New EAQMDS codes constructed from negacyclic codes
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New EAQMDS codes constructed from negacyclic codes Wan Jiang1,2 · Shixin Zhu1,2 · Xiaojing Chen3 Received: 15 May 2020 / Accepted: 6 November 2020 / Published online: 21 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we construct four families of entanglement-assisted quantum errorcorrecting codes (EAQECCs) with flexible parameters from negacyclic codes. By selecting different defining sets, our codes are new in the sense that their parameters are not covered by the codes available in the literature. Moreover, the EAQECCs we constructed are optimal and have larger minimum distance compared with the known codes with the same length. Keywords EAQECCs · Negacyclic codes · Defining sets · EAQMDS codes
1 Introduction Quantum error-correcting codes (QECCs) are useful in quantum information processing and quantum computation. After the pioneering work of Shor and Steane [1,2], many scholars focus on constructing quantum codes from classical error-correcting codes with certain inner product [3–7]. With the development of QECCs, there forms many research branch, such as asymmetric quantum codes and quantum convolutional codes. Later, there are many meaningful works about them in [8–10].
This research is supported by the National Natural Science Foundation of China (No. 61772168, 12001002) and the Natural Science Foundation of Anhui Province (No. 2008085QA04).
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Xiaojing Chen [email protected] Wan Jiang [email protected] Shixin Zhu [email protected]
1
School of Mathematics, Hefei University of Technology, Hefei 230601, Anhui, People’s Republic of China
2
Intelligent Interconnected Systems Laboratory of Anhui Province (Hefei University of Technology), Hefei, People’s Republic of China
3
School of Internet, Anhui University, Hefei 230039, Anhui, People’s Republic of China
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There is a gap when we construct QECCs, asymmetric quantum codes or quantum convolutional codes. That is the classical error-correcting codes need to be dualcontaining which is not easy satisfy all the time. This problem gets solved after Brun et al. [11] proposed EAQECCs which can be constructed from any classical codes if the sender and receiver have shared entangled states c in advance. Therefore, EAQECCs obtain extensive attention and there are many results about it in [12–16]. However, it is a challenge to determine the number of entangled states c. In order to solve this problem, Li et al. proposed the method of decomposing defining set and constructed EAQECCs from primitive quaternary BCH codes successfully [17,18]. In [19,20], their method is generalized to more general cases. Then the theory of EAQECCs has experienced tremendous growth and lots of entanglement-assisted quantum maximal distance separable (EAQMDS) codes with certain numbers of entangled states c have been constructed [21–24]. Recently, there is a brand try of constructing EAQMDS codes with flexible parameters. Qian and Zhang [25] obtained new EAQMDS codes of l
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