ZRM Codes

Quaternary ZRM(r,m) codes were defined so that their binary images, via Gray map, are Reed-Muller codes for some specific values of . In the literature, two different definitions of such codes can be found. They will be denoted ZRM(r,m) and ZRM^-(r,m) codes. In this correspondence, we show that both definitions are equivalent exactly for those values of r such that their binary images are Reed-Muller codes. Moreover, we prove that, for all r, these binary images are linear codes in the case of ZRM(r,m), but they are not if we use the definition of ZRM^-(r,m). In this last case, we compute the rank and the dimension of the kernel of these codes.