Johnson type bounds on constant dimension codes |
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Authors: | Shu-Tao Xia Fang-Wei Fu |
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Institution: | (1) Graduate School at Shenzhen, Tsinghua University, Shenzhen, Guangdong, 518055, People’s Republic of China;(2) National Mobile Communications Research Laboratory, Southeast University, Nanjing, Jiangsu, People’s Republic of China;(3) Chern Institute of Mathematics, Nankai University, Tianjin, 300071, People’s Republic of China;(4) Key Laboratory of Pure Mathematics and Combinatorics, Nankai University, Tianjin, 300071, People’s Republic of China |
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Abstract: | 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.
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Keywords: | Constant dimension codes Linear authentication codes Binary constant weight codes Johnson bounds Steiner structures Random network coding |
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