I have a question about the following theorem concerning a relation between error-correcting Hamming codes and Steiner triple designs.
Theorem. If $C$ is a Hamming $(2^m-1, 2^{2^m-1-m}, 3)$ code, and $A$ is the matrix whose columns contain the weight-3 codewords in $C$, then $A$ is the incidence matrix of a Steiner triple system on $(2^m-1)$ varieties.
Note that a Steiner triple has the same amount of varieties as blocks. This means that the incidence matrix $A$ must be square, with dimensions $2^m-1$. Because the columns of $A$ are the codewords in $C$ with weight 3, there are $2^m-1$ codewords in the Hamming code with weight 3.
Question. Why does a $(2^m-1, 2^{2^m-1-m}, 3)$ Hamming code contain $2^m-1$ codewords with weight 3?