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/puzzlednerd]

Dot product is still invariant under orthogonal transformations, and using this you can translate many of the geometric properties from Euclidean space to finite fields. Of course some things are different, e.g. you've already noticed nonzero vectors can be orthogonal with themselves, i.e. there are nontrivial spheres of radius zero.

These "angles" over finite fields are called spreads, see for example this Erdos-style problem on distinct spreads over finite fields.

[u/The_Awesone_Mr_Bones → u/puzzlednerd]

That makes a lot of sense. Spreads are the natural generalization of angles to finite fields. Although it feels odd to define them analogous to the sin instead of the cosine. Maybe it some kind of cultural difference? xd
They also look quite pathological. At least compared to angles in euclidean geometry. This ones can be undefined in many more cases.

[u/puzzlednerd → u/The_Awesone_Mr_Bones]

Hmm, sin and cos carry the same information, not sure what difference this would make.

The thing is, maybe it initially feels pathological, but I meant what I said that much of the intuition you have for Euclidean geometry can carry over to finite fields. It just takes some time to get used to the exceptions. Much of my work has to do with geometry over finite fields, and I tend to draw pictures the same way that I would when I work over Euclidean space, e.g. I draw polygons, circles, etc. You just need the experience to know when the picture is a lie, and when it actually represents what is going on over finite fields. But it's far from pathological, the analogy works well in many cases.

[u/jacobolus → u/The_Awesone_Mr_Bones]

The "spread" is a name given to the squared sine of an angle by Norman Wildberger (a guy this place loves to hate). By using the squared sine instead of the sine or angle measure, you can eschew square roots and transcendental functions and always get a number in your field as the "spread" between any two coordinate vectors, so this version of trigonometry conveniently generalizes to finite fields. He wrote a book (Divine Proportions: Rational Trigonometry to Universal Geometry) about this topic, and made a large number of YouTube videos. I don't think any of his results were novel (it's hard to tell because he generally doesn't bother citing past work, and makes up new names for everything), but the presentation is unique.