Lee Distance of cyclic codes of length $2^{\varsigma }$ over ${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}+u^2{\mathbb{F}}_{2^m}$

Abstract

Let $p$ be a prime number and $\varsigma$ and $m$ be a positive integers. Let $\mathcal{R}={\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}+u^2{\mathbb{F}}_{2^m}$ ($u^3=0$). Cyclic codes of length $2^{\varsigma }$ over $\mathcal{R}$ are precisely the ideals of the local ring $\frac{\mathcal{R}[x]}{⟨x^{2^{\varsigma }}-1⟩}$. The Gray map from a code of Lee weight over ${\mathbb{Z}}_4$ to a code with Hamming weight over ${\mathbb{F}}_2$ is known to preserve weight. In this paper, we determine the Lee distance of cyclic codes of length $2^{\varsigma }$ over $\mathcal{R}$.