Nearly Orthogonal Sets over Finite Fields
For a field đť”˝ and integers d and k, a set of vectors of đť”˝^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 Î´ = Î´ (p), such that for every field đť”˝ of characteristic p and for all integers k â‰Ą 2 and d â‰Ą k^{1/(p-1)}, there exists a k-nearly orthogonal set of at least d^{Î´ â‹… k^{1/(p-1)} / log k} vectors of đť”˝^d. In particular, for the binary field we obtain a set of d^Î©(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^Î©(log k / log log k)} (Alon and Szegedy, Graphs and Combin., 1999). The proof combines probabilistic and spectral arguments.
Nearly orthogonal sets
Finite fields
Theory of computation~Generating random combinatorial structures
Mathematics of computing~Spectra of graphs
Mathematics of computing~Extremal graph theory
39:1-39:11
Regular Paper
http://arxiv.org/abs/2402.08274
We are grateful to the anonymous reviewers for their useful suggestions.
Dror
Chawin
Dror Chawin
School of Computer Science, The Academic College of Tel Aviv-Yaffo, Israel
Research supported by the Israel Science Foundation (grant No. 1218/20).
Ishay
Haviv
Ishay Haviv
School of Computer Science, The Academic College of Tel Aviv-Yaffo, Israel
Research supported by the Israel Science Foundation (grant No. 1218/20).
10.4230/LIPIcs.SoCG.2024.39
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Dror Chawin and Ishay Haviv
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