四元ZRM码的研究

为了讨论二元Reed-Muller码的 Z₄ 线性, 文献中先后介绍了两类 Z₄ 线性码, 分别记为ZRM}(r, m)与QRM}(r, m), 它们在Gray映射下的二元像记为ZRM}(r, m)与 QRM}(r, m) . 该文系统地讨论了这两类 Z₄ 线性码. 计算了ZRM}(r, m)与QRM}(r, m)的类型, 证明当3≤r≤m-1时, ZRM}(r, m)是二元线性码, 而QRM}(r, m)是非线性的; 并且, 由QRM}(r, m)张成的二元线性码恰是ZRM}(r, m). 最后, 对于非线性码QRM}(r, m), 讨论了它的秩与核.

Study on Quaternary ZRM Codes

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).