The Lee Weight Distributions of a Class of Quaternary Codes

Let $\mathbb {Z}_{n}$ be the integer ring of residue classes modulo $n$ . A well known fact states that the Gray map from a code with Lee weight over $\mathbb {Z}_{4}$ to a code with Hamming weight over $\mathbb {Z}_{2}$ is weight-preserving. This leads that the study of Lee weight of quaternary codes plays an extremely important role in the research of coding theory. For any given positive integers $k_{1},k_{2}$ , and $\delta \in \{0,1,2,3\}$ , let $\mathcal {C}_{k_{1},k_{2}}^{\delta }$ be a quaternary code with the generator matrix $G_{k_{1},k_{2}}^{\delta }$ whose columns are all distinct nonzero vectors of the form $(c_{1},\cdots,c_{k_{1}},\cdots,c_{k_{1}+k_{2}})$ with $\sum _{i=1}^{k_{1}}c_{i}\equiv \delta \pmod {4}$ , $c_{j}\in \mathbb {Z}_{4}$ and $c_{t}\in \{0,2\}$ for each $1\le j\le k_{1}$ and $k_{1}+1\le t\le k_{1}+k_{2}$ . The objective of this paper is to establish the Lee weight distribution of $\mathcal {C}_{k_{1},k_{2}}^{\delta }$ for each $\delta $ . Moreover, the linearity and optimality for the Gray image of $\mathcal {C}_{k_{1},k_{2}}^{\delta }$ are characterized, from which we derive a family of optimal binary codes. Our result generalizes a work of Shi and Wang in 2014.