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Hermitian LCD codes over F q 2 + u F q 2 $\mathbb {F}_{q^{2}}+u \mathbb {F}_{q^{2}}$ and their applications to maximal entanglement EAQECCs
Cryptography and Communications ( IF 1.2 ) Pub Date : 2021-07-21 , DOI: 10.1007/s12095-021-00510-1
Heqian Xu 1, 2 , Wei Du 1
Affiliation  

Let \(R=\mathbb {F}_{q^{2}}+u \mathbb {F}_{q^{2}},\) where \(\mathbb {F}_{q^{2}}\) is the finite field with q2 elements, q is a power of a prime p, and u2 = 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 \(\mathbb {F}_{q^{2}}\) as Gray images of linear and constacyclic codes of length n over R. By using the explicit symplectic method in Galindo et al. (Quantum Inf. Process. 18(4), 116 9), we get the desired maximal entanglement EAQECCs.



中文翻译:

F q 2 + u F q 2 $\mathbb {F}_{q^{2}}+u \mathbb {F}_{q^{2}}$ 上的 Hermitian LCD 编码及其在最大纠缠 EAQECC 中的应用

\(R=\mathbb {F}_{q^{2}}+u \mathbb {F}_{q^{2}},\)其中\(\mathbb {F}_{q^{2 }}\)是具有q 2 个元素的有限域,q是素数p的幂,u 2 = 0。 本文得到一类最大纠缠纠缠辅助量子纠错码(EAQECCs)通过在R上使用长度为n 的(1 − u )-constacyclic Hermitian 线性互补对偶 (LCD) 码。首先,我们给出一个长度为n的线性代码CR 上的充分条件成为 Hermitian LCD 代码并声称不存在长度为n超过R的非自由 Hermitian LCD 代码。此外,假设\(\gcd (n, q)=1\),并且γR 中的一个单位,我们获得所有γ -constacyclic Hermitian LCD 代码。最后,我们得出长度为2的辛LCD码Ñ超过\(\ mathbb {F} _ {Q ^ {2}} \)作为线性的灰色图像和长度的常循环码Ñ超过ř。通过使用 Galindo 等人的显式辛方法。(Quantum Inf. Process. 18 (4), 116 9),我们得到了所需的最大纠缠 EAQECC。

更新日期:2021-07-22
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