Larger nearly orthogonal sets over finite fields

For a field F and integers d and k, a set A⊆F^d is called k-nearly orthogonal if its members are non-self-orthogonal and every k+1 vectors of A include an orthogonal pair. We prove that for every prime p there exists some δ=δ(p)>0, such that for every field F of characteristic p and for all integers k≥2 and d≥k, there exists a k-nearly orthogonal set of at least d^{δ⋅k/log⁡k} vectors of F^d. The size of the set is optimal up to the log⁡k term in the exponent. We further prove two extensions of this result. In the first, we provide a large set A of non-self-orthogonal vectors of F^d such that for every two subsets of A of size k+1 each, some vector of one of the subsets is orthogonal to some vector of the other. In the second extension, every k+1 vectors of the produced set A include ℓ+1 pairwise orthogonal vectors for an arbitrary fixed integer 1≤ℓ≤k. The proofs involve probabilistic and spectral arguments and the hypergraph container method.