Codes and anticodes in the Grassman graph
Journal of Combinatorial Theory Series A
IEEE Transactions on Information Theory
Linear authentication codes: bounds and constructions
IEEE Transactions on Information Theory
A Random Linear Network Coding Approach to Multicast
IEEE Transactions on Information Theory
Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams
IEEE Transactions on Information Theory
Cryptography and Coding '09 Proceedings of the 12th IMA International Conference on Cryptography and Coding
Construction and covering properties of constant-dimension codes
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 4
Packing and covering properties of subspace codes
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 4
Packing and covering properties of subspace codes for error control in random linear network coding
IEEE Transactions on Information Theory
Constant-rank codes and their connection to constant-dimension codes
IEEE Transactions on Information Theory
Recursive code construction for random networks
IEEE Transactions on Information Theory
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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.