Constant weight of a binary simplex code ($\mathrm{wt}(aH)=2^{l-1}(a\ne 0)$)

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 $2^{l-1}$.