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A NOTE ON k-GALOIS LCD CODES OVER THE RING $\mathbb {F}_{\mathbf{\mathit{q}}}\boldsymbol{+}\mathbf{\mathit{u}}\mathbb {F}_{\mathbf{\mathit{q}}}$

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

Published online by Cambridge University Press:  14 December 2020

RONGSHENG WU Open the ORCID record for RONGSHENG WU [Opens in a new window]  and
MINJIA SHI Open the ORCID record for MINJIA SHI [Opens in a new window]
RONGSHENG WU
Affiliation:
Ministry of Education Key Laboratory of Intelligent Computing and Signal Processing, School of Mathematical Sciences, Anhui University, Hefei, Anhui230601, P. R. China e-mail: wrs2510@163.com
MINJIA SHI*
Affiliation:
Ministry of Education Key Laboratory of Intelligent Computing and Signal Processing, School of Mathematical Sciences, Anhui University, Hefei, Anhui230601, P. R. China
*
e-mail: smjwcl.good@163.com
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Abstract

We study the k-Galois linear complementary dual (LCD) codes over the finite chain ring $R=\mathbb {F}_q+u\mathbb {F}_q$ with $u^2=0$ , where $q=p^e$ and p is a prime number. We give a sufficient condition on the generator matrix for the existence of k-Galois LCD codes over R. Finally, we show that a linear code over R (for $k=0, q> 3$ ) is equivalent to a Euclidean LCD code, and a linear code over R (for $0<k<e$ , $(p^{e-k}+1)\mid (p^e-1)$ and ${(p^e-1)}/{(p^{e-k}+1)}>1$ ) is equivalent to a k-Galois LCD code.

Keywords

Euclidean LCD codesGray mapk-Galois LCD codeslinear codeslinear complementary dual codes

MSC classification

Primary: 94B05: Linear codes, general
Secondary: 94B15: Cyclic codes
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 1 , August 2021 , pp. 154 - 161
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

This research is supported by the National Natural Science Foundation of China (12071001, 61672036), the Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20), the Academic Fund for Outstanding Talents in Universities (gxbjZD03) and the Natural Science Foundation of Anhui Provence (2008085QA04).

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