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Let R v , q = F q + v F q with v 2 = v . In this paper, we provide three methods of constructing quantum synchronizable codes (QSCs) by using the Euclidean sums of (1 - 2 v) -constacyclic codes over R v , q . Concrete examples are presented to enrich the variety of available QSCs. In addition, two of methods in our research are easier than available methods for constructing QSCs and producing more QSCs. [ABSTRACT FROM AUTHOR]

For any odd prime p ≠ 5 , the structures of cyclic codes of length 5 p s over F p m are applied to construct quantum error-correcting codes (briefly, QEC codes). Some new QEC codes are provided in the sense that their parameters are different from all the previous constructions. We give all quantum maximum-distance-separable (briefly, qMDS codes) constructed by the CSS construction. We also construct quantum synchronizable codes (briefly, QSCs). To enrich the variety of available QSCs, many new QSCs are constructed to illustrate our results. Among them, there are QSCs codes with shorter lengths and much larger minimum distances than known primitive narrow-sense BCH codes. [ABSTRACT FROM AUTHOR]

In this paper, we use the CSS and Steane's constructions to establish quantum error-correcting codes (briefly, QEC codes) from cyclic codes of length 3 p s over F p m . We obtain several new classes of QEC codes in the sense that their parameters are different from all the previous constructions. Among them, we identify all quantum MDS (briefly, qMDS) codes, i.e., optimal quantum codes with respect to the quantum Singleton bound. In addition, we construct quantum synchronizable codes (briefly, QSCs) from cyclic codes of length 3 p s over F p m . Furthermore, we give many new QSCs to enrich the variety of available QSCs. A lot of them are QSCs codes with shorter lengths and much larger minimum distances than known non-primitive narrow-sense BCH codes. [ABSTRACT FROM AUTHOR]

Quantum synchronizable codes are generally constructed from cyclic codes. In this paper, we extend the class of codes available for constructing quantum synchronizable codes. We prove how to construct quantum synchronizable codes using quasi-cyclic codes. We present the method for constructing quantum synchronizable codes from two families of repeated-root quasi-cyclic codes. In addition, we also prove the sufficient condition of Euclidean self-orthogonality and the minimum distance lower bound for the repeated-root quasi-cyclic codes with generators of the form (g(x), k(x)) and (f(x), f(x)). [ABSTRACT FROM AUTHOR]

Quantum synchronizable codes can be used to guard against both the interference of quantum noise on qubits and misalignment in block synchronization. In this paper, we utilize classical BCH codes to obtain three classes of quantum synchronizable codes. These codes can achieve the best achievable synchronization recovery capabilities. By computing the exact minimum distance of classical BCH codes, we determine the error-correcting performance towards Pauli errors of the resulting synchronizable codes. [ABSTRACT FROM AUTHOR]

Quantum synchronizable codes (QSCs) are special quantum error-correcting codes which can be used to correct the effects of quantum noise on qubits and misalignment in block synchronization. In this paper, we first construct a new class of QSCs from cyclic codes of length 2 n , by using the cyclotomic cosets. Besides, we give another new class of QSCs based on BCH codes of length an integer N satisfying some certain conditions. The synchronization capabilities of all these QSCs are always the upper bound. In addition, the lower bounds of correcting bit errors and phase errors for these codes are also given. [ABSTRACT FROM AUTHOR]

Quantum synchronizable codes (QSCs) are special quantum error-correcting codes that can correct the effects of both quantum noise on qubits and misalignment in block synchronization. In this paper, we propose two new methods to construct the QSCs from the Euclidean sums of cyclic or constacyclic codes over finite rings. The first one is derived from the Calderbank–Shor–Steane (CSS) construction applied to Euclidean sums of cyclic codes over finite chain rings. The second construction is derived from the CSS construction applied to Gray images of the Euclidean sums of constacyclic codes over semi-local rings F p + v F p with v 2 = v . By using the two methods, concrete examples are presented to construct new QSCs, whose synchronization capabilities attain the upper bound. [ABSTRACT FROM AUTHOR]

In this paper, we propose a family of quantum synchronizable codes from repeated-root cyclic codes and constacyclic codes. This family of quantum synchronizable codes are based on (λ(u + v)|u - v) construction which is constructed from constacyclic codes. Under this construction, we enrich the varieties of valid quantum synchronizable codes. We also prove that the obtained quantum synchronizable codes can achieve maximum synchronization error tolerance. Furthermore, quantum synchronizable codes based on (λ(u + v)|u - v) construction are shown to be able to have a better capability in correcting bit errors than those from projective geometry codes.

