Generalising the Scattered Property of Subspaces

Abstract

Let V be an r-dimensional \({\mathbb{F}_{{q^n}}}\)-vector space. We call an \({\mathbb{F}_{q}}\)-subspace U of V h-scattered if U meets the h-dimensional \({\mathbb{F}_{{q^n}}}\)-subspaces of V in \({\mathbb{F}_{q}}\)-subspaces of dimension at most h. In 2000 Blokhuis and Lavrauw proved that \({\dim_{\mathbb{F}_{q}}}\) Urn/2 when U is 1-scattered. Sub-spaces attaining this bound have been investigated intensively because of their relations with projective two-weight codes and strongly regular graphs. MRD-codes with a maximum idealiser have also been linked to rn/2-dimensional 1-scattered subspaces and to n-dimensional (r − 1)-scattered subspaces.

In this paper we prove the upper bound rn/(h + 1) for the dimension of h-scattered subspaces, h > 1, and construct examples with this dimension. We study their intersection numbers with hyperplanes, introduce a duality relation among them, and study the equivalence problem of the corresponding linear sets.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    S. Ball, A. Blokhuis and M. Lavrauw: Linear (q + 1)-fold blocking sets in PG(2,q4), Finite Fields Appl. 6 (2000), 294–301.

    MathSciNet  Article  Google Scholar 

  2. [2]

    D. Bartoli, B. Csajbók, G. Marino and R. Trombetti: Evasive subspaces, arXiv:2005.08401.

  3. [3]

    D. Bartoli, M. Giulietti, G. Marino and O. Polverino: Maximum scattered linear sets and complete caps in Galois spaces, Combinatorial 38(2) (2018), 255–278.

    MathSciNet  Article  Google Scholar 

  4. [4]

    A. Blokhuis and M. Lavrauw: Scattered spaces with respect to a spread in PG(n, q), Geom. Dedicata 81 (2000), 231–243.

    MathSciNet  Article  Google Scholar 

  5. [5]

    G. Bonoli and O. Polverino: \({\mathbb{F}_{q}}\)-linear blocking sets in PG(2,q4), Innov. Incidence Geom. 2 (2005), 35–56.

    MathSciNet  Article  Google Scholar 

  6. [6]

    P. J. Cameron: Notes on Counting: An Introduction to Enumerative Combinatorics, Cambridge University Press 2017.

  7. [7]

    L. Carlitz: Some inverse relations, Duke Mathematical Journal 40 (1973), 893–901.

    MathSciNet  MATH  Google Scholar 

  8. [8]

    B. Csajbók, G. Marino and O. Polverino: Classes and equivalence of linear sets in PG(1,qn), J. Combin. Theory Ser. A 157 (2018), 402–426.

    MathSciNet  Article  Google Scholar 

  9. [9]

    B. Csajbók, G. Marino and O. Polverino: A Carlitz type result for linearized polynomials, Ars Math. Contemp. 16 (2019), 585–608.

    MathSciNet  Article  Google Scholar 

  10. [10]

    B. Csajbók, G. Marino, O. Polverino and Y. Zhou: Maximum Rank-Distance codes with maximum left and right idealisers, Discrete Math. 343 (2020).

  11. [11]

    B. Csajbók, G. Marino, O. Polverino and F. Zullo: Maximum scattered linear sets and MRD-codes, J. Algebraic Combin. 46 (2017), 1–15.

    MathSciNet  Article  Google Scholar 

  12. [12]

    B. Csajbók and C. Zanella: On the equivalence of linear sets, Des. Codes Cryptogr. 81 (2016), 269–281.

    MathSciNet  Article  Google Scholar 

  13. [13]

    P. Delsarte: Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A 25 (1978), 226–241.

    MathSciNet  Article  Google Scholar 

  14. [14]

    E. Gabidulin: Theory of codes with maximum rank distance, Problems of information transmission, 21(3) (1985), 3–16.

    MathSciNet  MATH  Google Scholar 

  15. [15]

    G. Gasper and M. Rahman: Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications), Cambridge University Press (2004).

  16. [16]

    B. Huppert: Endliche Gruppen, volume 1, Springer Berlin-Heidelberg-New York, 1967.

    Book  Google Scholar 

  17. [17]

    M. Lavrauw: Scattered spaces in Galois Geometry, in: Contemporary Developments in Finite Fields and Applications, 2016, 195–216.

  18. [18]

    M. Lavrauw and G. Van de Voorde: Field reduction and linear sets in finite geometry, in: Topics in Finite Fields, AMS Contemporary Math, vol. 623, 271–293, American Mathematical Society, Providence (2015).

    MATH  Google Scholar 

  19. [19]

    G. Lunardon: MRD-codes and linear sets, J. Combin. Theory Ser. A 149 (2017), 1–20.

    MathSciNet  Article  Google Scholar 

  20. [20]

    G. Lunardon, P. Polito and O. Polverino: A geometric characterisation of linear k-blocking sets, J. Geom. 74 (2002), 120–122.

    MathSciNet  Article  Google Scholar 

  21. [21]

    G. Lunardon and O. Polverino: Translation ovoids of orthogonal polar spaces, Forum Math. 16 (2004), 663–669.

    MathSciNet  Article  Google Scholar 

  22. [22]

    G. Lunardon, R. Trombetti and Y. Zhou: On kernels and nuclei of rank metric codes, J. Algebraic Combin. 46 (2017), 313–340.

    MathSciNet  Article  Google Scholar 

  23. [23]

    V. Napolitano and F. Zullo: Codes with few weights arising from linear sets, Advances in Mathematics of Communications, accepted.

  24. [24]

    O. Polverino: Linear sets in finite projective spaces, Discrete Math. 310 (2010), 3096–3107.

    MathSciNet  Article  Google Scholar 

  25. [25]

    A. Ravagnani: Rank-metric codes and their duality theory, Des. Codes Cryptogr. 80 (2016), 197–216.

    MathSciNet  Article  Google Scholar 

  26. [26]

    B. Segre: Teoria di Galois, fibrazioni proiettive e geometrie non Desarguesiane, Ann. Mat. Pura Appl. 64 (1964), 1–76.

    MathSciNet  Article  Google Scholar 

  27. [27]

    J. Sheekey: MRD codes: constructions and connections, Combinatorics and finite fields: Difference sets, polynomials, pseudorandomness and applications, Radon Series on Computational and Applied Mathematics, K.-U. Schmidt and A. Winterhof (eds.).

  28. [28]

    J. Sheekey and G. Van de Voorde: Rank-metric codes, linear sets and their duality, Des. Codes Cryptogr. 88 (2020), 655–675.

    MathSciNet  Article  Google Scholar 

  29. [29]

    G. Zini and F. Zullo: Scattered subspaces and related codes, submitted, arXiv:2007.04643.

  30. [30]

    F. Zullo: Linear codes and Galois geometries, Ph.D thesis, Università degli Studi della Campania “Luigi Vanvitelli”.

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Olga Polverino.

Additional information

The research was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA — INdAM). The first author was partially supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the National Research, Development and Innovation Office — NKFIH under the grants PD 132463 and K 124950. The last two authors were supported by the project “VALERE: VAnviteLli pEr la RicErca” of the University of Campania “Luigi Vanvitelli”.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Csajbók, B., Marino, G., Polverino, O. et al. Generalising the Scattered Property of Subspaces. Combinatorica 41, 237–262 (2021). https://doi.org/10.1007/s00493-020-4347-y

Download citation

Mathematics Subject Classification (2010)

  • 05B25
  • 51E20
  • 33C20