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Classification of large partial plane spreads in \({{\,\mathrm{PG}\,}}(6,2)\) and related combinatorial objects

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Abstract

The partial plane spreads in \({{\,\mathrm{PG}\,}}(6,2)\) of maximum possible size 17 and of size 16 are classified. Based on this result, we obtain the classification of the following closely related combinatorial objects: vector space partitions of \({{\,\mathrm{PG}\,}}(6,2)\) of type \((3^{16} 4^1)\), binary \(3\times 4\) MRD codes of minimum rank distance 3, and subspace codes with the optimal parameters \((7,17,6)_2\) and \((7,34,5)_2\).

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Acknowledgements

We thank the anonymous referees for their suggestions and careful reading. 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 second and the third author are members of the Action IC1104 Random Network Coding and Designs over GF(q). 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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Authors and Affiliations

  1. Department of Information and Electronic Engineering, Zhejiang University, 38 Zheda Road, 310027, Hangzhou, China

    Thomas Honold

  2. Mathematisches Institut, Universität Bayreuth, 95440, Bayreuth, Germany

    Michael Kiermaier & Sascha Kurz

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  1. Thomas Honold
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Correspondence to Michael Kiermaier.

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Honold, T., Kiermaier, M. & Kurz, S. Classification of large partial plane spreads in \({{\,\mathrm{PG}\,}}(6,2)\) and related combinatorial objects. J. Geom. 110, 5 (2019). https://doi.org/10.1007/s00022-018-0459-6

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