How to find mutually orthogonal vectors with unknown?

u is the row of this sheet. e.g. u1=[-1,+1,+1,...,-1], u2=[X21, X22,...-1]

All Xs are unknown number(either -1 or +1). I want to find every Xs such that every u is orthogonal to each other. For example, dot(u1,u2)=0.

Answer 1

[John D'Errico]

In theory, Gram-Schmidt would be sufficient.

https://en.wikipedia.org/wiki/Gram–Schmidt_process

In practice, if the number of dimensions are at all large, the computation will become significant. You won't be able to do it in a reasonable time. And "large" does not need to be very large in this context. 10 dimensions will surely be a problem, so you are surely wasting your time if you hope to find a result for 22 dimensions. Symbolic computations like this become truly massive when the dimensionality of the problem increases.