Johnson type bounds on constant dimension codes

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 WangXing-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. keywords: Constant dimension codes, linear authentication codes, binary constant weight codes, Johnson bounds, Steiner structures, random network coding. This research is supported in part by the NSFC-GDSF Joint Fund under Grant No. U0675001, and the open research fund of National Mobile Communications Research Laboratory, Southeast University. S.-T. Xia is with the Graduate School at Shenzhen of Tsinghua University, Shenzhen, Guangdong 518055, P. R. China. He is also with the National Mobile Communications Research Laboratory, Southeast University, P.R. China. E-mail: [email protected] F.-W. Fu is with the Chern Institute of Mathematics, and The Key Laboratory of Pure Mathematics and Combinatorics, Nankai University, Tianjin 300071, P.R. China. Email: [email protected]