Classification of $q$-Ary Perfect Quantum Codes

Li, Zhuo; Xing, Lijuan

doi:10.1109/tit.2012.2217475

Cited by 4 publications

(5 citation statements)

References 19 publications

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“…To illustrate this, we note that there is a significant amount of modern articles which aim to generalise the results in Table 1 to extended codes or to different metrics, but they always treat the case when q is a prime power. For example, under this assumption, Li and Xing [12] classified perfect quantum codes and Gubitosi, Portela, and Qureshi [13] showed analogous results for the Niederreiter-Rosenbloom-Tsfasman metric.…”

Section: 4mentioning

confidence: 78%

“…All perfect codes over prime power alphabets are either Hamming codes or have parameters (n, M, e, q) = (11, 3 6 , 2, 3), (23, 2 12 , 3, 2).…”

Section: Prime Power ≥1mentioning

confidence: 99%

“…Using the algorithm that we have previously presented, we obtain all solutions (x, y) ∈ Z 2 to each of the generalised Ramanujan-Nagell equations. By (12), we can then recover n as:…”

Section: Resolving the Diophantine Equationsmentioning

confidence: 99%

“…Li and Xing [12] classified perfect quantum codes over q-dimensional qudits, where q is a prime power. In addition, they proved an analogue of Lloyd's theorem for quantum codes.…”

Section: Future Lines Of Workmentioning

confidence: 99%

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Perfect Codes over Non-Prime Power Alphabets: An Approach Based on Diophantine Equations

Cazorla García

2024

Mathematics

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Perfect error-correcting codes allow for an optimal transmission of information while guaranteeing error correction. For this reason, proving their existence has been a classical problem in both pure mathematics and information theory. Indeed, the classification of the parameters of e-error correcting perfect codes over q-ary alphabets was a very active topic of research in the late 20th century. Consequently, all parameters of perfect e-error-correcting codes were found if e≥3, and it was conjectured that no perfect 2-error-correcting codes exist over any q-ary alphabet, where q>3. In the 1970s, this was proved for q a prime power, for q=2r3s and for only seven other values of q. Almost 50 years later, it is surprising to note that there have been no new results in this regard and the classification of 2-error-correcting codes over non-prime power alphabets remains an open problem. In this paper, we use techniques from the resolution of the generalised Ramanujan–Nagell equation and from modern computational number theory to show that perfect 2-error-correcting codes do not exist for 172 new values of q which are not prime powers, substantially increasing the values of q which are now classified. In addition, we prove that, for any fixed value of q, there can be at most finitely many perfect 2-error-correcting codes over an alphabet of size q.

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Section: 4mentioning

confidence: 78%

“…All perfect codes over prime power alphabets are either Hamming codes or have parameters (n, M, e, q) = (11, 3 6 , 2, 3), (23, 2 12 , 3, 2).…”

Section: Prime Power ≥1mentioning

confidence: 99%

“…Using the algorithm that we have previously presented, we obtain all solutions (x, y) ∈ Z 2 to each of the generalised Ramanujan-Nagell equations. By (12), we can then recover n as:…”

Section: Resolving the Diophantine Equationsmentioning

confidence: 99%

“…Li and Xing [12] classified perfect quantum codes over q-dimensional qudits, where q is a prime power. In addition, they proved an analogue of Lloyd's theorem for quantum codes.…”

Section: Future Lines Of Workmentioning

confidence: 99%

See 2 more Smart Citations

Perfect Codes over Non-Prime Power Alphabets: An Approach Based on Diophantine Equations

Cazorla García

2024

Mathematics

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show abstract

“…We can assert that this framework is a very useful and potential tool in studying the problems about quantum error-correcting codes. In fact, the idea of this framework has been used to solve the problem about the classification of perfect quantum codes [20]. Recently, we realized that this tool might have potential application in finding out various bounds on the parameters of the most general quantum error-correcting codes.…”

Section: Discussionmentioning

confidence: 99%

Universal Framework for Quantum Error-Correcting Codes

Xing

2021

Entropy

Self Cite

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We present a universal framework for quantum error-correcting codes, i.e., a framework that applies to the most general quantum error-correcting codes. This framework is based on the group algebra, an algebraic notation associated with nice error bases of quantum systems. The nicest thing about this framework is that we can characterize the properties of quantum codes by the properties of the group algebra. We show how it characterizes the properties of quantum codes as well as generates some new results about quantum codes.

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Quantum data-syndrome codes: Subsystem and impure code constructions

Nemec

2023

Quantum Inf Process

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Classification of $q$-Ary Perfect Quantum Codes

Abstract: We solve the problem of the classification of perfect quantum codes. We prove that the only nontrivial perfect quantum codes are those with the parameters ( ( 2 2 2

Cited by 4 publications

References 19 publications

Perfect Codes over Non-Prime Power Alphabets: An Approach Based on Diophantine Equations

Perfect Codes over Non-Prime Power Alphabets: An Approach Based on Diophantine Equations

Universal Framework for Quantum Error-Correcting Codes

Quantum data-syndrome codes: Subsystem and impure code constructions

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