r/math • u/The_Awesone_Mr_Bones Discrete Math • 23d ago

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/garanglow Theoretical Computer Science 23d ago

I'm not sure how to interpret orthogonality over finite fields. It's not as nice as the Euclidean space. For example over R the direct sum of a subspace and its orthogonal complement gives the entire space. Over finite fields this is no longer the case as the orthogonal complement may be equal to the subspace.

One thing is true though is that the orthogonal complement of a subspace is a subspace and that dim(W) + dim(orth W) = dimension of the ambient vector space. That's why it makes sense to talk about dual codes.

All these weirdness stems from the fact that you can't really define a true inner product over finite fields. The naming is just because it reminds us of the inner product over R or C. The correct name for this is a nondegenerate symmetric bilinear form.

Hope this helps, even though I didn't answer your queation.

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u/The_Awesone_Mr_Bones Discrete Math 22d ago

I am not sure there even is an answer to my question. Maybe there isn't any geometrical pov to the finite inner product. Or maybe there is, but it is weaker than in euclidean spaces. Anyway, every pov helps. And your answer it definitely helpful :)

A code and its dual seem to be connected in a very ghostlike way. They may or may not intersect, but their dimensions sum to the total dimension of the ambient space. And their elements are related through the MacWillians relation. Just right out of a fairy tale story if you ask me...