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Problemy Peredachi Informatsii, 2021, Volume 57, Issue 3, Pages 3–16
DOI: https://doi.org/10.31857/S0555292321030013 (Mi ppi2344) 		 
This article is cited in 6 scientific papers (total in 6 papers)

Coding Theory

On perfect and Reed–Muller codes over finite fields

A. M. Romanov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
Full-text PDF (232 kB) Citations (6)
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DOI: https://doi.org/10.31857/S0555292321030013
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.
Keywords: Reed–Muller code, affine Reed–Muller code, projective Reed–Muller code, Hamming code, perfect code, quasi-perfect code, MDS code, finite field.
Funding agency Grant number
Siberian Branch of Russian Academy of Sciences I.5.1., проект № 0314-2019-0017
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.
Received: 30.06.2020
Revised: 12.04.2021
Accepted: 04.06.2021
English version:
Problems of Information Transmission, 2021, Volume 57, Issue 3, Pages 199–211
DOI: https://doi.org/10.1134/S0032946021030017
Bibliographic databases:
Document Type: Article
UDC: 621.391 : 519.725
Language: Russian
Citation: A. M. Romanov, “On perfect and Reed–Muller codes over finite fields”, Probl. Peredachi Inf., 57:3 (2021), 3–16; Problems Inform. Transmission, 57:3 (2021), 199–211
Citation in format AMSBIB
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\by A.~M.~Romanov
\paper On perfect and Reed--Muller codes over finite fields
\jour Probl. Peredachi Inf.
\yr 2021
\vol 57
\issue 3
\pages 3--16
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\crossref{https://doi.org/10.31857/S0555292321030013}
\transl
\jour Problems Inform. Transmission
\yr 2021
\vol 57
\issue 3
\pages 199--211
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    • This publication is cited in the following 6 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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