Generalized Reed–Muller codes over $${\mathbb{Z}_q}$$

Abstract

We have given a generalization of Reed---Muller codes over the prime power integer residue ring $${\mathbb{Z}_q}$$ . These codes are analogs of generalized Reed---Muller (GRM) codes over finite fields. We mainly focus on primitive GRM codes, which are basically a generalization of Quaternary Reed---Muller (QRM) codes. We have also given a multivariate representation of these codes. Non-primitive GRM codes over $${\mathbb{Z}_q}$$ are also briefly discussed. It has been shown that GRM codes over $${\mathbb{Z}_q}$$ are free extended cyclic codes. A trace description of these codes is also given. We have obtained formulas for their ranks and also obtained expressions for their minimum Hamming distances.