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 Fundam. Prikl. Mat., 2013, Volume 18, Issue 4, Pages 137–154 (Mi fpm1535)

Basic Reed–Muller codes as group codes

I. N. Tumaykin

Lomonosov Moscow State University, Moscow, Russia

Abstract: Reed–Muller codes are one of the most well-studied families of codes, however, there are still open problems regarding their structure. Recently, a new ring-theoretic approach has emerged that provides a rather intuitive construction of these codes. This approach is centered around the notion of basic Reed–Muller codes. We recall that Reed–Muller codes over a prime field are radical powers of a corresponding group algebra. In this paper, we prove that basic Reed–Muller codes in the case of a nonprime field of arbitrary characteristic are distinct from radical powers. This implies the same result for regular codes. Also we show how to describe the inclusion graph of basic Reed–Muller codes and radical powers via simple arithmetic equations.

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English version:
Journal of Mathematical Sciences (New York), 2015, 206:6, 699–710

Bibliographic databases:

UDC: 512.552.7+512.624.95

Citation: I. N. Tumaykin, “Basic Reed–Muller codes as group codes”, Fundam. Prikl. Mat., 18:4 (2013), 137–154; J. Math. Sci., 206:6 (2015), 699–710

Citation in format AMSBIB
\Bibitem{Tum13} \by I.~N.~Tumaykin \paper Basic Reed--Muller codes as group codes \jour Fundam. Prikl. Mat. \yr 2013 \vol 18 \issue 4 \pages 137--154 \mathnet{http://mi.mathnet.ru/fpm1535} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3431838} \transl \jour J. Math. Sci. \yr 2015 \vol 206 \issue 6 \pages 699--710 \crossref{https://doi.org/10.1007/s10958-015-2347-z} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84956716218} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. I. N. Tumaykin, “Group ring ideals related to Reed–Muller codes”, J. Math. Sci., 233:5 (2018), 745–748
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