= Orthogonal generators of doubly even codes
Suppose the generators of a <binary linear code> have <Hamming weights> divisible by four and have pairwise zero binary <inner products>. Their <linear span> is <doubly even>: for any two vectors,
$$
\operatorname{wt}(u+v)=\operatorname{wt}(u)+\operatorname{wt}(v)-2|\operatorname{supp}(u)\cap\operatorname{supp}(v)|.
$$
The binary <inner product> says the intersection size is even. Induction on the number of generators in a sum proves divisibility by four.
// Target: algebra.bigb
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