Johnson type bounds on constant dimension codes
- First Online:
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.
KeywordsConstant dimension codes Linear authentication codes Binary constant weight codes Johnson bounds Steiner structures Random network coding
Mathematics Subject Classifications (2000)94B65 94A62
Unable to display preview. Download preview PDF.
- Koetter R., Kschischang F.: Coding for errors and erasures in random network coding. In: Proceedings of the IEEE International Symposium on Information Theory, pp. 791–795. Nice, France. Full version is available online at http://www.arxiv.org/abs/cs.IT/0703061 (2007).
- MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1981).Google Scholar
- Ho T., Koetter R., Médard M., Karger D., Effros M.: The benefits of coding over routing in a randomized setting. In: Proceedings of the IEEE International Symposium on Information Theory, p. 442. Yokohama, Japan (2003).Google Scholar
- Tonchev V.D.: Codes and designs. In: Pless V.C., Huffman W.C. (eds.) Handbook of Coding Theory, Chapter 15. North-Holland, Amsterdam (1998).Google Scholar