Real norms on vector spaces over finite fields

Question

I am interested in functions of the form $\psi: F^n \to \mathbb{R}^+$, where $F$ is a finite field, that have norm-like properties, e.g., $\psi(x+y) \le \psi(x) + \psi(y)$. Does anybody know if there is any literature on this area?

Answer 1

If you are truly only interested in the triangle inequality, then there is the Hamming weight $w(x_1,x_2,\ldots,x_n)=m$ where $m$ is simply the number of non-zero components. This gives you a metric. Mind you, the space $F^n$ is finite, so any metric on it is going to give you the discrete topology. Adding any kind of norm-like requirements (on top of the triangle inequality) is problematic for several reason, as others have pointed out.

The Hamming weight obviously depends on the choice of basis, which may restrict its usefulness (depending on what you wanted to do with this 'norm').

I'm sad to say there isn't an awful lot of analysis happening in this space.

Answer 2

The triangle inequality might allow some interesting functions, but positive homogeneity doesn't. A finite field has a prime characteristic, $p$. Positive homogeneity says $\psi(rx)=|r|\psi(x)$ for $r\in\mathbb{Q}$. To satisfy this property, $\psi(0)=\psi(px)=p\psi(x)$, which first says $\psi(0)=0$ and then $\psi(x)=0$ for all $x\in F$.