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Linking Reversed and Dual Codes of Quasi-Cyclic Codes
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Summary :
It is known that quasi-cyclic (QC) codes over the finite field Fq correspond to certain Fq[x]-modules. A QC code C is specified by a generator polynomial matrix G whose rows generate C as an Fq[x]-module. The reversed code of C, denoted by R, is the code obtained by reversing all codewords of C while the dual code of C is denoted by C⊥. We call C reversible, self-orthogonal, and self-dual if R = C, C⊥ ⊇ C, and C⊥ = C, respectively. In this study, for a given C, we find an explicit formula for a generator polynomial matrix of R. A necessary and sufficient condition for C to be reversible is derived from this formula. In addition, we reveal the relations among C, R, and C⊥. Specifically, we give conditions on G corresponding to C⊥ ⊇ R, C⊥ ⊆ R, and C = R = C⊥. As an application, we employ these theoretical results to the construction of QC codes with best parameters. Computer search is used to show that there exist various binary reversible self-orthogonal QC codes that achieve the upper bounds on the minimum distance of linear codes.
- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E105-A No.3 pp.381-388
- Publication Date
- 2022/03/01
- Publicized
- 2021/07/30
- Online ISSN
- 1745-1337
- DOI
- 10.1587/transfun.2021TAP0010
- Type of Manuscript
- Special Section PAPER (Special Section on Information Theory and Its Applications)
- Category
- Coding Theory
Authors
Ramy TAKI ELDIN
Ain Shams University
Hajime MATSUI
Toyota Technological Institute
Keyword
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Ramy TAKI ELDIN, Hajime MATSUI, "Linking Reversed and Dual Codes of Quasi-Cyclic Codes" in IEICE TRANSACTIONS on Fundamentals,
vol. E105-A, no. 3, pp. 381-388, March 2022, doi: 10.1587/transfun.2021TAP0010.
Abstract: It is known that quasi-cyclic (QC) codes over the finite field Fq correspond to certain Fq[x]-modules. A QC code C is specified by a generator polynomial matrix G whose rows generate C as an Fq[x]-module. The reversed code of C, denoted by R, is the code obtained by reversing all codewords of C while the dual code of C is denoted by C⊥. We call C reversible, self-orthogonal, and self-dual if R = C, C⊥ ⊇ C, and C⊥ = C, respectively. In this study, for a given C, we find an explicit formula for a generator polynomial matrix of R. A necessary and sufficient condition for C to be reversible is derived from this formula. In addition, we reveal the relations among C, R, and C⊥. Specifically, we give conditions on G corresponding to C⊥ ⊇ R, C⊥ ⊆ R, and C = R = C⊥. As an application, we employ these theoretical results to the construction of QC codes with best parameters. Computer search is used to show that there exist various binary reversible self-orthogonal QC codes that achieve the upper bounds on the minimum distance of linear codes.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.2021TAP0010/_p
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@ARTICLE{e105-a_3_381,
author={Ramy TAKI ELDIN, Hajime MATSUI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Linking Reversed and Dual Codes of Quasi-Cyclic Codes},
year={2022},
volume={E105-A},
number={3},
pages={381-388},
abstract={It is known that quasi-cyclic (QC) codes over the finite field Fq correspond to certain Fq[x]-modules. A QC code C is specified by a generator polynomial matrix G whose rows generate C as an Fq[x]-module. The reversed code of C, denoted by R, is the code obtained by reversing all codewords of C while the dual code of C is denoted by C⊥. We call C reversible, self-orthogonal, and self-dual if R = C, C⊥ ⊇ C, and C⊥ = C, respectively. In this study, for a given C, we find an explicit formula for a generator polynomial matrix of R. A necessary and sufficient condition for C to be reversible is derived from this formula. In addition, we reveal the relations among C, R, and C⊥. Specifically, we give conditions on G corresponding to C⊥ ⊇ R, C⊥ ⊆ R, and C = R = C⊥. As an application, we employ these theoretical results to the construction of QC codes with best parameters. Computer search is used to show that there exist various binary reversible self-orthogonal QC codes that achieve the upper bounds on the minimum distance of linear codes.},
keywords={},
doi={10.1587/transfun.2021TAP0010},
ISSN={1745-1337},
month={March},}
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TY - JOUR
TI - Linking Reversed and Dual Codes of Quasi-Cyclic Codes
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 381
EP - 388
AU - Ramy TAKI ELDIN
AU - Hajime MATSUI
PY - 2022
DO - 10.1587/transfun.2021TAP0010
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E105-A
IS - 3
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - March 2022
AB - It is known that quasi-cyclic (QC) codes over the finite field Fq correspond to certain Fq[x]-modules. A QC code C is specified by a generator polynomial matrix G whose rows generate C as an Fq[x]-module. The reversed code of C, denoted by R, is the code obtained by reversing all codewords of C while the dual code of C is denoted by C⊥. We call C reversible, self-orthogonal, and self-dual if R = C, C⊥ ⊇ C, and C⊥ = C, respectively. In this study, for a given C, we find an explicit formula for a generator polynomial matrix of R. A necessary and sufficient condition for C to be reversible is derived from this formula. In addition, we reveal the relations among C, R, and C⊥. Specifically, we give conditions on G corresponding to C⊥ ⊇ R, C⊥ ⊆ R, and C = R = C⊥. As an application, we employ these theoretical results to the construction of QC codes with best parameters. Computer search is used to show that there exist various binary reversible self-orthogonal QC codes that achieve the upper bounds on the minimum distance of linear codes.
ER -
English
