A family of optimal linear codes of length $2^{s}$ over the ring $\frac{\mathbb{F}_2[u]}{\langle u^2 \rangle}$ with Lee distance $4$

Abstract

Optimal linear codes over finite rings are of significant interest in coding theory. In this paper, we introduce a method for constructing an infinite class of Lee metric codes of type $4^{m_1}2^{m_2}$ with length $2^s$ over the ring $R=\frac{\mathbb{F}_2[u]}{\langle u^2 \rangle}$. These codes are constructed by taking a union of certain particular cosets of the ideal $I=\langle u(x+1)\rangle $ in the ring  $ S= \frac{R[x]}{\langle x^{2^s}+1 \rangle}, \text{ for }s\geq 2$. We discuss their algebraic structure and demonstrate that these codes are linear with Lee distance  $4$ and are optimal with respect to the Lee sphere packing bound. Furthermore, we conclude that these codes are even codes for the Lee metric and also provide the complete Lee weight distribution of these codes.