Designs, Codes and Cryptography

, Volume 50, Issue 2, pp 163–172 | Cite as

Johnson type bounds on constant dimension codes

  • Shu-Tao Xia
  • Fang-Wei Fu


Very recently, an operator channel was defined by Koetter and Kschischang when they studied random network coding. They also introduced constant dimension codes and demonstrated that these codes can be employed to correct errors and/or erasures over the operator channel. Constant dimension codes are equivalent to the so-called linear authentication codes introduced by Wang, Xing and Safavi-Naini when constructing distributed authentication systems in 2003. In this paper, we study constant dimension codes. It is shown that Steiner structures are optimal constant dimension codes achieving the Wang-Xing-Safavi-Naini bound. Furthermore, we show that constant dimension codes achieve the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures. Then, we derive two Johnson type upper bounds, say I and II, on constant dimension codes. The Johnson type bound II slightly improves on the Wang-Xing-Safavi-Naini bound. Finally, we point out that a family of known Steiner structures is actually a family of optimal constant dimension codes achieving both the Johnson type bounds I and II.


Constant dimension codes Linear authentication codes Binary constant weight codes Johnson bounds Steiner structures Random network coding 

Mathematics Subject Classifications (2000)

94B65 94A62 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Graduate School at ShenzhenTsinghua UniversityShenzhenPeople’s Republic of China
  2. 2.National Mobile Communications Research LaboratorySoutheast UniversityNanjingPeople’s Republic of China
  3. 3.Chern Institute of MathematicsNankai UniversityTianjinPeople’s Republic of China
  4. 4.Key Laboratory of Pure Mathematics and CombinatoricsNankai UniversityTianjinPeople’s Republic of China

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