Document Open Access Logo

Nearly Orthogonal Sets over Finite Fields

Authors Dror Chawin, Ishay Haviv

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2024.39.pdf
  • Filesize: 0.57 MB
  • 11 pages

Document Identifiers

Related Versions

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

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.

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

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.

References

  1. Noga Alon, H. Tracy Hall, Christian Knauer, Rom Pinchasi, and Raphael Yuster. On graphs and algebraic graphs that do not contain cycles of length 4. J. Graph Theory, 68(2):91-102, 2011. Google Scholar
  2. Noga Alon and Michael Krivelevich. Constructive bounds for a Ramsey-type problem. Graphs and Combinatorics, 13(3):217-225, 1997. Google Scholar
  3. Noga Alon and Mario Szegedy. Large sets of nearly orthogonal vectors. Graphs and Combinatorics, 15(1):1-4, 1999. Google Scholar
  4. Igor Balla. Orthonormal representations, vector chromatic number, and extension complexity. arXiv, abs/2310.17482, 2023. Google Scholar
  5. Igor Balla, Shoham Letzter, and Benny Sudakov. Orthonormal representations of H-free graphs. Discret. Comput. Geom., 64(3):654-670, 2020. Google Scholar
  6. Alexander Barg, Moshe Schwartz, and Lev Yohananov. Storage codes on triangle-free graphs with asymptotically unit rate. arXiv, abs/2212.12117, 2022. Google Scholar
  7. Alexander Barg and Gilles Zémor. High-rate storage codes on triangle-free graphs. IEEE Trans. Inform. Theory, 68(12):7787-7797, 2022. Google Scholar
  8. Elwyn Ralph Berlekamp. On subsets with intersections of even cardinality. Canad. Math. Bull., 12(4):471-474, 1969. Google Scholar
  9. Piotr Berman and Georg Schnitger. On the complexity of approximating the independent set problem. Inf. Comput., 96(1):77-94, 1992. Preliminary version in STACS'89. Google Scholar
  10. Anna Blasiak, Robert Kleinberg, and Eyal Lubetzky. Broadcasting with side information: Bounding and approximating the broadcast rate. IEEE Trans. Inform. Theory, 59(9):5811-5823, 2013. Google Scholar
  11. Bruno Codenotti, Pavel Pudlák, and Giovanni Resta. Some structural properties of low-rank matrices related to computational complexity. Theor. Comput. Sci., 235(1):89-107, 2000. Google Scholar
  12. Louis Deaett. The minimum semidefinite rank of a triangle-free graph. Linear Algebra Appl., 434(8):1945-1955, 2011. Google Scholar
  13. Paul Erdős and George Szekeres. A combinatorial problem in geometry. Compositio Mathematica, 2:463-470, 1935. Google Scholar
  14. Uriel Feige. Randomized graph products, chromatic numbers, and the Lovász ϑ-function. Combinatorica, 17(1):79-90, 1997. Preliminary version in STOC'95. Google Scholar
  15. Peter Frankl and Vojtěch Rödl. Forbidden intersections. Trans. Amer. Math. Soc., 300(1):259-286, 1987. Google Scholar
  16. Zoltán Füredi and Richard P. Stanley. Sets of vectors with many orthogonal pairs. Graphs and Combinatorics, 8(4):391-394, 1992. Google Scholar
  17. Alexander Golovnev and Ishay Haviv. The (generalized) orthogonality dimension of (generalized) Kneser graphs: Bounds and applications. Theory of Computing, 18(22):1-22, 2022. Preliminary version in CCC'21. Google Scholar
  18. Willem H. Haemers. An upper bound for the Shannon capacity of a graph. In László Lovász and Vera T. Sós, editors, Algebraic Methods in Graph Theory, volume 25/I of Colloquia Mathematica Societatis János Bolyai, pages 267-272. Bolyai Society and North-Holland, 1978. Google Scholar
  19. Ishay Haviv. On minrank and forbidden subgraphs. ACM Trans. Comput. Theory (TOCT), 11(4):20, 2019. Preliminary version in RANDOM'18. Google Scholar
  20. Hexiang Huang and Qing Xiang. Construction of storage codes of rate approaching one on triangle-free graphs. Des. Codes Cryptogr., 91(12):3901-3913, 2023. Google Scholar
  21. László Lovász. On the Shannon capacity of a graph. IEEE Trans. Inform. Theory, 25(1):1-7, 1979. Google Scholar
  22. László Lovász. Graphs and Geometry, volume 65. Colloquium Publications, 2019. Google Scholar
  23. Ali Mohammadi and Giorgis Petridis. Almost orthogonal subsets of vector spaces over finite fields. Eur. J. Comb., 103:103515, 2022. Google Scholar
  24. Jaroslav Nešetřil and Moshe Rosenfeld. Embedding graphs in Euclidean spaces, an exploration guided by Paul Erdős. Geombinatorics, 6:143-155, 1997. Google Scholar
  25. René Peeters. Orthogonal representations over finite fields and the chromatic number of graphs. Combinatorica, 16(3):417-431, 1996. Google Scholar
  26. Pavel Pudlák. Cycles of nonzero elements in low rank matrices. Combinatorica, 22(2):321-334, 2002. Google Scholar
  27. Moshe Rosenfeld. Almost orthogonal lines in E^d. DIMACS Series in Discrete Math., 4:489-492, 1991. Google Scholar
  28. Le Anh Vinh. On the number of orthogonal systems in vector spaces over finite fields. Electr. J. Comb., 15(32):1-4, 2008. Google Scholar
  29. Alan Zame. Orthogonal sets of vectors over Z_m. J. Comb. Theory, 9(2):136-143, 1970. Google Scholar
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact