GR(7²,2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self-dual codes of length 12. Furthermore, over GR(11²,2) we have MDS self-dual codes of lengths up to 12.

Original language	English
Pages (from-to)	247-258
Number of pages	12
Journal	Designs, Codes, and Cryptography
Volume	45
Issue number	2
DOIs	https://doi.org/10.1007/s10623-007-9117-y
State	Published - Nov 2007

Bibliographical note

Funding Information:
Acknowledgments The authors thank the anonymous referees for comments and suggestions which helped to improve the paper. J.-L. Kim was supported in part by a Project Completion Grant from the University of Louisville and Y. Lee was supported by NSERC.

Keywords

Galois ring
MDS code
Self-dual code

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10.1007/s10623-007-9117-y

Cite this

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title = "Construction of MDS self-dual codes over Galois rings",

abstract = "The purpose of this paper is to construct nontrivial MDS self-dual codes over Galois rings. We consider a building-up construction of self-dual codes over Galois rings as a GF(q)-analogue of (Kim and Lee, J Combin Theory ser A, 105:79-95). We give a necessary and sufficient condition on which the building-up construction holds. We construct MDS self-dual codes of lengths up to 8 over GR(32,2), GR(33,2) and GR(34,2), and near-MDS self-dual codes of length 10 over these rings. In a similar manner, over GR(52,2), GR(53,2) and GR(72,2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self-dual codes of length 12. Furthermore, over GR(112,2) we have MDS self-dual codes of lengths up to 12.",

keywords = "Galois ring, MDS code, Self-dual code",

author = "Kim, {Jon Lark} and Yoonjin Lee",

note = "Funding Information: Acknowledgments The authors thank the anonymous referees for comments and suggestions which helped to improve the paper. J.-L. Kim was supported in part by a Project Completion Grant from the University of Louisville and Y. Lee was supported by NSERC.",

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AU - Kim, Jon Lark

AU - Lee, Yoonjin

N1 - Funding Information: Acknowledgments The authors thank the anonymous referees for comments and suggestions which helped to improve the paper. J.-L. Kim was supported in part by a Project Completion Grant from the University of Louisville and Y. Lee was supported by NSERC.

PY - 2007/11

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N2 - The purpose of this paper is to construct nontrivial MDS self-dual codes over Galois rings. We consider a building-up construction of self-dual codes over Galois rings as a GF(q)-analogue of (Kim and Lee, J Combin Theory ser A, 105:79-95). We give a necessary and sufficient condition on which the building-up construction holds. We construct MDS self-dual codes of lengths up to 8 over GR(32,2), GR(33,2) and GR(34,2), and near-MDS self-dual codes of length 10 over these rings. In a similar manner, over GR(52,2), GR(53,2) and GR(72,2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self-dual codes of length 12. Furthermore, over GR(112,2) we have MDS self-dual codes of lengths up to 12.

AB - The purpose of this paper is to construct nontrivial MDS self-dual codes over Galois rings. We consider a building-up construction of self-dual codes over Galois rings as a GF(q)-analogue of (Kim and Lee, J Combin Theory ser A, 105:79-95). We give a necessary and sufficient condition on which the building-up construction holds. We construct MDS self-dual codes of lengths up to 8 over GR(32,2), GR(33,2) and GR(34,2), and near-MDS self-dual codes of length 10 over these rings. In a similar manner, over GR(52,2), GR(53,2) and GR(72,2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self-dual codes of length 12. Furthermore, over GR(112,2) we have MDS self-dual codes of lengths up to 12.

KW - Galois ring

KW - MDS code

KW - Self-dual code

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DO - 10.1007/s10623-007-9117-y

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JO - Designs, Codes, and Cryptography

JF - Designs, Codes, and Cryptography

IS - 2

ER -

Construction of MDS self-dual codes over Galois rings

Abstract

Bibliographical note

Keywords

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