Lifted Hamming, Simplex and Reed-Muller 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 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.