One-Lee weight and two-Lee weight ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-additive codes

In this paper, we study one-Lee weight and two-Lee weight codes over ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$, where $u^2=0$. Some properties of one-Lee weight ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-additive codes are given, and a complete classification of one-Lee weight ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-additive formally self-dual codes is obtained. The structure of two-Lee weight projective ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$ codes is determined. Some optimal binary linear codes are obtained directly from one-Lee weight and two-Lee weight ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-additive codes via the extended Gray map.