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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A new construction of quantum error-correcting codes


Authors: Keqin Feng and Chaoping Xing
Journal: Trans. Amer. Math. Soc. 360 (2008), 2007-2019
MSC (2000): Primary 11T71, 94B60, 05A18
Published electronically: October 23, 2007
MathSciNet review: 2366972
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Abstract: In this paper, we present a characterization of (binary and non-binary) quantum error-correcting codes. Based on this characterization, we introduce a method to construct $ p$-ary quantum codes using Boolean functions satisfying a system of certain quadratic relations. As a consequence of the construction, we are able to construct quantum codes of minimum distance $ 2$. In particular, we produce a class of binary quantum $ ((n,2^{n-2}-\frac 12{n-1\choose (n-1)/2},2))$-codes for odd length $ n\ge 5$. For $ n\ge 11$, this improves the result by Rains in Quantum codes of minimal distance two, 1999, showing the existence of binary quantum $ ((n,3\cdot2^{n-4},2))$-codes for odd $ n\ge 5$. Moreover, our binary quantum $ ((n,2^{n-2}-\frac 12{n-1\choose (n-1)/2},2))$-codes of odd length achieve the Singleton bound asymptotically.

Finally, based on our characterization some propagation rules of quantum codes are proposed and the rules are similar to those in classical coding theory. It turns out that some new quantum codes are found through these propagation rules.


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Additional Information

Keqin Feng
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
Email: kfeng@math.tsinghua.edu.cn

Chaoping Xing
Affiliation: Division of Mathematical Sciences, Nanyang Technological University, Singapore 637616, Republic of Singapore

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04242-0
Received by editor(s): June 30, 2005
Received by editor(s) in revised form: November 7, 2005, and December 20, 2005
Published electronically: October 23, 2007
Additional Notes: This work was supported by the National Scientific Research Project 973 of China
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.