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The geometric approach to the existence of some quaternary Griesmer codes

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Abstract

In this paper we prove the nonexistence of the hypothetical arcs with parameters (395, 100), (396, 100), (448, 113), and (449, 113) in \({{\,\mathrm{PG}\,}}(4,4)\). This rules out the existence of Griesmer codes with parameters \([395,5,295]_4\), \([396,5,296]_4\), \([448,5,335]_4\), \([449,5,336]_4\) and solves four instances of the main problem of coding theory for \(q=4\), \(k=5\). The proof relies on the characterization of (100, 26)- and (113, 29)-arcs in \({{\,\mathrm{PG}\,}}(3,4)\) and is entirely computer-free.

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Acknowledgements

The research of the first author was supported by the Science Research Fund of Sofia University under Contract 80-10-81/15.04.2019. The research of the second author was supported by the Fund for Strategic Development of the New Bulgarian University under Contract 1428/28.03.2019.

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Authors and Affiliations

  1. Faculty of Mathematics and Informatics, Sofia University “St. Kl. Ohridski”, 5 J. Bourchier Blvd., 1164, Sofia, Bulgaria

    Assia Rousseva

  2. New Bulgarian University, 21 Montevideo str., 1618, Sofia, Bulgaria

    Ivan Landjev

  3. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 Acad G. Bonchev str., 1113, Sofia, Bulgaria

    Ivan Landjev

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  1. Assia Rousseva
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  2. Ivan Landjev
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Correspondence to Assia Rousseva.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography 2019”.

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Rousseva, A., Landjev, I. The geometric approach to the existence of some quaternary Griesmer codes. Des. Codes Cryptogr. 88, 1925–1940 (2020). https://doi.org/10.1007/s10623-020-00777-0

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  • DOI: https://doi.org/10.1007/s10623-020-00777-0

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