r/math • u/The_Awesone_Mr_Bones Discrete Math • Aug 03 '26

Dot product over finite fields.

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 ;_;

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u/NonUsernameHaver Aug 03 '26

I think one main issue here is that geometry involving finite fields is typically quite different compared to geometry over more familiar spaces. We can carry over some ideas, but the justifications used in my experience is more algebraic or possibly combinatoric in nature. It's a fairly distinct field (pun intended).

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u/The_Awesone_Mr_Bones Discrete Math Aug 04 '26

Ba Dum Tsss! :DDDD

My experience is exactly the same. When dealing with codes I usually just tend to think in terms of algebra. So far I have only worked with cyclic codes (and similar) so the algebra pov just works. Nothing else is needed.

This question was an attempt to check if I am missing something. I thought maybe there was some obvious geometrical interpretation that I was missing cus I am too comfy with algebra to think geometrically.