A generator matrix of a linear code \({\mathcal {C}}\) over \({\textrm{GF}}(q)\) is also a matrix of the same rank k over any extension field \({\textrm{GF}}(q^\ell )\) and generates a linear code of the same length, same dimension and same minimum distance over \({\textrm{GF}}(q^\ell )\) , denoted by \({\mathcal {C}}(q|q^\ell )\) and called a lifted code of \({\mathcal {C}}\) . Although \({\mathcal {C}}\) and their lifted codes \({\mathcal {C}}(q|q^\ell )\) 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, and studies the 2-designs of the lifted projective Reed–Muller codes, lifted Hamming codes and lifted Simplex codes. In addition, this paper settles the weight distributions of the lifted Reed–Muller codes of certain orders, and investigates the 3-designs supported by these lifted codes. As a by-product, an infinite family of three-weight projective codes over \({\textrm{GF}}(4)\) is obtained.
