Nearly Orthogonal Sets over Finite Fields
Dror Chawin, Ishay Haviv
TL;DR
The paper addresses the size of $k$-nearly orthogonal sets in $\mathbb{F}^d$ for finite fields, establishing field-dependent lower bounds that surpass real-field benchmarks. It extends the probabilistic tensor-product framework of Alon–Szegedy to finite fields, augmented with spectral tools to control the structure of pairwise non-orthogonal vectors, and uses a box-based union bound to certify large sets. For the binary field, it proves $\alpha(d,k,\mathbb{F}_2)\ge d^{\delta\cdot k/\log k}$, and for prime characteristic $p$, it shows $\alpha(d,k,\mathbb{F})\ge d^{\delta\cdot k^{1/(p-1)}/\log k}$ with $d\ge k^{1/(p-1)}$, plus a bipartite analogue. The results have implications for orthogonal representations and related graph parameters, including bounds on $\xi_{\mathbb F}(G)$ and ratios involving clique covers, thereby enriching the understanding of how finite-field structure influences near-orthogonality and associated combinatorial objects.
Abstract
For a field $\mathbb{F}$ and integers $d$ and $k$, a set of vectors of $\mathbb{F}^d$ is called $k$-nearly orthogonal if its members are non-self-orthogonal and every $k+1$ of them include an orthogonal pair. We prove that for every prime $p$ there exists a positive constant $δ= δ(p)$, such that for every field $\mathbb{F}$ of characteristic $p$ and for all integers $k \geq 2$ and $d \geq k^{1/(p-1)}$, there exists a $k$-nearly orthogonal set of at least $d^{δ\cdot k^{1/(p-1)}/ \log k}$ vectors of $\mathbb{F}^d$. In particular, for the binary field we obtain a set of $d^{Ω( k /\log k)}$ vectors, and this is tight up to the $\log k$ term in the exponent. For comparison, the best known lower bound over the reals is $d^{Ω( \log k / \log \log k)}$ (Alon and Szegedy, Graphs and Combin., 1999). The proof combines probabilistic and spectral arguments.