"Show that the Hamming code $H_r$ is unique, i.e. any linear code with parameters $[2^r − 1, 2^r − 1 − r, 3]$ is equivalent to $H_r$."
I see that Hamming codes are perfect and packs the whole space, hence any code with the same parameters will be equal to it.
But I can not explain in mathematical words and prove the statement exactly.
Can anyone help me?
Any linear code $C$ with the parameters of an Hamming code is actually equivalent to a Hamming code. This is easy to see: Let $H$ be a parity-check matrix for $C$. It is a $r × \frac{q^r−1}{q-1}$ matrix. By $d(C)=3$, we know any two columns of $H$ are linearly independent. Hence the vectors, corresponding to any two columns, generate different $1$-dimensional vector space in $\mathbb{F}_q^r$. We already know that $\mathbb{F}_q^r$ contains $\frac{q^r−1}{q-1}$ distinct subspaces of dimension $1$ and $\frac{q^r−1}{q-1}$ is the number of columns of $H$. So by the definition of Hamming code, $C$ is a Hamming code, i.e., equivalent to it.
We could also use the classification of perfect codes by van Lint and Tietäväinen, because any such linear code with these parameters is perfect. Since it is non-trivial, and not a Golay-code, it must be equivalent to a Hamming code.