eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
39:1
39:11
10.4230/LIPIcs.SoCG.2024.39
article
Nearly Orthogonal Sets over Finite Fields
Chawin, Dror
1
Haviv, Ishay
1
School of Computer Science, The Academic College of Tel Aviv-Yaffo, Israel
For a field 𝔽 and integers d and k, a set of vectors of 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 𝔽 of characteristic p and for all integers k ≥ 2 and d ≥ k^{1/(p-1)}, there exists a k-nearly orthogonal set of at least d^{δ ⋅ k^{1/(p-1)} / log k} vectors of 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.39/LIPIcs.SoCG.2024.39.pdf
Nearly orthogonal sets
Finite fields