New bounds for linear codes of covering radii 2 and 3

Abstract

The length function q(r, R) is the smallest length of a q-ary linear code of covering radius R and codimension r. In this work we obtain new upper bounds on q(2t + 1,2), q(3t + 1,3), q(3t + 2,3), t ≥ 1. In particular, we prove that

$$\ell_{q}(3,2)\le\sqrt{q(3\ln q+\ln\ln q)}+\sqrt{\frac{q}{3\ln q}}+ 3~\text{ for all } q, $$

\(\ell _{q}(3,2)\le 1.05\sqrt {3q\ln q}\) for q ≤ 321007, \(\ell _{q}(4,3)<2.8\sqrt [3]{q\ln q}\) for q ≤ 6229, and \(\ell _{q}(5,3)<3\sqrt [3]{q^{2}\ln q}\) for q ≤ 761. The new bounds on q(2t + 1,2), q(3t + 1,3), q(3t + 2,3), t > 1, are then obtained by lift-constructions. For q a non-square the new bound on q(2t + 1,2) improves the previously known ones. For many values of q≠(q)3 and r ≠ 3t we provide infinite families of [n, nr]q3 codes showing that \(\ell _{q}(r,3)\thickapprox c\sqrt [3]{\ln q}\cdot q^{(r-3)/3}\), where c is a universal constant. As far as it is known to the authors, such families have not been previously described in the literature.

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Acknowledgements

The research of D. Bartoli, M. Giulietti, S. Marcugini, and F. Pambianco was supported in part by Ministry for Education, University and Research of Italy (MIUR) (Project “Geometrie di Galois e strutture di incidenza”), by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INDAM), and by University of Perugia (Projects ”Configurazioni Geometriche e Superfici Altamente Simmetriche” and ”Codici lineari e strutture geometriche correlate”, Base Research Fund 2015). The research of A.A. Davydov was carried out at the IITP RAS at the expense of the Russian Foundation for Sciences (project 14-50-00150). This work has been carried out using computing resources of the federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC Kurchatov Institute, http://ckp.nrcki.ru/. The authors would like to thank the anonymous referees for their helpful comments and suggestions which improved this paper.

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Bartoli, D., Davydov, A.A., Giulietti, M. et al. New bounds for linear codes of covering radii 2 and 3. Cryptogr. Commun. 11, 903–920 (2019). https://doi.org/10.1007/s12095-018-0335-0

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Keywords

  • Covering codes
  • Saturating sets
  • The length function
  • Upper bounds
  • Projective spaces

Mathematics Subject Classification (2010)

  • 94B05
  • 51E21
  • 51E22