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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 -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 . In particular, we produce a class of binary quantum -codes for odd length . For , this improves the result by Rains in *Quantum codes of minimal distance two*, 1999, showing the existence of binary quantum -codes for odd . Moreover, our binary quantum -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.