Let Let be a matrix whose columns are all the nonzero elements of The Hamming Linear Code is the linear code with Parity Check Matrix (original is ) Lemma The Hamming Linear Code has minimum distance and is a Perfect Code Error correcting code. Proof Any columns of are linearly independent, but there are some columns that are linearly dependent so (see Parity Check Matrix Lemma) To be perfect, require Here , so it works out