Self-dual and LCD double circulant and double negacirculant codes over $${\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$

Yadav, Shikha; Islam, Habibul; Prakash, Om; Solé, Patrick

doi:10.1007/s12190-021-01499-9

Cited by 13 publications

(3 citation statements)

References 29 publications

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“…Notably, they studied self-dual and LCD double circulant codes over this ring and obtained a class of asymptotically good linear codes by using the Artin conjecture. Similar researches have been presented in [17] and [19].…”

supporting

confidence: 85%

Linear complementary dual codes and double circulant codes over a semi-local ring

Cheng

Cao

Qian

2022

AMC

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<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ q $\end{document}</tex-math></inline-formula> be an odd prime power and <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{F}_q $\end{document}</tex-math></inline-formula> be the finite field with <inline-formula><tex-math id="M3">\begin{document}$ q $\end{document}</tex-math></inline-formula> elements. In this paper, suppose ring <inline-formula><tex-math id="M4">\begin{document}$ R = \mathbb{F}_{q}+ \mu \mathbb{F}_{q}+ \nu \mathbb{F}_{q}+ \mu \nu \mathbb{F}_{q} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ \mu \nu = \nu \mu, \mu^{2} = \mu, \nu^{2} = \nu. $\end{document}</tex-math></inline-formula> We first give a Gray map from <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula> onto <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{F}_q^{4} $\end{document}</tex-math></inline-formula> and consider a decomposition of the ring <inline-formula><tex-math id="M8">\begin{document}$ R $\end{document}</tex-math></inline-formula>. Additionally, we investigate linear complementary dual (LCD) codes over the ring <inline-formula><tex-math id="M9">\begin{document}$ R $\end{document}</tex-math></inline-formula>. Some conditions for such linear codes over <inline-formula><tex-math id="M10">\begin{document}$ R $\end{document}</tex-math></inline-formula> to be linear complementary dual are given. Furthermore, based on the Artin conjecture, we get a class of good codes by calculating the total number of LCD double circulant codes over <inline-formula><tex-math id="M11">\begin{document}$ R $\end{document}</tex-math></inline-formula>.</p>

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confidence: 85%

Linear complementary dual codes and double circulant codes over a semi-local ring

Cheng

Cao

Qian

2022

AMC

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“…The enumeration and performance of such codes have explicitly been presented there.Further, these codes are investigated over another non-chain rings F q + vF q + v 2 F q , v 3 = v in [37] and F q [v]/⟨v 2 − 1⟩ in [36]. Recently, we [35] investigated self-dual and LCD double circulant and double negacirculant codes over F q +uF q +vF q where u 2 = u, v 2 = v, uv = vu = 0. In continuation, here we consider a class of non-chain rings R q = F q + uF q + u 2 F q + • • • + u q−1 F q , u q = u of size q q , where q = p m and p is an odd prime.…”

Section: Introductionmentioning

confidence: 99%

Self-dual and LCD double circulant codes over a class of non-local rings

et al. 2022

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Let F q be a finite field of order q = p m where p is an odd prime. This paper presents the study of self-dual and LCD double circulant codes over a class of finite commutative non-chain rings R q = F q +uF q +u 2 F q +• • •+u q−1 F q where u q = u. Here, the whole contribution is two-folded. Firstly, we enumerate self-dual and LCD double circulant codes of length 2n over R q , where n is an odd integer. Then by considering a dual-preserving Gray map ϕ, we show that Gray images of such codes are asymptotically good. Secondly, we investigate the algebraic structure of 1-generator quasi-cyclic (QC) codes over R q for q = 3. In that context, we present their generator polynomials along with their minimal generating sets and minimum distance bounds. Here, it is proved that ϕ(C) is an sq-QC code of length nq over F q if C is an s-QC code of length n over R q .

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“…Therefore, it is still open to investigate the structure of LCD codes over finite commutative non-chain rings. Very recently, these codes have been explored over a non-chain ring F q + uF q + vF q by Yadav et al [34]. Here, we consider a class of finite commutative non-chain rings with unity R e,q = F q [u]/ u e − 1 where q = et + 1 and e ≥ 2, t ≥ 1 are integers.…”

Section: Introductionmentioning

confidence: 99%

Cyclic codes over a non-chain ring $R_{e,q}$ and their application to LCD codes

Islam,

Martínez-Moro,

Prakash

2021

Preprint

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Let F q be a finite field of order q, a prime power integer such that q = et + 1 where t ≥ 1, e ≥ 2 are integers.In this paper, we study cyclic codes of length n over a non-chain ring R e,q = F q [u]/ u e −1 . We define a Gray map ϕ and obtain many maximum-distance-separable (MDS) and optimal F q -linear codes from the Gray images of cyclic codes. Under certain conditions we determine linear complementary dual (LCD) codes of length n when gcd(n, q) = 1 and gcd(n, q) = 1, respectively. It is proved that a cyclic code C of length n is an LCD code if and only if its Gray image ϕ(C) is an LCD code of length 4n over F q . Among others, we present the conditions for existence of free and non-free LCD codes. Moreover, we obtain many optimal LCD codes as the Gray images of non-free LCD codes over R e,q .

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Self-dual and LCD double circulant and double negacirculant codes over $${\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$

Cited by 13 publications

References 29 publications

Linear complementary dual codes and double circulant codes over a semi-local ring

Linear complementary dual codes and double circulant codes over a semi-local ring

Self-dual and LCD double circulant codes over a class of non-local rings

Cyclic codes over a non-chain ring $R_{e,q}$ and their application to LCD codes

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