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Cryptography and Communications

BCH codes with explicit minimum distance and applications in quantum synchronizable codes

Xueting Wang, Junling Zhou

Published Jun 16, 2025• 0 citations
Publisher
DOI
10.1007/s12095-025-00815-5

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In this paper, we construct a new family of quantum synchronizable codes from repeated-root cyclic codes of lengths psand lpsover Fq, where s 1 andl ≥ 2 are integers, and p ≥ 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 ps and l ps 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.
Let p be an odd prime, and Fp(m)is the finite field of pmelements. In this paper, all maximum distance separable (briefly, MDS) cyclic and negacyclic codes of length 2psover Fp(m)are established. As an application, all quantum MDS (briefly, qMDS) codes are constructed from cyclic and negacyclic codes of length 2psover 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 2psover 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.