Multipartite Nearly Orthogonal Sets Over Finite Fields

For a field F F and integers d , k and ℓ , a set A ⊆ Fᵈ A ⊆ F d is called (k, ) ( k , ℓ ) -nearly orthogonal if all vectors in A are non-self-orthogonal and every $$k+1$$ k + 1 vectors in A contain + 1 ℓ + 1 pairwise orthogonal vectors. Recently, Haviv, Mattheus, Milojević and Wigderson have improved the lower bound on nearly orthogonal sets over finite fields, using counting arguments and a hypergraph container lemma. They showed that for every prime p and an integer ℓ , there is a constant δ (p, ) δ ( p , ℓ ) such that for every field F F of characteristic p and for all integers d ≥ k ≥ + 1 d ≥ k ≥ ℓ + 1 , Fᵈ F d contains a (k, ) ( k , ℓ ) -nearly orthogonal set of size dδ k / log k d δ k / log k . This nearly matches an upper bound ( arraycd+k\\ karray) d + k <