Are there any books/papers talking about inner product on vectors over finite fields?

Question

Are there any books/papers talking about inner product on vectors over finite fields? In particular, I'd like to learn things on $F_p^n$, or simply $F_2^n$. I read some proofs using the inner product on $F_2^n$ in some papers, but I can only understand the general ideas. However, since it's possible that $v\cdot v = 0$ even if $v \neq 0$ for some $v\in F_2^n$, I don't know how they exclude this special case in their analysis. And I have no idea whether there are some other special cases should be considered. I also have some other concerns, so I think maybe I should read some related stuffs to so as to fully answer all of my concerns.

Answer 1

Recommend Gerstein, who has an ongoing interest in forms over finite fields $F$ as well as rings $F[t]$

Answer 2

There is lots of literature on "inner products" on vector spaces over finite fields. Though they are not called that, because you cannot rule out vectors being self-orthogonal, and you don't have a notion of positive/negative, distance, etc.

They go under the name of bilinear/sesquilinear forms (also related to quadratic forms); you can also find details of the geometries, known as finite polar spaces. These have many interesting connections to finite simple groups. They are also related to important families of distance regular graphs

I think a good book to start with is Grove's Classical Groups and Geometric Algebra, for a good algebraic description. These notes by Peter Cameron are also good, although challenging: http://www.maths.qmul.ac.uk/~pjc/pps/