On a Problem Concerning the Quantum Hamming Bound for Impure Quantum Codes

A famous open problem in the theory of quantum error-correcting codes is whether or not the parameters of an impure quantum code can violate the quantum Hamming bound for pure quantum codes. We partially solve this problem. We demonstrate that there exists a threshold N ( d , m ) such that an arbitr...

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Bibliographic Details
Published in:IEEE transactions on information theory 2010-09, Vol.56 (9), p.4731-4734
Main Authors: Li, Zhuo, Xing, Lijuan
Format: Article
Language:English
Subjects:
Applied sciences
Binary codes
Code standards
Coding, codes
Electronic mail
Error correction
Error correction codes
Exact sciences and technology
Hamming codes
Impure quantum codes
Information systems
Information theory
Information, signal and communications theory
Intserv networks
Polynomials
quantum error-correcting codes
quantum Hamming bound
Quantum mechanics
Quantum theory
Signal and communications theory
State-space methods
Telecommunications and information theory
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Summary:A famous open problem in the theory of quantum error-correcting codes is whether or not the parameters of an impure quantum code can violate the quantum Hamming bound for pure quantum codes. We partially solve this problem. We demonstrate that there exists a threshold N ( d , m ) such that an arbitrary ((n , K , d )) m quantum code must obey the quantum Hamming bound whenever n ≥ N ( d , m ). We list some values of N ( d , m ) for small d and binary quantum codes.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2010.2054610