This article is cited in 1 scientific paper (total in 1 paper)
Basic Reed–Muller codes as group codes
I. N. Tumaykin
Lomonosov Moscow State University, Moscow, Russia
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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Journal of Mathematical Sciences (New York), 2015, 206:6, 699–710
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
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\paper Basic Reed--Muller codes as group codes
\jour Fundam. Prikl. Mat.
\jour J. Math. Sci.
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I. N. Tumaykin, “Group ring ideals related to Reed–Muller codes”, J. Math. Sci., 233:5 (2018), 745–748
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