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Partial Spreads and Vector Space Partitions

  • Thomas Honold
  • Michael Kiermaier
  • Sascha Kurz
Chapter
Part of the Signals and Communication Technology book series (SCT)

Abstract

Constant-dimension codes with the maximum possible minimum distance have been studied under the name of partial spreads in Finite Geometry for several decades. Not surprisingly, for this subclass typically the sharpest bounds on the maximal code size are known. The seminal works of Beutelspacher and Drake & Freeman on partial spreads date back to 1975 and 1979, respectively. From then until recently, there was almost no progress besides some computer-based constructions and classifications. It turns out that vector space partitions provide the appropriate theoretical framework and can be used to improve the long-standing bounds in quite a few cases. Here, we provide a historic account on partial spreads and an interpretation of the classical results from a modern perspective. To this end, we introduce all required methods from the theory of vector space partitions and Finite Geometry in a tutorial style. We guide the reader to the current frontiers of research in that field, including a detailed description of the recent improvements.

Notes

Acknowledgements

The authors would like to acknowledge the financial support provided by COST – European Cooperation in Science and Technology. The first author was also supported by the National Natural Science Foundation of China under Grant 61571006. The third author was supported in part by the grant KU 2430/3-1 – Integer Linear Programming Models for Subspace Codes and Finite Geometry from the German Research Foundation.

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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Zhejiang UniversityHangzhouChina
  2. 2.University of BayreuthBayreuthGermany

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