Infinite families $2$-designs from binary projective three-weight codes

Combinatorial designs are closely related to linear codes. In recent year, there are a lot of $t$-designs constructed from certain linear codes. In this paper, we aim to construct $2$-designs from binary three-weight codes. For any binary three-weight code $\mathcal{C}$ with length $n$, let $A_{n}(\mathcal{C})$ be the number of codewords in $\mathcal{C}$ with Hamming weight $n$, then we show that $\mathcal{C}$ holds $2$-designs when $\mathcal{C}$ is projective and $A_{n}(\mathcal{C})=1$. Furthermore, by extending some certain binary projective two-weight codes and basing on the defining set method, we construct two classes of binary projective three-weight codes which are suitable for holding $2$-designs.