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.

翻译：组合设计与线性码密切相关。近年来，许多$t$-设计从特定线性码中构造而来。本文旨在从二进制三重量码中构造$2$-设计。对于任意长度为$n$的二进制三重量码$\mathcal{C}$，设$A_{n}(\mathcal{C})$表示$\mathcal{C}$中汉明重量为$n$的码字数量，我们证明：当$\mathcal{C}$为射影码且$A_{n}(\mathcal{C})=1$时，$\mathcal{C}$包含$2$-设计。此外，通过扩展某些特定的二进制射影二重量码，并基于定义集方法，我们构造了两类适用于承载$2$-设计的二进制射影三重量码。