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 for these codes are already known. Our technique differs from the previously used method in the following. There, the codes in question were represented as kernels of some homomorphisms; in other words, these were defined by some kind of parity-check relations. 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 Q S in a fixed set of q such subspaces, whose sums also present Reed–Solomon codes.
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(V. T. Markov) Supported by RFBR, grant No. 11-01-00794-a.
(C. Martínez) Supported by MTM2010-18370-C04-01 grant.
(A. A. Nechaev) Supported by the Grants Council (under RF President) for State Aid of Leading Scientific Schools, grants NSh-8.2010.10 and NSh-6260.2012.10.
Translated from Algebra i Logika, Vol. 51, No. 3, pp. 297-320, May-June, 2012.
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Couselo, E., González, S., Markov, V.T. et al. Ideal representation of Reed–Solomon and Reed–Muller codes. Algebra Logic 51, 195–212 (2012). https://doi.org/10.1007/s10469-012-9183-8
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DOI: https://doi.org/10.1007/s10469-012-9183-8