Ideal representation of Reed–Solomon and Reed–Muller codes

Abstract

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