Group ring ideals related to Reed–Muller codes

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. It is known that basic Reed–Muller codes $\mathcal{M}_{\pi}(m,k)$ over a prime field are powers of the radical $\mathfrak{R}_S$ 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 $\mathfrak{R}_S \mathcal{M}_{\pi}(m,k)$ that arise while studying the inclusions of the basic codes and radical powers.