Abstract 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 (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 d^ k / k d δ k / log k. This nearly matches an upper bound (array{cd+k\\ karray}) d + k <
Nenadov et al. (Tue,) studied this question.