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Reed–Muller codes are one of the most well-studied families of codes; however, there are till 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. It is known that basic Reed–Muller codes ℳπ(m, k) over a prime field are powers of the radical RS of a corresponding group algebra and over a nonprime field there are no such equalities, except for trivial ones. In this paper, we consider the ideals ℜSℳπ(m, k) that arise while studying the inclusions of the basic codes and radical powers.

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.