 Binary inner product

For vectors $x,y\in {\mathbb{F}}_2^n$, the binary inner product is the bilinear formIts value is the parity of the coordinates at which both vectors are one. For example, $(1,1,0)\cdot (1,0,1)=1$, whereas $(1,1)\cdot (1,1)=0$ because $1+1=0$ in the finite field ${\mathbb{F}}_2$.

The terminology does not impose the positive definiteness axiom of a real or complex inner product. A nonzero vector can have zero self-pairing, as the second example shows. The form is nevertheless a nondegenerate bilinear form: a nonzero coordinate of $x$ is detected by pairing with the corresponding standard basis vector. This pairing defines orthogonality and dual codes in coding theory.