Hamming code of length $8$ self dual I'm reading a paper by N.J.A Sloane on self dual codes, and he introduces the binary Hamming code of length $8$ with generator matrix  $$ G = \begin{bmatrix}
1&1&1&1&1&1&1&1\\
0&1&1&1&0&1&0&0\\
0&0&1&1&1&0&1&0\\
0&0&0&1&1&1&0&1 
\end{bmatrix}$$
Where if we assign labels to the columns from left to right; $\infty, 0,1,2,3,4,5,6$, then the second row has $1's$ where the column is labelled by a quadratic residue mod $7$. Then we cyclic shift to get the remaining rows.
What is being done here? I assume we just ignore the first row and first column as this just extends the matrix. I've looked in to quadratic residue codes but I can't seem to find the link between them and this generator matrix? Could anyone explain what's being done here?
The paper I'm referring to can be found here; https://arxiv.org/abs/math/0612535.
 A: There is a quadratic residue (QR) code of length $p$
over the finite field $GF(m)$ whenever $p$
and $m$ are primes, $p$ is odd, and 
$m$ is a quadratic residue modulo $p$.
It just happens that the (7,4,3) Hamming Code, and the (23,12,7) Golay Code, both over $GF(2)$ (thus $m=2$)
can also be expressed as QR codes.
As $p$ increases, QR codes are no longer very efficient, with their minimum distance asymptotically $O(\sqrt{p})$ but for short lengths they produce the above two perfect codes.
Edit: The online magma calculator is quite useful for computations in coding theory. You can access it here.
Entering 

QRCode(GF(2),7); HammingCode(GF(2),3);

gives

[7, 4, 3] "Quadratic Residue code" Linear Code over GF(2)

and

[7, 4, 3] "Hamming code (r = 3)" Linear Code over GF(2)

and displays the same generator matrix for both.
A: The code is self-dual. The generator matrix is also the parity check matrix. If you consider the matrix $G$ as parity check matrix, then deleting the first row and the first column gives the parity check matrix of the 7,4,3 Hamming code.
