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Entanglement-assisted quantum error-correcting (EAQEC) codes could generalize and improve performance of standard quantum error-correcting (QEC) codes to a great extent. In this paper, series of EAQEC codes of length n = q - 1 a (q + 1) are constructed from cyclic codes and negacyclic codes, where q is a prime power and a is a positive integer such that a ∣ (q - 1) . It turns out that the number of required entanglement bits can take almost all possible values. Consequently, our EAQEC codes have flexible parameters and most of them are new. For given the same length, our construction contain and extend those known consequences in Grassl et al. (Int J Quantum Inf 2(1):55–64, 2004), Jin et al. (IEEE Trans Inf Theory 56:4735–4740, 2010), Kai et al. (IEEE Trans Inf Theory 60:2080–2086, 2014), Jin and Xing (IEEE Trans Inf Theory 60:2921–2925, 2014), Chen et al. (IEEE Trans Inf Theory 61:1474–1484, 2015), Zhang and Ge (IEEE Trans Inf Theory 61:5224–5228, 2015; Des Codes Cryptogr 83(3):503–517, 2016), Shi et al. (Cryptogr Commun 10(6):1165–1182, 2018; Finite Fields Appl 46:347–362, 2017), Fan et al. (Quantum Inf Comput 16:423–434, 2016), Lu et al. (Quantum Inf Process 17(69):1–23, 2018), Li et al. (Int J Quantum Inf 17(1):1950022, 2019), Liu et al. (Quantum Inf Process 17(210):1–19, 2018), Fang et al. (Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs. arXiv:https://arxiv.org/abs/1812.09019v3). Above all, all our codes are maximum-distance-separable (MDS) if their minimum distance d ≤ n + 2 2 . [ABSTRACT FROM AUTHOR]

In this paper, we investigate narrow-sense and non-narrow-sense negacyclic Bose–Chaudhuri–Hocquenghem (NBCH) codes of length n = q m - 1 a (q m + 1) over F q 2 closely, where q is an odd prime power, m ≥ 3 is an odd integer and a ∣ (q m - 1) is an even integer. To derive accurate maximum designed distance of Hermitian dual containing NBCH codes, we define 2 ≤ a ≤ 2 q 2 - q - 1 for narrow-sense codes with δ m , a N and 2 ≤ a < 2 (q - 1) for non-narrow-sense codes with δ m , a NN . For given a, our maximum designed distance improves over the distance δ m A of Aly et al. (IEEE Trans Inf Theory 53:1183–1188, 2007) to a great extent, that is, δ m , a N = δ m , a NN = a + 2 2 δ m A . After determining dimensions of such Hermitian dual containing NBCH codes, we construct many new quantum codes via Hermitian construction naturally, whose parameters are better than the ones in the literature. [ABSTRACT FROM AUTHOR]

In this paper, by investigating q 2 -cyclotomic coset modulo rn in detail, where q is a prime power, n = q 2 + 1 and r ∣ (q + 1) , series of entanglement-assisted quantum error correction (EAQEC) codes with flexible parameters of length n are constructed from constacyclic codes (including cyclic codes). Most of our EAQEC codes are new and have large minimum distance. As to EAQEC codes constructed from cyclic codes, their all possible parameters are determined completely. When minimum distance d ≤ n + 2 2 , all of our constructed EAQEC codes are entanglement-assisted quantum MDS (EAQMDS) codes. Those previously known EAQMDS codes with the same length in Fan et al. (Quantum Inf Comput 16:423–434, 2016), Chen et al. (Quantum Inf Process 16(303):1–22, 2017), Lu et al. (Finite Fields Their Appl 53:309–325, 2018), Mustafa and Emre (Comput Appl Math 38(75):1–13, 2019) and Qian and Zhang (Quantum Inf Process 18(71):1–12, 2019) are special cases of ours. Besides, some maximum entanglement EAQEC codes and maximum entanglement EAQMDS codes are derived as well. [ABSTRACT FROM AUTHOR]

In this paper, by investigating the Hermitian dual-containing conditions of constacyclic codes with lengths n = q + 1 r (q 2 + 1) and n = q - 1 b (q 2 + 1) , where r ∣ q + 1 and b ∣ q - 1 , we construct two classes of quantum codes from non-narrow-sense constacyclic codes. Most of these new quantum codes have better parameters than quantum twisted codes and quantum BCH codes, some of them are new with relatively larger distance and can not be constructed in the literature. [ABSTRACT FROM AUTHOR]

In this work, our main objective is to construct quantum codes from quasi-twisted (QT) codes. At first, a necessary and sufficient condition for Hermitian self-orthogonality of QT codes is introduced by virtue of the Chinese remainder theorem. Then, we utilize these self-orthogonal QT codes to provide quantum codes via the famous Hermitian construction. Moreover, we present a new construction method of q-ary quantum codes, which can be viewed as an effective generalization of the Hermitian construction. General QT codes that are not self-orthogonal are also employed to construct quantum codes. As the computational results, some binary, ternary and quaternary quantum codes are constructed and their parameters are determined, which all cannot be deduced by the quantum Gilbert–Varshamov bound. In the binary case, a small number of quantum codes are derived with strictly improved parameters compared with the current records. In the ternary and quaternary cases, our codes fill some gaps or have better performances than the current results. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum error-correcting codes expand the usual paradigm of quantum error correction by allowing two parties to make use of pre-shared entanglement. This entanglement can increase either the rate of communication or the number of correctable errors. By employing generalized Reed–Solomon codes, we construct several classes of entanglement-assisted quantum maximum distance separable (EAQMDS) codes in this paper. Consequently, the results show that many of these EAQMDS codes have much larger minimum distance than ones available in the literature. Meanwhile, some of these EAQMDS codes are new in the sense that the parameters of these codes are not covered by the previously known ones, whose required number of maximally entangled states is more flexible. [ABSTRACT FROM AUTHOR]

In this paper, two families of non-narrow-sense (NNS) BCH codes of lengths n=q2m-1q2-1 and n=q2m-1q+1 (m≥3) over the finite field Fq2 are studied. The maximum designed distances δmaxnew of these dual-containing BCH codes are determined by a careful analysis of properties of the cyclotomic cosets. NNS BCH codes which achieve these maximum designed distances are presented, and a sequence of nested NNS BCH codes that contain these BCH codes with maximum designed distances are constructed and their parameters are computed. Consequently, new nonbinary quantum BCH codes are derived from these NNS BCH codes. The new quantum codes presented here include many classes of good quantum codes, which have parameters better than those constructed from narrow-sense BCH codes, negacyclic and constacyclic BCH codes in the literature. [ABSTRACT FROM AUTHOR]