Hamming Codes

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