Generalized Reed-Muller codes over Z(q)
We have given a generalization of Reed-Muller codes over the prime power integer residue ring 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 Z(q) are also briefly discussed. It has been shown that GRM codes over 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.
grm codes codes over rings galois rings kerdock codes