On polycyclic codes over a finite chain ring

Fotue-Tabue, Alexandre; Martı́nez-Moro, Edgar; Blackford, J.T.

doi:10.3934/amc.2020028

Cited by 12 publications

(10 citation statements)

References 10 publications

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“…We find that the Hamming distance of C ⊥ H is equal to 1. According to Theorem 9, we get a symplectic LCD code ϕ(C) over F 4 with parameters [10,8] which provides a [[5, 4, 1; 1]] maximal entanglement EAQECC over F 4 .…”

Section: Corollarymentioning

confidence: 99%

Hermitian LCD codes over $\mathbb {F}_{q^{2}}+u \mathbb {F}_{q^{2}}$ and their applications to maximal entanglement EAQECCs

2021

Cryptogr. Commun.

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Let R = F q 2 + uF q 2 , where F q 2 is the finite field with q 2 elements, q is a power of a prime p, and u 2 = 0. In this paper, a class of maximal entanglement entanglement-assisted quantum error-correcting codes (EAQECCs) is obtained by employing (1 − u)-constacyclic Hermitian linear complementary dual (LCD) codes of length n over R. First, we give a sufficient condition for a linear code C of length n over R to be a Hermitian LCD code and claim that there does not exist a non-free Hermitian LCD code of length n over R. Also, assume that gcd(n, q) = 1, and γ is a unit in R, we obtain all γ -constacyclic Hermitian LCD codes. Finally, we derive symplectic LCD codes of length 2n over F q 2 as Gray images of linear and constacyclic codes of length n over R. By using the explicit symplectic method in [9], we get the desired maximal entanglement EAQECCs.

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Section: Corollarymentioning

confidence: 99%

Hermitian LCD codes over $\mathbb {F}_{q^{2}}+u \mathbb {F}_{q^{2}}$ and their applications to maximal entanglement EAQECCs

2021

Cryptogr. Commun.

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“…[9, 4 2 , 6] Good (X 6 + X 3 + 1) [9, 4 3 , 3] Good (X − 1)(X 2 + X + 1) [9, 4 6 , 2] Good (X 8 + X 7 + 3X 6 + 3X 4 + 3X 2 + X + 1) [17, 4 9 , 7] Optimal (X − 1)(X 8 [31, 4 21 , 6] Optimal g 2 g 3 g 4 g 5 g 6 g 7 g 9 g 10 g 12 g 13 [63, 4 13 , 36] Optimal g 1 g 3 g 4 g 5 g 6 g 7 g 9 g 10 g 12 g 13 [63, 4 14 , 34] Optimal g 3 g 4 g 6 g 7 g 8 g 10 g 11 g 12 g 13 [63, 4 15 , 21] Optimal g 1 g 6 g 7 g 10 g 11 g 12 g 13 [63, 4 20 , 18] Optimal g 1 g 5 g 6 g 7 g 9 g 13 [63, 4 32 , 16] Optimal g 1 g 2 g 7 g 8 g 10 g 11 g 12 g 13 [63, 4 24 , 14] Optimal g 2 g 3 g 4 g 6 g 10 g 12 [63, 4 37 , 12] Optimal g 1 g 2 g 3 g 4 g 10 g 12 [63, 4 42 , 10] Optimal g 2 g 5 g 6 g 7 g 9 g 13 [63, 4 31 , 9] Optimal g 3 g 4 g 7 g 8 g 11 g 13 [63, 4 33 , 7] Optimal g 1 g 8 g 11 [63, 4 50 , 6] Optimal Example 4.4. The factorization of X 15 − 1 over 8 into a product of basic irreducible polynomials over 8 is given by…”

Section: Generators Of Cmentioning

confidence: 99%

Do non-free LCD codes over finite commutative Frobenius rings exist?

Bhowmick

Fotue-Tabue

Martı́nez-Moro

et al. 2020

Des. Codes Cryptogr.

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In this paper, we clarify some aspects on LCD codes in the literature. We first prove that a non-free LCD code does not exist over finite commutative Frobenius local rings. We then obtain a necessary and sufficient condition for the existence of LCD code over finite commutative Frobenius rings. We later show that a free constacyclic code over finite chain ring is LCD if and only if it is reversible, and also provide a necessary and sufficient condition for a constacyclic code to be reversible over finite chain rings. We illustrate the minimum Lee-distance of LCD codes over some finite commutative chain rings and demonstrate the results with examples. We also got some new optimal 4 codes of different lengths which are cyclic LCD codes over 4 .

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“…In recent ten years, polycyclic codes and sequential codes have been extensively studied in [2,9,11,14]. Later, in 2020, Fotue-Tabue et al [6] studied polycyclic codes and their annihilator dual codes over finite chain rings.…”

Section: Introductionmentioning

confidence: 99%

On the polycyclic codes over $ \mathbb{F}_q+u\mathbb{F}_q $

Qi¹

2024

AMC

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<p style='text-indent:20px;'>In this article, we mainly study the polycyclic codes over <inline-formula><tex-math id="M2">\begin{document}$ S $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ S = \mathbb{F}_q+u\mathbb{F}_q $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ u^2 = u $\end{document}</tex-math></inline-formula>. First, the annihilator self-dual codes, annihilator self-orthogonal codes and annihilator <inline-formula><tex-math id="M5">\begin{document}$ {{{\rm{LCD}}}} $\end{document}</tex-math></inline-formula> codes over <inline-formula><tex-math id="M6">\begin{document}$ S $\end{document}</tex-math></inline-formula> are also introduced and studied. Next, we define a Gray map from <inline-formula><tex-math id="M7">\begin{document}$ S^n $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M8">\begin{document}$ \mathbb{F}^{2n}_q $\end{document}</tex-math></inline-formula> and investigate the structure properties of polycyclic codes over <inline-formula><tex-math id="M9">\begin{document}$ S $\end{document}</tex-math></inline-formula> using the decomposition method. The Hamming distances of the Gray images are also determined by their decompositions. Finally, we obtain some good codes based on the results.</p>

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On polycyclic codes over a finite chain ring

Cited by 12 publications

References 10 publications

Hermitian LCD codes over $\mathbb {F}_{q^{2}}+u \mathbb {F}_{q^{2}}$ and their applications to maximal entanglement EAQECCs

Hermitian LCD codes over $\mathbb {F}_{q^{2}}+u \mathbb {F}_{q^{2}}$ and their applications to maximal entanglement EAQECCs

Do non-free LCD codes over finite commutative Frobenius rings exist?

On the polycyclic codes over $ \mathbb{F}_q+u\mathbb{F}_q $

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