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 the lifted Hamming codes and lifted Simplex codes as well as the lifted Reed-Muller codes of certain orders, and investigates the $2$-designs and $3$-designs supported by these lifted codes. Infinite families of $2$-designs and $3$-designs are obtained. In addition, an infinite family of two-weight projective codes and two infinite families of three-weight projective codes are presented.

翻译：线性码 $\C$ 在 $\gf(q)$ 上的生成矩阵，在任意扩域 $\gf(q^\ell)$ 上同样是一个秩为 $k$ 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础理论，确定了提升汉明码、提升单纯形码以及特定阶数提升里德-穆勒码的重量分布，并研究了由这些提升码支撑的 $2$-设计和 $3$-设计。由此获得了无限族的 $2$-设计和 $3$-设计。此外，本文还提出了一个无限族的二重量射影码和两个无限族的三重量射影码。