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

Generalized Extended Hamming Codes over Galois Ring of Characteristic $2^{n}$