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New bounds for linear codes of covering radii 2 and 3

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Abstract

The length function ℓq(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 ℓq(2t + 1,2), ℓq(3t + 1,3), ℓq(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 ℓq(2t + 1,2), ℓq(3t + 1,3), ℓq(3t + 2,3), t > 1, are then obtained by lift-constructions. For q a non-square the new bound on ℓq(2t + 1,2) improves the previously known ones. For many values of q≠(q′)3 and r ≠ 3t we provide infinite families of [n, n − r]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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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Perugia University, Perugia, 06123, Italy

    Daniele Bartoli, Massimo Giulietti, Stefano Marcugini & Fernanda Pambianco

  2. Institute for Information Transmission Problems (Kharkevich institute), Russian Academy of Sciences, Moscow, 127051, Russia

    Alexander A. Davydov

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  1. Daniele Bartoli
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Correspondence to Alexander A. Davydov.

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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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