New covering codes of radius R, codimension tR and \(tR+\frac{R}{2}\), and saturating sets in projective spaces

Abstract

The length function \(\ell _q(r,R)\) is the smallest length of a q-ary linear code of codimension r and covering radius R. In this work we obtain new constructive upper bounds on \(\ell _q(r,R)\) for all \(R\ge 4\), \(r=tR\), \(t\ge 2\), and also for all even \(R\ge 2\), \(r=tR+\frac{R}{2}\), \(t\ge 1\). The new bounds are provided by infinite families of new covering codes with fixed R and increasing codimension. The new bounds improve upon the known ones. We propose a general regular construction (called “Line+Ovals”) of a minimal \(\rho \)-saturating \(((\rho +1)q+1)\)-set in the projective space \(\mathrm {PG}(2\rho +1,q)\) for all \(\rho \ge 0\). Such a set corresponds to an \([Rq+1,Rq+1-2R,3]_qR\) locally optimal code of covering radius \(R=\rho +1\). Basing on combinatorial properties of these codes regarding to spherical capsules, we give constructions for code codimension lifting and obtain infinite families of new surface-covering codes with codimension \(r=tR\), \(t\ge 2\). In addition, we obtain new 1-saturating sets in the projective plane \(\mathrm {PG}(2,q^2)\) and, basing on them, construct infinite code families with fixed even radius \(R\ge 2\) and codimension \(r=tR+\frac{R}{2}\), \(t\ge 1\).

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Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments and suggestions.

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Correspondence to Alexander A. Davydov.

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The research of A.A. Davydov was done at IITP RAS and supported by the Russian Government (Contract No 14.W03.31.0019). The research of S. Marcugini and F. Pambianco was supported in part by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INDAM) and by University of Perugia, (Project: “Curve algebriche in caratteristica positiva e applicazioni”, Base Research Fund 2018).

Communicated by K. Metsch.

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Davydov, A.A., Marcugini, S. & Pambianco, F. New covering codes of radius R, codimension tR and \(tR+\frac{R}{2}\), and saturating sets in projective spaces. Des. Codes Cryptogr. 87, 2771–2792 (2019). https://doi.org/10.1007/s10623-019-00649-2

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Keywords

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

Mathematics Subject Classification

  • 51E21
  • 51E22
  • 94B05