On Reed–Muller codes over and some applications

In this work, the main purpose is to extend some well known binary and quaternary codes to the ring F 2 + u F 2 . Reed–Muller, Goethals, Delsarte–Goethals codes are extended, their properties and relations to binary and quaternary versions are studied. Double error correcting families of codes as Goethals and shortened Goethals codes over F 2 + u F 2 are also obtained. As an application of this extension, we also present a new algebraic method of obtaining polar codes from codes over F 2 + u F 2 . We introduce two new polar-like codes from codes over this ring, which we call RM 1 and RM 2 codes. Finally polar codes for the binary erasure channel (BEC) and Reed–Muller codes are compared in terms of trellis complexity.