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Entanglement-assisted quantum error-correcting (EAQEC) codes are a generalization of quantum error-correcting (QEC) codes, which can be constructed from arbitrary classical linear codes by relaxing the dual-containing condition and by using preshared entangled states between the sender and the receiver. In this paper, we investigate EAQEC codes of length n = 2 (q 2 + 1) a , where q is an odd prime power, a = m 2 + 1 and m is an odd integer. The resulting EAQEC codes are entanglement-assisted quantum maximum-distance-separable (EAQMDS) codes when the minimum distance d ≤ n + 2 2 . [ABSTRACT FROM AUTHOR]

Maximum distance separable (MDS) quantum error-correcting codes (QECCs) and MDS entanglement-assisted QECCs (EAQECCs) play significant roles in quantum information theory. In this paper, we construct several new families of MDS QECCs and MDS EAQECCs by utilizing Hermitian self-orthogonal generalized Reed–Solomon codes. These newly obtained MDS QECCs contain some known classes of MDS QECCs as subclasses and some of them have larger minimum distance. In addition, many q-ary MDS QECCs and MDS EAQECCs in our constructions have length exceeding q + 1 and minimum distance surpassing q 2 + 1 . [ABSTRACT FROM AUTHOR]

By generalizing the stabilizer quantum error-correcting codes, entanglement-assisted quantum error-correcting (EAQEC) codes were introduced, which could be derived from any classical linear codes via the relaxation of self-orthogonality conditions with the aid of pre-shared entanglement between the sender and the receiver. In this paper, three classes of entanglement-assisted quantum error-correcting maximum-distance-separable (EAQMDS) codes are constructed through generalized Reed–Solomon codes. Under our constructions, the minimum distances of our EAQMDS codes are much larger than those of the known EAQMDS codes of the same lengths that consume the same number of bits. Furthermore, some of the lengths of the EAQMDS codes are not divisors of q 2 - 1 , which are completely new and unlike all those known lengths existed before. [ABSTRACT FROM AUTHOR]

The hull of a linear code is the intersection of itself with its dual code with respect to certain inner product. Both Euclidean and Hermitian hulls are of theorical and practical significance. In this paper, we construct several new classes of maximum distance separable (MDS) codes via (extended) generalized Reed-Solomon (GRS) codes and determine their Euclidean or Hermitian hulls. As a consequence, four new classes of MDS codes with Hermitian hulls of flexible dimensions and six new classes of MDS codes with Euclidean hulls of flexible dimensions are constructed. As applications, for the former, we further construct four new families of entanglement-assisted quantum error-correcting codes (EAQECCs) and four new families of MDS EAQECCs of length n > q + 1 . Meanwhile, many of the distance parameters of our MDS EAQECCs are greater than ⌈ q 2 ⌉ or q; for the latter, we show some examples on Euclidean self-orthogonal and one-dimensional Euclidean hull MDS codes. In addition, two new general methods for constructing extended GRS codes with (k - 1) -dimensional Hermitian hull and Hermitian self-orthogonal extended GRS codes are also provided. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum error-correcting codes as a generalization of stabilizer quantum error-correcting (QEC) codes can improve the performance of stabilizer QEC codes and can be constructed from arbitrary classical linear codes by relaxing the dual-containing condition and using pre-shared entanglement states between the sender and the receiver. In this paper, we construct some families of entanglement-assisted quantum maximum distance separable codes with parameters [ [ q 2 - 1 a , q 2 - 1 a - 2 (d - 1) + c , d ; c ] ] q , where q is an odd prime power with the form q = a m ± l , a = l 2 - 1 or a = l 2 - 1 2 , l is an odd integer, and m is a positive integer. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature. [ABSTRACT FROM AUTHOR]

Galois hulls are generalizations of Euclidean and Hermitian hulls, which have attracted interest because of their important applications in determining the complexity of some algorithms about linear codes and in constructing entanglement-assisted quantum error-correcting codes (EAQECCs). In this paper, the whole contribution is two-folded. On one hand, we propose explicit methods to construct linear codes of larger lengths with Galois hulls of arbitrary dimensions from given Galois self-orthogonal codes. Conditions required in our approach are proven to be relatively weak. On the other hand, we apply these results to construct EAQECCs. Two bounds for EAQECCs constructed from linear codes with prescribed dimensional Galois hull are given and EAQECCs with rates greater than or equal to 1 2 and positive net rates can be obtained. We also present many interesting examples to explain visually how these two aspects work. [ABSTRACT FROM AUTHOR]

In this paper, a family of new entanglement-assisted quantum error-correcting codes (EAQECCs) is constructed from constacyclic codes with length n = q 2 + 1 a by giving a new method to select defining sets, where a = m 2 + 1 , m > 2 is even and q is an odd prime power with a ∣ (q ± m) . It is worth pointing out that those EAQECCs are entanglement-assisted quantum maximum distance separable (EAQMDS) codes when d ≤ n + 2 2 . All of them are new in the sense that their parameters are not covered by the previously known ones. Moreover, they have minimum distance larger than q + 1 . Compared with the codes with the same length listed in Table 1, our codes have larger minimum distance. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum maximum distance separable (MDS) codes form a significant class of quantum codes. By using constacyclic codes, we construct some new classes of q-ary entanglement-assisted quantum error-correcting MDS codes. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we construct four families of entanglement-assisted quantum error-correcting 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. [ABSTRACT FROM AUTHOR]

The entanglement-assisted stabilizer formalism can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes. In this work, we construct some new entanglement-assisted quantum maximum distance separable codes with length n = q 2 + 1 5 from cyclic codes. Compared with all the previously known parameters with the same length, all of them have flexible parameters and larger minimum distance. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum maximum distance separable (MDS) codes form a significant class of quantum codes. By using generalized Reed–Solomon (GRS) codes and extended GRS codes, we construct some new classes of q-ary entanglement-assisted quantum error-correcting MDS codes. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, where we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several new families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. Moreover, several new classes of q-ary MDS EAQECCs of length $ {n} > {q}+1$ are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we construct optimal or almost optimal dual-containing cyclic codes from cyclotomic classes of order r. Based on these cyclic codes constructed, we obtain many new quantum codes comparing with the known literatures. Furthermore, we construct some quantum synchronizable codes with good error-correcting ability towards bit errors and phase errors by a pair of cyclic codes with special containing property. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum error-correcting codes are a generalization of standard stabilizer quantum error-correcting codes, which can be possibly constructed from any classical codes by relaxing the duality condition and utilizing pre-shared entanglement between the sender and the receiver. In this paper, we construct seven new families of entanglement-assisted quantum maximum-distance-separable codes from cyclic codes by exploiting less pre-shared entangled states. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we give the structures of all ( 1 + u + v ) -constacyclic codes of length 2 s over F 2 + u F 2 + v F 2 + u v F 2 , we also obtain the number of codewords, and the dual of each constacyclic code. Among other results, we study the form of such constacyclic codes that are self-dual and give the number of these codes. We end by giving all self-dual constacyclic codes of length 2, 4 and 8. [ABSTRACT FROM AUTHOR]

We give a construction for quantum codes from linear and cyclic codes over $\mathbb{F}_4 + u\mathbb{F}_4$. We derive Hermitian self-orthogonal codes over $\mathbb{F}_4$ as Gray images of linear and cyclic codes over $\mathbb{F}_4 + u\mathbb{F}_4$. In particular, we use two binary codes associated with a cyclic code over $\mathbb{F}_4 + u\mathbb{F}_4$ of odd length to determine the parameters of the corresponding quantum code. [ABSTRACT FROM AUTHOR]