Larger Nearly Orthogonal Sets over Finite Fields
Ishay Haviv, Sam Mattheus, Aleksa Milojević, Yuval Wigderson
TL;DR
The paper studies large sets in $\mathbb{F}^d$ whose members are non-self-orthogonal and exhibit a strong almost-orthogonality property: any $k$-element subset contains $\ell+1$ pairwise orthogonal vectors. It develops a probabilistic-tensor-product construction over fields of finite characteristic and leverages spectral and container-method tools for pseudo-random graphs to achieve lower bounds of the form $d^{\delta\,k/\log k}$ for both $\beta(d,k,\mathbb{F})$ and $\alpha(d,k,\ell,\mathbb{F})$, with $\delta=\delta(p,\ell)$ depending only on the characteristic. The approach hinges on counting substructures in orthogonality graphs via the Alon–Szegedy framework, bi-independent-set bounds, and the hypergraph container method, enabling near-optimal (up to a $\log k$ factor in the exponent) size of large nearly orthogonal sets over finite fields. As a byproduct, the paper obtains two extensions: a large cross-orthogonality guarantee between two $k+1$-subsets and a version where every $k+1$ vectors contain $\ell+1$ pairwise orthogonal vectors, both relying on the same probabilistic-containment machinery.
Abstract
For a field $\mathbb{F}$ and integers $d$ and $k$, a set ${\cal A} \subseteq \mathbb{F}^d$ is called $k$-nearly orthogonal if its members are non-self-orthogonal and every $k+1$ vectors of ${\cal A}$ include an orthogonal pair. We prove that for every prime $p$ there exists some $δ= δ(p)>0$, such that for every field $\mathbb{F}$ of characteristic $p$ and for all integers $k \geq 2$ and $d \geq k$, there exists a $k$-nearly orthogonal set of at least $d^{δ\cdot k/\log k}$ vectors of $\mathbb{F}^d$. 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 ${\cal A}$ of non-self-orthogonal vectors of $\mathbb{F}^d$ such that for every two subsets of ${\cal A}$ 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 ${\cal A}$ include $\ell+1$ pairwise orthogonal vectors for an arbitrary fixed integer $1 \leq \ell \leq k$. The proofs involve probabilistic and spectral arguments and the hypergraph container method.