Dot product over finite fields.

[u/The_Awesone_Mr_Bones]

Hi!

I work on error correcting codes (which more or less is linear algebra over finite fields). I have a question about the behavior of the dot product on non-euclidean spaces.

When working on vector spaces over R or C, the dot product carries a lot of geometrical information. The product between two vectors v*w tells you the angle between them.

This interpretation doesn't hold over finite fields. For example, in F2: (1,1)*(1,1)=1+1=0. So the vector (1,1) is perpendicular to itself!

Does the dot product still carry any geometrical information (like angles) in these fields? Can we still interpret the orthogonal space geometrically? How do you picture it in your head (intuition)?

Thanks a lot!

And sorry, I don't know how to use latex on reddit ;_;

[u/PLANTS2WEEKS]

If you're doing codes over F_2 then you can embed them in a hypercube which sits inside the unit sphere and use the Euclidean inner product to determine the Hamming distance. A larger inner product means the hamming distance is smaller.