A generator matrix of a linear code $\C$ over $\gf(q)$ is also a matrix of the same rank $k$ over any extension field $\gf(q^\ell)$ and generates a linear code of the same length, same dimension and same minimum distance over $\gf(q^\ell)$, denoted by $\C(q|q^\ell)$ and called a lifted code of $\C$. Although $\C$ and their lifted codes $\C(q|q^\ell)$ have the same parameters, they have different weight distributions and different applications. Few results about lifted linear codes are known in the literature. This paper proves some fundamental theory for lifted linear codes, settles the weight distributions of lifted Hamming codes and lifted Simplex codes, and investigates the $2$-designs supported by the lifted Hamming and Simplex codes. Infinite families of $2$-designs are obtained. In addition, an infinite family of two-weight codes and an infinite family of three-weight codes are presented.

翻译：线性码 $\C$ 在 $\gf(q)$ 上的生成矩阵，在任何扩域 $\gf(q^\ell)$ 上也是具有相同秩 $k$ 的矩阵，并在 $\gf(q^\ell)$ 上生成长度、维数和最小距离均相同的线性码，记作 $\C(q|q^\ell)$，称为 $\C$ 的提升码。尽管 $\C$ 与其提升码 $\C(q|q^\ell)$ 具有相同的参数，但它们具有不同的重量分布和不同的应用。文献中关于提升线性码的结果很少。本文证明了提升线性码的一些基本理论，确定了提升汉明码和提升单形码的重量分布，并研究了由提升汉明码和单形码支撑的 $2$-设计。获得了无限族的 $2$-设计。此外，还给出了一个无限族的二重量码和一个无限族的三重量码。