Cyclic Reed–Muller Codes and Their Decoding

A large number of cyclic codes with a majority decoding scheme are constructed on the basis of the so-called quasi-separated checks. A general method of decoding for these codes (orthogonalization in $L$ steps) was described by Massey. In this paper another approach to the decoding of these codes which leads to considerably simpler decoding schemes is considered. This approach is based on the analogy of Reed–Muller and $M(n,k)$-codes, constructed by means of finite projective geometries. Cyclic analogs of Reed–Muller codes are described, and a decoding method similar to Reed's method is deduced.