= Systematic-matrix proof of extended Golay distance
Suppose a binary <symmetric matrix> $A$ satisfies $A^2=I$, has one row of <Hamming weight> eleven and eleven rows of <Hamming weight> seven, and every sum of two distinct rows has <Hamming weight> six. Then $(I\mid A)$ generates a <self-dual code>. Its rows have weights twelve or eight; their mutual orthogonality makes every generated word <doubly even>. A weight-four word $(u,uA)$ would have left-half weight one, two, three or four. The first two cases are excluded by the stated row weights; weight three would force $uA$ to be a unit vector and hence $u$ to be a row of $A$; weight four would force $uA=0$. Thus weight four is impossible and the <minimum Hamming distance> is eight.
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