Study On Quaternary Reed-Muller Codes

The theory of quaternary linear codes is one of the important subjects of classical error-correcting coding theory in recent ten years, which is closely related with the study on constructions and properties of binary nonlinear codes. In this paper, we discuss ZRM.(r, m) codes and ZRM^*(r,m) codes, and their binary images ZRM(r,m) codes and ZRM^*(r,m) codes via Gray map .Solutions presented in this paper can be summarized as follows:1. Discuss the types of ZRM(r,m) codes and ZRM^*(r,m) codes. When 2 ≤ r ≤ m + 1, the type of ZRM(r,m) is 4^k₁2^k₂-k₁ where k₁ = 1 + (₁^m + ... + (_4-1^m), k₂ = 1 + (₁^m) + ... + (_t^m), t = min{2r - 2, m}; the type of ZRM^*(r,m) is 4^k₁′)2₍(k₂)′-(k₁)′, wherek₁′ = n-(₁^m) + ... + (_r-1^m), (k₂)′ = 1 + (₁^m) + ... + (_r^m).2. Prove that the binary images are linear codes in the case of ZRM(r, m), but they are not in the case of ZRM^*(r, m). Moreover, since the linear codes spanned by ZRM^*(r,m) are ZRM(r,m), we can conclude that the rank of ZRM^*(r,m) is k₁ +k₂. At last, we dicussthe kernel of ZRM* (r, m) and get the dimension of kernel of ZRM^* (r, m) is 1+ (₁^m) + .... +(_r^m) + (m + l).3. Give a new method to prove that RM(r,m) is a Z₄-nonlinear code for 3 ≤ r ≤ m - 2. Furthermore, prove that ZRM(r,m - 1) is the minimum quaternary code such that φ{ZRM(r,m - 1)) (?) RM{r,m).