Geodesic
Basic Reed–Muller codes as group codes
Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 4, pp. 137-154
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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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I. N. Tumaykin. Basic Reed–Muller codes as group codes. Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 4, pp. 137-154. http://geodesic.mathdoc.fr/item/FPM_2013_18_4_a10/

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