Binary simplex code

The binary simplex code is the dual of the binary Hamming code. For redundancy $r$ it has parameters $[2^r-1,r,2^{r-1}]$.

Its relation to the Hamming code is duality, not equality of the code families.

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 $2^{l-1}$.