Hermitian self-dual abelian codes

Hermitian self-dual abelian codes in a group ring Fq2 [G], where Fq2 is a finite field of order q2 and G is a finite abelian group, are studied. Using the well-known discrete Fourier transform decomposition for a semi-simple group ring, a characterization of Hermitian self-dual abelian codes in Fq2...

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Main Authors: Jitman, Somphong, Ling, San, Solé, Patrick
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2014
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Online Access:https://hdl.handle.net/10356/101987
http://hdl.handle.net/10220/19795
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Institution: Nanyang Technological University
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spelling sg-ntu-dr.10356-1019872023-02-28T19:43:42Z Hermitian self-dual abelian codes Jitman, Somphong Ling, San Solé, Patrick School of Physical and Mathematical Sciences DRNTU::Science::Mathematics::Algebra Hermitian self-dual abelian codes in a group ring Fq2 [G], where Fq2 is a finite field of order q2 and G is a finite abelian group, are studied. Using the well-known discrete Fourier transform decomposition for a semi-simple group ring, a characterization of Hermitian self-dual abelian codes in Fq2 [G] is given, together with an alternative proof of necessary and sufficient conditions for the existence of such a code in Fq2 [G], i.e., there exists a Hermitian self-dual abelian code in Fq2 [G] if and only if the order of G is even and q = 2l for some positive integer l. Later on, the study is further restricted to the case where F22l [G] is a principal ideal group ring, or equivalently, G = A Z2k with 2 - jAj. Based on the characterization obtained, the number of Hermitian self-dual abelian codes in F22l [A Z2k ] can be determined easily. When A is cyclic, this result answers an open problem of Jia et al. concerning Hermitian self-dual cyclic codes. In many cases, F22l [A Z2k ] contains a unique Hermitian self-dual abelian code. The criteria for such cases are determined in terms of l and the order of A. Finally, the distribution of finite abelian groups A such that a unique Hermitian self-dual abelian code exists in F22l [A Z2] is established, together with the distribution of odd integers m such that a unique Hermitian self-dual cyclic code of length 2m over F22l exists. Accepted version 2014-06-16T04:35:01Z 2019-12-06T20:48:00Z 2014-06-16T04:35:01Z 2019-12-06T20:48:00Z 2014 2014 Journal Article Jitman, S., Ling, S., & Solé, P. (2014). Hermitian self-dual abelian codes. IEEE Transactions on information theory, 60(3), 1496-1507. https://hdl.handle.net/10356/101987 http://hdl.handle.net/10220/19795 10.1109/TIT.2013.2296495 en IEEE transactions on information theory © 2014 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. The published version is available at: [http://dx.doi.org/10.1109/TIT.2013.2296495]. 14 p. application/pdf
institution Nanyang Technological University
building NTU Library
continent Asia
country Singapore
Singapore
content_provider NTU Library
collection DR-NTU
language English
topic DRNTU::Science::Mathematics::Algebra
spellingShingle DRNTU::Science::Mathematics::Algebra
Jitman, Somphong
Ling, San
Solé, Patrick
Hermitian self-dual abelian codes
description Hermitian self-dual abelian codes in a group ring Fq2 [G], where Fq2 is a finite field of order q2 and G is a finite abelian group, are studied. Using the well-known discrete Fourier transform decomposition for a semi-simple group ring, a characterization of Hermitian self-dual abelian codes in Fq2 [G] is given, together with an alternative proof of necessary and sufficient conditions for the existence of such a code in Fq2 [G], i.e., there exists a Hermitian self-dual abelian code in Fq2 [G] if and only if the order of G is even and q = 2l for some positive integer l. Later on, the study is further restricted to the case where F22l [G] is a principal ideal group ring, or equivalently, G = A Z2k with 2 - jAj. Based on the characterization obtained, the number of Hermitian self-dual abelian codes in F22l [A Z2k ] can be determined easily. When A is cyclic, this result answers an open problem of Jia et al. concerning Hermitian self-dual cyclic codes. In many cases, F22l [A Z2k ] contains a unique Hermitian self-dual abelian code. The criteria for such cases are determined in terms of l and the order of A. Finally, the distribution of finite abelian groups A such that a unique Hermitian self-dual abelian code exists in F22l [A Z2] is established, together with the distribution of odd integers m such that a unique Hermitian self-dual cyclic code of length 2m over F22l exists.
author2 School of Physical and Mathematical Sciences
author_facet School of Physical and Mathematical Sciences
Jitman, Somphong
Ling, San
Solé, Patrick
format Article
author Jitman, Somphong
Ling, San
Solé, Patrick
author_sort Jitman, Somphong
title Hermitian self-dual abelian codes
title_short Hermitian self-dual abelian codes
title_full Hermitian self-dual abelian codes
title_fullStr Hermitian self-dual abelian codes
title_full_unstemmed Hermitian self-dual abelian codes
title_sort hermitian self-dual abelian codes
publishDate 2014
url https://hdl.handle.net/10356/101987
http://hdl.handle.net/10220/19795
_version_ 1759857889266106368