Lifted Hamming and Simplex Codes and Their Designs

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.