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Designs, Codes and Cryptography

, Volume 86, Issue 2, pp 251–268 | Cite as

A new series of large sets of subspace designs over the binary field

  • Michael Kiermaier
  • Reinhard Laue
  • Alfred Wassermann
Article
Part of the following topical collections:
  1. Special Issue on Network Coding and Designs

Abstract

In this article, we show the existence of large sets \({\text {LS}}_2[3](2,k,v)\) for infinitely many values of k and v. The exact condition is \(v \ge 8\) and \(0 \le k \le v\) such that for the remainders \(\bar{v}\) and \(\bar{k}\) of v and k modulo 6 we have \(2 \le \bar{v} < \bar{k} \le 5\). The proof is constructive and consists of two parts. First, we give a computer construction for an \({\text {LS}}_2[3](2,4,8)\), which is a partition of the set of all 4-dimensional subspaces of an 8-dimensional vector space over the binary field into three disjoint 2-\((8, 4, 217)_2\) subspace designs. Together with the already known \({\text {LS}}_2[3](2,3,8)\), the application of a recursion method based on a decomposition of the Graßmannian into joins yields a construction for the claimed large sets.

Keywords

Large set Subspace design Recursion Method of Kramer and Mesner 

Mathematics Subject Classification

05B05 05B25 51E05 

Notes

Acknowledgements

The authors would like to acknowledge the financial support provided by COST—European Cooperation in Science and Technology. The authors are members of the Action IC1104 Random Network Coding and Designs over GF(q).

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Michael Kiermaier
    • 1
  • Reinhard Laue
    • 2
  • Alfred Wassermann
    • 1
  1. 1.Mathematisches InstitutUniversität BayreuthBayreuthGermany
  2. 2.Institut für InformatikUniversität BayreuthBayreuthGermany

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