New Bounds for Linear Codes of Covering Radius 2

  • Daniele Bartoli
  • Alexander A. Davydov
  • Massimo Giulietti
  • Stefano Marcugini
  • Fernanda Pambianco
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10495)

Abstract

The length function \(\ell _q(r,R)\) is the smallest length of a q-ary linear code of covering radius R and codimension r. New upper bounds on \(\ell _q(r,2)\) are obtained for odd \(r\ge 3\). In particular, using the one-to-one correspondence between linear codes of covering radius 2 and saturating sets in the projective planes over finite fields, we prove that
$$\begin{aligned} \ell _q(3,2)\le \sqrt{q(3\ln q+\ln \ln q)}+\sqrt{\frac{q}{3\ln q}}+3 \end{aligned}$$
and then obtain estimations of \(\ell _q(r,2)\) for all odd \(r\ge 5\). The new upper bounds are smaller than the previously known ones. Also, the new bounds hold for all q, not necessary large, whereas the previously best known estimations are proved only for q large enough.

Keywords

Covering codes Saturating sets The length function Upper bounds Projective spaces 

Notes

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”) and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INDAM). 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).

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Daniele Bartoli
    • 1
  • Alexander A. Davydov
    • 2
  • Massimo Giulietti
    • 1
  • Stefano Marcugini
    • 1
  • Fernanda Pambianco
    • 1
  1. 1.Department of Mathematics and Computer SciencePerugia UniversityPerugiaItaly
  2. 2.Institute for Information Transmission Problems (Kharkevich Institute), Russian Academy of SciencesMoscowRussian Federation

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