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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]

By studying the properties of q 2 -cyclotomic cosets, the maximum designed distances of Hermitian dual-containing constacyclic Bose–Chaudhuri–Hocquenghem (BCH) codes with length n = 2 (q m + 1) are determined, where q is an odd prime power and m ≥ 3 is an integer. Further, their dimensions are calculated precisely for the given designed distance. Consequently, via Hermitian Construction, many new quantum codes could be obtained from these codes, which are not covered in the literature. [ABSTRACT FROM AUTHOR]

In this paper, two families of Hermitian dual-containing Bose-Chau- dhuri-Hocquenghem (BCH) codes with length n = a ⋅ q 2 + 1 2 and n = b (q2 + 1) are studied, where odd a∣(q − 1) for odd prime power q and b∣ (q + 1) for even prime power q, respectively. Using these non-narrow-sense BCH codes, some new quantum BCH codes are constructed. Most of these quantum codes have better parameters 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]

In this paper, we consider a family of one-generator quasi-cyclic codes and their applications in quantum codes construction. We give a sufficient condition for self-orthogonality with respect to Hermitian inner product. By virtue of the well-known MacWilliams equations, some binary and ternary stabilizer quantum codes with good parameters are constructed. Furthermore, we present a lower bound on the Hermitian dual distance of these involved codes. As the computational results, some good stabilizer quantum codes over small finite fields are obtained. [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]

Let $q$ be a prime power and $m\geq 3$ be odd. Suppose that $n=\frac {q^{2m}-1}{a}$ with $a|(q^{m}+1)$ and $3\leq a \leq 2(q^{2}-q+1)$. This paper mainly determines the actual maximum designed distance of Hermitian dual-containing Bose-Chaudhuri-Hocquenghem (BCH) codes over $\mathbb {F}_{q^{2}}$ of length $n$. Firstly, we give the maximum designed distance $\delta _{m,a}^{R}$ of narrow-sense Hermitian dual-containing BCH codes. Secondly, we show that there are also non-narrow-sense ones of designed distance up to $\delta _{m,a}^{R}$. It is worth mentioning that our maximum designed distance $\delta _{m,a}^{R}>\lceil \frac {a}{2}\rceil \delta _{m}^{A}$ , where $\delta _{m}^{A}$ is given by Aly et al. (IEEE Trans. Inf. Theory, vol. 53, no. 3, pp. 1183-1188, 2007). Thus, many families of Hermitian dual-containing BCH codes with relatively large designed distance are obtained. Using the Hermitian construction to them, we can subsequently construct different classes of nonprimitive quantum codes, which are new in the sense that their parameters are not covered in the literature. [ABSTRACT FROM AUTHOR]