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Nearly Orthogonal Sets over Finite Fields

Authors Dror Chawin, Ishay Haviv

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Author Details

Dror Chawin
  • School of Computer Science, The Academic College of Tel Aviv-Yaffo, Israel
Ishay Haviv
  • School of Computer Science, The Academic College of Tel Aviv-Yaffo, Israel

Funding

  • Chawin, Dror: Research supported by the Israel Science Foundation (grant No. 1218/20).
  • Haviv, Ishay: Research supported by the Israel Science Foundation (grant No. 1218/20).

Acknowledgements

We are grateful to the anonymous reviewers for their useful suggestions.

Cite As Get BibTex

Dror Chawin and Ishay Haviv. Nearly Orthogonal Sets over Finite Fields. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 39:1-39:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.39

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Generating random combinatorial structures
  • Mathematics of computing → Spectra of graphs
  • Mathematics of computing → Extremal graph theory
Keywords
  • Nearly orthogonal sets
  • Finite fields

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