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A new distance-regular graph of diameter 3 on 1024 vertices

  • Minjia ShiEmail author
  • Denis S. Krotov
  • Patrick Solé
Article
  • 27 Downloads

Abstract

The dodecacode is a nonlinear additive quaternary code of length 12. By puncturing it at any of the twelve coordinates, we obtain a uniformly packed code of distance 5. In particular, this latter code is completely regular but not completely transitive. Its coset graph is distance-regular of diameter three on \(2^{10}\) vertices, with new intersection array \(\{33,30,15;1,2,15\}\). The automorphism groups of the code, and of the graph, are determined. Connecting the vertices at distance two gives a strongly regular graph of (previously known) parameters \((2^{10}, 495,238, 240)\). Another strongly regular graph with the same parameters is constructed on the codewords of the dual code. A non trivial completely regular binary code of length 33 is constructed.

Keywords

Distance-regular graphs Completely regular codes Uniformly packed codes Additive quaternary codes 

Mathematics Subject Classification

05E30 94B05 

Notes

Acknowledgements

We thank Jack Koolen, Alexander Makhnev, and Bill Martin for helpful discussions and the anonymous referees for useful comments.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesAnhui UniversityHefeiChina
  2. 2.Sobolev Institute of MathematicsNovosibirskRussia
  3. 3.4CNRS/LAGAUniversity of Paris 8Saint-DenisFrance

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