  In this paper, we introduce generalized extended Hamming codes over Galois
rings $GR(2^n,m)$ of characteristic $2^n$ with extension degree $m$.
Furthermore we prove that the minimum Hamming weight of generalized extended
Hamming codes over $GR(2^n,m)$ is 4 and the minimum Lee weight of generalized
extended Hamming codes over $GR(8,m)$ is 6 for all $m \geq 3$.
