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Journal of Interdisciplinary Mathematics

Quantum synchronizable codes through generalized cyclotomy of order 2r over Z2n

Manju Pruthi, Arun Kumar Yadav, Pooja Soni

Published Jul 4, 2026• 0 citations
Publisher

Abstract

In this work, we introduce the construction of a generalized cyclotomic class of order 2 r over Z 2n , and the work further shows that the resulting codes are cyclic codes containing their dual, along with their augmented counterparts.In addition, the research provides a comprehensive analysis of two categories of quantum synchronization capabilities that can

DOI
10.47974/jim-2382

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In this work, we introduce the construction of a generalized cyclotomic class of order 2 r over Z 2n , and the work further shows that the resulting codes are cyclic codes containing their dual, along with their augmented counterparts.In addition, the research provides a comprehensive analysis of two categories of quantum synchronization capabilities that can
Quantum synchronizable codes can correct the ef-fects of quantum noise on qubits as well as misalignment in block synchronization. In this paper, dual-containing cyclic codes of length2pq2pq2pqandp2p^{2}p2are constructed by generalized cyclotomy, where some of the cyclic codes of lengthp2p^{2}p2are optimal or almost optimal. Based on these cyclic codes and their augmented codes, we obtain two new classes of quantum synchronizable codes whose synchronization capabilities are the best.
Quantum Synchronizable Codes From Cyclotomic Classes of Order Two over Z2q\mathbb{Z}_{2q}Z2q​
Quantum synchronizable codes are kinds of quantum error-correcting codes that can not only correct the effects of quantum noise on qubits but also the misalignment in block synchronization. This paper contributes to constructing two classes of quantum synchronizable codes by the cyclotomic classes of order two over Z2q\mathbb{Z}_{2q}Z2q​, whose synchronization capabilities can reach the upper bound under certain conditions. Moreover, the quantum synchronizable codes possess good error-correcting capability towards bit errors and phase errors.
Quantum synchronizable codes are quantum error-correcting codes designed to correct the effects of both quantum noise and block synchronization errors. While it is known that quantum synchronizable codes can be constructed from cyclic codes that satisfy special properties, only a few classes of cyclic codes have been proved to give promising quantum synchronizable codes. In this paper, using quadratic residue codes and their supercodes, we give a simple construction for quantum synchronizable codes whose synchronization capabilities attain the upper bound. The method is applicable to cyclic codes of prime length.
Quantum synchronizable codes are kinds of quantum error-correcting codes that can not only correct the effects of quantum noise on qubits but also the misalignment in block synchronization. In this paper, the quantum synchronizable codes constructed are CSS quantum error-correcting codes whose synchronization capabilities reach the upper bound. And we use cyclic codes gained by sextic cyclotomic classes to construct two classes of quantum synchronizable codes. Moreover, the quantum synchronizable codes are posses good error-correcting capability towards bit error and phase error, since the cyclic codes we used are optimal or almost optimal.
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.
A new criterion for the existence of Hermitian dual-containing cyclic codes is obtained based on a characterization of q-cyclotomic cosets modulo n. This criterion admits a method of constructing new Hermitian dual-containing cyclic codes (hence new quantum codes) from the old ones, and enables us to construct several classes of quantum maximum-distance-separable (MDS) codes from classical Hermitian dual-containing cyclic codes. We analyse that our codes are good in certain sense. Besides, some enumerative conclusions are also presented as a result of the characterization of q-cyclotomic cosets modulo n.