Ideal representations of Reed–Solomon and Reed–Muller codes

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 of these codes are already known. Our technique differs from the previously used method in the following. There, the codes in question are represented as kernels of some homomorphisms; in other words, the codes are defined by some kind of parity check relation. 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 $_QS$ in a fixed set of $q$ such subspaces, whose sums also present Reed–Solomon codes.