Galois LCD codes over finite fields

Liu, Xiusheng; Fan, Yun; Liu, Hualu

doi:10.1016/j.ffa.2017.10.001

Cited by 60 publications

(43 citation statements)

References 18 publications

(51 reference statements)

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“…we can conclude that B is good. Based on the preceding discussion, the following proposition (which was proven in (Liu et al 2018 The following theorem shows that Galois LCD QC codes are asymptotically good over some finite fields and the proof is similar to that of Theorems 3.3 and 3.7 in Güneri et al (2016).…”

Section: Asymptotically Good Galois Lcd Qc Codesmentioning

confidence: 81%

Quasi-cyclic codes: algebraic properties and applications

2020

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In this paper, we provide criteria for the reversibility and conjugate-reversibility of 1-generator quasi-cyclic codes. The Chinese remainder theorem is used to provide a characterization for generalized quasi-cyclic codes to be Galois linear complementary-dual and Galois self-dual. Using the approach proposed by Güneri and Özbudak (IEEE Trans Inf Theory 59(2):979-985, 2013), a new concatenated structure for quasi-cyclic codes is given. We show that Galois linear complementary-dual quasi-cyclic codes are asymptotically good over some finite fields. In addition, DNA codes are given which have more codewords than previously known codes.

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Section: Asymptotically Good Galois Lcd Qc Codesmentioning

confidence: 81%

Quasi-cyclic codes: algebraic properties and applications

2020

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“…Fan and Zhang [18] studied l-Galois self-dual constacyclic codes over finite fields. Liu, Fan and Liu [23] introduced the l-Galois LCD codes and constructed some classes of l-Galois LCD constacyclic MDS codes. Liu and Pan [22] studied Galois hulls of linear codes over finite fields.…”

Section: Introductionmentioning

confidence: 99%

On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $

Liu

2022

AMC

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<p style='text-indent:20px;'>In this paper, we generalize the notion of self-orthogonal codes to <inline-formula><tex-math id="M1">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-self-orthogonal codes over an arbitrary finite ring. Then, we study the <inline-formula><tex-math id="M2">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-self-orthogonality of constacyclic codes of length <inline-formula><tex-math id="M3">\begin{document}$ p^s $\end{document}</tex-math></inline-formula> over the finite commutative chain ring <inline-formula><tex-math id="M4">\begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ p $\end{document}</tex-math></inline-formula> is a prime, <inline-formula><tex-math id="M6">\begin{document}$ u^2 = 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula> is an arbitrary ring automorphism of <inline-formula><tex-math id="M8">\begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}</tex-math></inline-formula>. We characterize the structure of <inline-formula><tex-math id="M9">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-dual code of a <inline-formula><tex-math id="M10">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-constacyclic code of length <inline-formula><tex-math id="M11">\begin{document}$ p^s $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M12">\begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}</tex-math></inline-formula>. Further, the necessary and sufficient conditions for a <inline-formula><tex-math id="M13">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-constacyclic code to be <inline-formula><tex-math id="M14">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-self-orthogonal are provided. In particular, we determine all <inline-formula><tex-math id="M15">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-self-dual constacyclic codes of length <inline-formula><tex-math id="M16">\begin{document}$ p^s $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M17">\begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}</tex-math></inline-formula>. In the end of this paper, when <inline-formula><tex-math id="M18">\begin{document}$ p $\end{document}</tex-math></inline-formula> is an odd prime, we extend the results to constacyclic codes of length <inline-formula><tex-math id="M19">\begin{document}$ 2 p^s $\end{document}</tex-math></inline-formula>.</p>

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“…Recently, k-Galois dual codes were introduced in [4] for studying Galois constacyclic codes (a generalisation of Euclidean constacyclic and Hermitian constacyclic codes). The k-Galois LCD codes over finite fields have been studied in [8]. A necessary and sufficient condition for linear codes to be k-Galois LCD was obtained and several classes of k-Galois LCD maximum distance separable codes were exhibited.…”

Section: Introductionmentioning

confidence: 99%

“…A remarkable result for LCD codes was established by Carlet et al [1], showing that any linear code over F q is equivalent to a Euclidean LCD code for q ≥ 4, and any linear code over F q 2 is equivalent to a Hermitian LCD code for q ≥ 3. Later, these results were generalised to k-Galois codes over finite fields by using Gröbner bases [8]. A natural question arises as to how to characterise k-Galois LCD codes over a finite chain ring R. Another interesting problem is to study the connection between k-Galois LCD codes over finite fields and linear codes in the context of finite chain rings.…”

Section: Introductionmentioning

confidence: 99%

A NOTE ON k-GALOIS LCD CODES OVER THE RING

Shi

2020

Bull. Aust. Math. Soc.

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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.

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Galois LCD codes over finite fields

Cited by 60 publications

References 18 publications

Quasi-cyclic codes: algebraic properties and applications

Quasi-cyclic codes: algebraic properties and applications

On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $

A NOTE ON k-GALOIS LCD CODES OVER THE RING

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