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On Perfect and Reed–Muller Codes over Finite Fields

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Abstract

We consider error-correcting codes over a finite field with \(q\) elements (\(q\)-ary codes). We study relations between single-error-correcting \(q\)-ary perfect codes and \(q\)-ary Reed–Muller codes. For \(q\ge 3\) we find parameters of affine Reed–Muller codes of order \((q-1)m-2\). We show that affine Reed–Muller codes of order \((q-1)m-2\) are quasi-perfect codes. We propose a construction which allows to construct single-error-correcting \(q\)-ary perfect codes from codes with parameters of affine Reed–Muller codes. A modification of this construction allows to construct \(q\)-ary quasi-perfect codes with parameters of affine Reed–Muller codes.

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Acknowledgment

The author is sincerely grateful to a reviewer for valuable remarks and suggestions, which helped him to considerably improve the original version of the paper.

Funding

The research was supported by the Fundamental Scientific Research Program no. I.5.1 of the Siberian Branch of the Russian Academy of Sciences, project no. 0314-2019-0017.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

    A. M. Romanov

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  1. A. M. Romanov
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Translated from Problemy Peredachi Informatsii, 2021, Vol. 57, No. 3, pp. 3–16 https://doi.org/10.31857/S0555292321030013.

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Romanov, A. On Perfect and Reed–Muller Codes over Finite Fields. Probl Inf Transm 57, 199–211 (2021). https://doi.org/10.1134/S0032946021030017

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