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Designs, Codes and Cryptography

, Volume 87, Issue 2–3, pp 375–391 | Cite as

Classifying optimal binary subspace codes of length 8, constant dimension 4 and minimum distance 6

  • Daniel HeinleinEmail author
  • Thomas Honold
  • Michael Kiermaier
  • Sascha Kurz
  • Alfred Wassermann
Article
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography

Abstract

We determine the maximum size \(A_2(8,6;4)\) of a binary subspace code of packet length \(v=8\), minimum subspace distance \(d=6\), and constant dimension \(k=4\) to be 257. There are two isomorphism types of optimal codes. Both of them are extended LMRD codes. In finite geometry terms, the maximum number of solids in \({\text {PG}}(7,2)\) mutually intersecting in at most a point is 257. The result was obtained by combining the classification of substructures with integer linear programming techniques. This result implies that the maximum size \(A_2(8,6)\) of a binary mixed-dimension subspace code of packet length 8 and minimum subspace distance 6 is 257 as well.

Keywords

Network coding Constant-dimension codes Subspace distance Classification Integer linear programming 

Mathematics Subject Classification

51E20 94B65 05B25 51E23 

Notes

Acknowledgements

The authors would like to thank the High Performance Computing group of the University of Bayreuth for providing the excellent computing cluster and especially Bernhard Winkler for his support. This work was supported by the grants KU 2430/3-1, WA 1666/9-1—“Integer Linear Programming Models for Subspace Codes and Finite Geometry”—from the German Research Foundation and by Grant No. 61571006—“Research on Subspace Codes and Related Combinatorial Structures”—from the National Natural Science Foundation of China. The authors would like to thank the editor and the anonymous referees for their remarks improving the presentation of our results.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BayreuthBayreuthGermany
  2. 2.ZJU-UIUC InstituteZhejiang UniversityHainingChina

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