Let p be an odd prime, and Fp(m) is the finite field of pm elements. In this paper, all maximum distance separable (briefly, MDS) cyclic and negacyclic codes of length 2ps over Fp(m) are established. As an application, all quantum MDS (briefly, qMDS) codes are constructed from cyclic and negacyclic codes of length 2ps over finite fields using the Calderbank Shor-Steane (briefly, CSS) and Hermitian constructions. These codes are new in the sense that their parameters are different from all the previous constructions. Furthermore, quantum synchronizable codes (briefly, QSCs) are obtained from cyclic codes of length 2ps over Fp(m). To enrich the variety of available QSCs, many new QSCs are constructed to illustrate our results. Among them, there are QSCs codes with shorter lengths and much larger minimum distances than known primitive narrow-sense Bose-Chaudhuri-Hocquenghem (briefly, BCH) codes.

In this paper, we establish all quantum error-correcting codes (briefly, QEC codes) from cyclic and negacyclic codes of length 4 p s over F p m using CSS and Steane's constructions. Some new QEC codes are given in the sense that their parameters are different from all the previous constructions. In addition, if we fix the length and the Hamming distance of a QEC code, some QEC codes are better than all QEC codes in the sense that their dimensions are larger than the dimensions of all the previous QEC codes. We also investigate quantum maximum distance separable (briefly, qMDS) codes from cyclic and negacyclic codes of length 4 p s over finite fields using the CSS and Hermitian constructions. We provide all qMDS codes constructed from dual codes of cyclic and negacyclic codes of length 4 p s over finite fields using the Hermitian construction. We also construct quantum synchronizable codes (briefly, QSCs) from cyclic codes of length 4 p s over F p m . To enrich the variety of available QSCs, many new QSCs are constructed to illustrate our results. Among them, there are QSCs codes with shorter lengths and much larger minimum distances than known non-primitive narrow-sense BCH codes. [ABSTRACT FROM AUTHOR]

Quantum synchronizable codes (QSCs) are special quantum error-correcting codes that can correct the effects of both quantum noise on qubits and misalignment in block synchronization. In this paper, using the factorizations of cyclotomic polynomials Φ p 1 p 2 (x) , where p1 and p2 are distinct odd primes, we give a new method to construct QSCs whose synchronization capabilities can reach the best attainable tolerance against misalignment. [ABSTRACT FROM AUTHOR]

In this paper, we provide two methods of constructing quantum synchronizable codes from cyclic or constacyclic codes over finite rings. The first one is derived from the Calderbank-Shor-Steane (briefly, CSS) construction applied to dual-containing codes over finite chain rings. The second construction is derived from the CSS construction applied to Gray images of the constacyclic codes over semi-local rings F p + v F p with v 2 = v . By using two methods, concrete examples are presented to construct new quantum synchronizable codes. [ABSTRACT FROM AUTHOR]

Quantum synchronizable codes (QSCs) are special quantum error-correcting codes that correct the effects of both quantum noise on qubits and misalignment in block synchronization. In this letter, we construct a family of QSCs from cyclic codes. The best attainable misalignment tolerance can also be achieved under certain constraints. And we give two classes of specific constructions of QSCs, which achieve the best attainable misalignment tolerance. Moreover, the second class of QSCs possess good error-correcting capability towards bit errors and phase errors. [ABSTRACT FROM AUTHOR]

Due to the wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting research topic in coding theory. In this paper, let p be a prime with p ≥ 7. We determine the Hamming distances of all repeated-root cyclic codes of length 5 p s over F q , where q = p m and both s and m are positive integers. Furthermore, we find all MDS cyclic codes of length 5 p s and take quantum synchronizable codes from repeated-root cyclic codes of length 5 p s . By comparing the minimum distances of 5 p s -length repeated-root cyclic codes to BCH codes of close lengths, we illustrated that quantum synchronizable codes constructed from repeated-root cyclic codes have in general better performance in correcting Pauli errors than those from BCH codes. [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]

Quantum synchronizable codes (QSCs) are special quantum error-correcting codes (QECCs) that correct the effects of both quantum noise on qubits and misalignment in block synchronization. In this letter, we construct two classes of QSCs from the cyclic codes which are obtained by the cyclotomic classes of order four. The QSCs constructed are Calderbank–Shor–Steane QECCs which can tolerate maximum number of misalignment errors. Moreover, the QSCs usually possess good error-correcting capability towards bit error and phase error, due to many cyclic codes employed to construct QSCs are optimal or almost optimal. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new family of quantum synchronizable codes from repeated-root cyclic codes of lengths p^s and lp^s over \mathbb Fq , where $s\geq 1~and l\geq 2$ are integers, and $p\geq 3$ is the odd characteristic. Within some loose limitations, these synchronizable codes can possess the best possible capability in synchronization recovery, and therefore, enriches the variety of good quantum synchronizable codes. Furthermore, by using known techniques in classical coding theory which convert the computation of the minimum distance of a repeated-root cyclic code to that of a shorter simple-root cyclic code, we prove that the repeated-root cyclic codes of lengths p^s and lp^s are in general better than narrow-sense BCH codes of close lengths in terms of minimum distances, and thereby enable the obtained synchronizable codes to correct more Pauli errors. [ABSTRACT FROM PUBLISHER]