= Constant weight of a binary simplex code
{title2=$\operatorname{wt}(aH)=2^{l-1}\quad(a\ne0)$}
When the columns of $H$ are every nonzero vector in $\mathbb F_2^l$, a nonzero row vector $a$ defines a nonzero <linear functional> $v\mapsto a\cdot v$. Exactly half the $2^l$ input vectors take value $1$: translation by any vector on which the functional is $1$ pairs its zero and one fibres. Removing the zero input removes only a zero value. Thus every nonzero word in the <binary simplex code> has <Hamming weight> $2^{l-1}$.
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