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 Algebra Logika: Year: Volume: Issue: Page: Find

 Algebra Logika, 2012, Volume 51, Number 3, Pages 297–320 (Mi al536)

Ideal representations of Reed–Solomon and Reed–Muller codes

E. Couseloa, S. Gonzáleza, V. T. Markovb, C. Martíneza, A. A. Nechaevc

a University of Oviedo, Oviedo, Spain
b Moscow State University, Moscow, Russia
c Moscow State University, Moscow, Russia

Abstract: Reed–Solomon codes and Reed–Muller codes are represented as ideals of the group ring $S=QH$ of an elementary Abelian $p$-group $H$ over a finite field $Q=\mathbb F_q$ of characteristic $p$. Such representations of these codes are already known. Our technique differs from the previously used method in the following. There, the codes in question are represented as kernels of some homomorphisms; in other words, the codes are defined by some kind of parity check relation. Here, we explicitly specify generators for the ideals presenting the codes. In this case Reed–Muller codes are obtained by applying the trace function to some sums of one-dimensional subspaces of $_QS$ in a fixed set of $q$ such subspaces, whose sums also present Reed–Solomon codes.

Keywords: Reed–Muller codes, Reed–Solomon codes, group ring, elementary Abelian $p$-group.

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English version:
Algebra and Logic, 2012, 51:3, 195–212

Bibliographic databases:

UDC: 519.725+512.552.7
Revised: 18.04.2012

Citation: E. Couselo, S. González, V. T. Markov, C. Martínez, A. A. Nechaev, “Ideal representations of Reed–Solomon and Reed–Muller codes”, Algebra Logika, 51:3 (2012), 297–320; Algebra and Logic, 51:3 (2012), 195–212

Citation in format AMSBIB
\Bibitem{CouGonMar12} \by E.~Couselo, S.~Gonz\'alez, V.~T.~Markov, C.~Mart{\'\i}nez, A.~A.~Nechaev \paper Ideal representations of Reed--Solomon and Reed--Muller codes \jour Algebra Logika \yr 2012 \vol 51 \issue 3 \pages 297--320 \mathnet{http://mi.mathnet.ru/al536} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3013906} \zmath{https://zbmath.org/?q=an:06121536} \transl \jour Algebra and Logic \yr 2012 \vol 51 \issue 3 \pages 195--212 \crossref{https://doi.org/10.1007/s10469-012-9183-8} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000309471100001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84866148448} 

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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, “Basic Reed–Muller codes as group codes”, J. Math. Sci., 206:6 (2015), 699–710
2. I. N. Tumaikin, “Basic Reed–Muller codes and their connections with powers of radical of group algebra over a non-prime field”, Moscow University Mathematics Bulletin, 68:6 (2013), 295–298
3. I. N. Tumaykin, “Group ring ideals related to Reed–Muller codes”, J. Math. Sci., 233:5 (2018), 745–748
4. K. V. Vedenev, V. M. Deundyak, “The structure of finite group algebra of a semidirect product of abelian groups and its applications”, Chebyshevskii sb., 20:3 (2019), 107–123
5. K. V. Vedenev, V. M. Deundyak, “Relationship between Codes and Idempotents in a Dihedral Group Algebra”, Math. Notes, 107:2 (2020), 201–216
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