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GALOIS LCD CODES OVER $\boldsymbol {\mathbb {F}_q+u\mathbb {F}_q+v\mathbb {F}_q+uv\mathbb {F}_q}$

Part of: Theory of error-correcting codes and error-detecting codes

Published online by Cambridge University Press:  15 December 2022

ASTHA AGRAWAL Open the ORCID record for ASTHA AGRAWAL [Opens in a new window] ,
GYANENDRA K. VERMA Open the ORCID record for GYANENDRA K. VERMA [Opens in a new window]  and
R. K. SHARMA Open the ORCID record for R. K. SHARMA [Opens in a new window]
ASTHA AGRAWAL*
Affiliation:
Department of Mathematics, Indian Institute of Technology Delhi, New Delhi 110016, India
GYANENDRA K. VERMA
Affiliation:
Department of Mathematics, Indian Institute of Technology Delhi, New Delhi 110016, India e-mail: gkvermaiitdmaths@gmail.com
R. K. SHARMA
Affiliation:
Department of Mathematics, Indian Institute of Technology Delhi, New Delhi 110016, India e-mail: rksharmaiitd@gmail.com
*
e-mail: asthaagrawaliitd@gmail.com
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Abstract

Wu and Shi [‘A note on k-Galois LCD codes over the ring $\mathbb {F}_q + u\mathbb {F}_q$ ’, Bull. Aust. Math. Soc. 104(1) (2021), 154–161] studied $ k $ -Galois LCD codes over the finite chain ring $\mathcal {R}=\mathbb {F}_q+u\mathbb {F}_q$ , where $u^2=0$ and $ q=p^e$ for some prime p and positive integer e. We extend the results to the finite nonchain ring $ \mathcal {R} =\mathbb {F}_q+u\mathbb {F}_q+v\mathbb {F}_q+uv\mathbb {F}_q$ , where $u^2=u,v^2=v $ and $ uv=vu $ . We define a correspondence between the $ l $ -Galois dual of linear codes over $ \mathcal {R} $ and the $ l $ -Galois dual of their component codes over $ \mathbb {F}_q $ . Further, we construct Euclidean LCD and $ l $ -Galois LCD codes from linear codes over $ \mathcal {R} $ . We prove that any linear code over $ \mathcal {R} $ is equivalent to a Euclidean code over $\mathbb {F}_q$ with $ q>3 $ and an $ l $ -Galois LCD code over $ \mathcal {R}$ with $0<l<e$ and $p^{e-l}+1\mid p^e-1$ . Finally, we investigate MDS codes over $ \mathcal {R}$ .

Keywords

linear codeEuclidean LCD codel-Galois LCD codeGray mapMDS code

MSC classification

Primary: 94B05: Linear codes, general
Secondary: 94B99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 107 , Issue 2 , April 2023 , pp. 330 - 341
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first and second authors are supported by UGC, New Delhi, Govt. of India under grant DEC18-417932 and CSIR, New Delhi, Govt. of India under F. No. 09/086(1407)/2019-EMR-I, respectively.

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