Abstract:

In the literature two classes of Z₄ linear codes were defined to discuss the Z₄ linearity of binary Reed-Muller codes, they are denoted by ZRM(r, m) and QRM(r, m), and their binary images under the Gray map are denoted by ZRM(r, m) and QRM(r, m) respectively. In this correspondence, the types of ZRM}(r, m) and QRM(r, m) are computed respectively. When 3 ≤ r ≤ m-1, it is shown that the binary image ZRM(r, m) is linear while QRM(r, m) is nonlinear. Moreover, the linear code spanned by QRM(r, m) is proved to be ZRM(r, m). Finally, the rank and the kernel are determined for the nonlinear code QRM(r, m).

Key words: Reed-Muller code, Gray map, Binary image,  ZRM code,  QRM code