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IEEE Transactions on Information Theory

Non-Binary Quantum Synchronizable Codes From Repeated-Root Cyclic Codes

Lan Luo, Zhi Ma

Published Jan 19, 2018• 35 citations
Publisher

Abstract

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.

DOI
10.1109/tit.2018.2795479

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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.
Many good quantum error-correcting codes were constructed from cyclic codes. However, it is a difficult problem to determine the true minimum distance of quantum cyclic codes for large length n. In this work, we construct nonbinary quantum cyclic codes and asymmetric quantum cyclic codes that are derived from repeated-root cyclic codes for an arbitrary length ps, and determine the true minimum distance of all those codes. Some proposed quantum cyclic codes are optimal. Additionally, some proposed asymmetric quantum cyclic codes have better parameters than the ones available in the literature.
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 6ps over Fpm.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 6ps over Fpm. 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.
Quantum synchronizable error-correcting codes are special quantum error-correcting codes that are designed to correct both the effect of quantum noise on qubits and misalignment in block synchronization. It is known that, in principle, such a code can be constructed through a combination of a classical linear code and its subcode if the two are both cyclic and dual-containing. However, finding such classical codes that lead to promising quantum synchronizable error-correcting codes is not a trivial task. In fact, although there are two families of classical codes that are proved to produce quantum synchronizable codes with good minimum distances and highest possible tolerance against misalignment, their code lengths have been restricted to primes and Mersenne numbers. In this paper, examining the incidence vectors of projective spaces over the finite fields of characteristic 2, we give quantum synchronizable codes from cyclic codes whose lengths are not primes or Mersenne numbers. These projective geometric codes achieve good performance in quantum error correction and possess the best possible ability to recover synchronization, thereby enriching the variety of good quantum synchronizable codes. We also extend the current knowledge of cyclic codes in classical coding theory by explicitly giving generator polynomials of the finite geometric codes and completely characterizing the minimum weight nonzero codewords. In addition to the codes based on projective spaces, we carry out a similar analysis on the well-known cyclic codes from Euclidean spaces that are known to be majority logic decodable and determine their exact minimum distances.
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.
On a Family of Quantum Synchronizable Codes Based on the (λ(u+v)∣u−v)(\lambda(u + v)|u - v)(λ(u+v)∣u−v) Construction
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.