Simple terminological question: the equation $x_1+\dots+x_n = 0$ over $\mathbb{F}_2^n$ is called a subspace. I'm wondering if we could also call it a hyperplane, a half-space or neither?
The equality sign seems to suggest it's a hyperplane, but the fact that it always cuts the space exactly in half seems to suggest it's a half-space. Just curious!
It's certainly a hyperplane – to my mind "hyperplane" is just another word for affine subspace, usually exactly one dimension lower than the enclosing space, as in this case. It's also half a space, but I would find calling it a "half-space" confusing. It doesn't fall under Wikipedia's definition of a half-space, which matches my use of the term: A half of a space separated from the other half by a plane in a symmetric way, either including (closed) or not including (open) that plane, but not being that plane.