New classes of entanglement-assisted quantum MDS codes
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New classes of entanglement-assisted quantum MDS codes Renjie Jin1 · Derong Xie1 · Jinquan Luo1 Received: 29 October 2019 / Accepted: 17 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, two new classes of entanglement-assisted quantum MDS codes (EAQMDS codes for short) with length n being a factor of q 2 ± 1 are presented via cyclic codes over finite fields of odd characteristic. Among our constructions, there are several EAQMDS codes with new parameters which have never been reported. Moreover, some of them have much larger minimum distance than known results. Keywords MDS code · EAQEC code · EAQMDS code · Cyclic code
1 Introduction Quantum information can protect messages between sender and receiver avoiding decoherence by encoding it into quantum error-correcting codes. Entanglementassisted quantum error-correcting codes (EAQEC codes for short) are crucial to quantum information theory (see [1–4,12]). Recently, construction of good quantum codes via classical codes is a hot topic for quantum information and quantum computing (see [16,17,20,22,24]). EAQEC codes use preexisting entanglement between the sender and the receiver to improve information rate. Many researchers have been devoted to obtaining EAQEC codes via classical liner codes, such as negacyclic codes and generalized Reed–Solomon codes. It has been shown that EAQEC codes have some advantages over standard stabilizer codes. For example, only a dual-containing classical linear quaternary code can be transformed into a standard stabilizer code, but
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Jinquan Luo [email protected] Renjie Jin [email protected] Derong Xie [email protected]
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School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China 0123456789().: V,-vol
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any classical linear quaternary code can be transformed into an EAQEC code. Some of them can be summarized as follows. In [13,23], some new EAQEC codes with good parameters via cyclic and constacyclic codes are constructed. In [5], new decomposition of negacyclic codes are proposed, by which four new classes of EAQEC codes have been constructed. In [7], Fan et al. constructed some classes of EAQMDS codes based on classical maximum distance separable (MDS for short) codes by exploiting one or more pre-shared maximally entangled states. In [22], Qian and Zhang constructed some new classes of MDS linear complementary dual (LCD) codes with respect to Hermitian inner product. As applications, they have constructed new families of EAQMDS codes. In [9], Guenda et al. showed that the number of shared pairs required to construct an EAQEC code is related to the hull of classical codes. Using this fact, they put forward new methods to construct EAQEC codes requiring desirable amounts of entanglements. The E A-Singleton bound for an [[n, k, d; c]]q EAQEC code is 2(d − 1) ≤ n − k + c. A q-ary EAQEC code attaining this bound is said to be an EAQMDS code
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