Advertisement

Transitive PSL(2,11)-invariant k-arcs in PG(4,q)

  • Torger Olson
  • Eric Swartz
Article
  • 5 Downloads

Abstract

A k-arc in the projective space \(\mathrm{PG}(n,q)\) is a set of k projective points such that no subcollection of \(n+1\) points is contained in a hyperplane. In this paper, we construct new 60-arcs and 110-arcs in \(\mathrm{PG}(4,q)\) that do not arise from rational or elliptic curves. We introduce computational methods that, when given a set \(\mathcal {P}\) of projective points in the projective space of dimension n over an algebraic number field \({\mathbb {Q}}(\xi )\), determines a complete list of primes p for which the reduction modulo p of \(\mathcal {P}\) to the projective space \(\mathrm{PG}(n,p^h)\) may fail to be a k-arc. Using these methods, we prove that there are infinitely many primes p such that \(\mathrm{PG}(4,p)\) contains a \(\mathrm{PSL}(2,11)\)-invariant 110-arc, where \(\mathrm{PSL}(2,11)\) is given in one of its natural irreducible representations as a subgroup of \(\mathrm{PGL}(5,p)\). Similarly, we show that there exist \(\mathrm{PSL}(2,11)\)-invariant 110-arcs in \(\mathrm{PG}(4,p^2)\) and \(\mathrm{PSL}(2,11)\)-invariant 60-arcs in \(\mathrm{PG}(4,p)\) for infinitely many primes p.

Keywords

Finite projective space k-arc \(\mathrm{PSL}(2, 11)\) 

Mathematics Subject Classification

51E20 20G40 

Notes

Acknowledgements

The authors would like to thank John Bamberg for helpful discussions about the \({\mathrm{GAP}}\) package FinInG and Jan De Beule for \({\mathrm{GAP}}\) code that allows for calculations using FinInG beyond what is normally possible in \({\mathrm{GAP}}\). The authors would also like to thank the anonymous referees for many useful suggestions that improved the readability of this paper.

References

  1. 1.
    Abbott R., Bray J., Linton S., Nickerson S., Norton S., Parker R., Suleiman I., Tripp J., Walsh P., Wilson R.: Atlas of finite group representations—version 3. http://brauer.maths.qmul.ac.uk/Atlas/v3/.
  2. 2.
    Bamberg J., Betten A., Cara Ph, De Beule J., Lavrauw M., Neunhöffer M.: Finite incidence geometry. FInInG-a GAP package, version 1(4) (2017).Google Scholar
  3. 3.
    Bartoli D., Giuletti M., Platoni I.: On the covering radius of MDS codes. IEEE Trans. Inf. Theory 61(2), 801–811 (2015).MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bray J.N., Holt D.F., Roney-Dougal C.M.: The maximal subgroups of the low-dimensional finite classical groups. With a foreword by M. Liebeck. London Mathematical Society Lecture Note Series, 407. Cambridge University Press, Cambridge (2013).Google Scholar
  5. 5.
    Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A.: Atlas of finite groups. Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G, Thackray (1985).Google Scholar
  6. 6.
    Giulietti M., Korchmáros G., Marcugini S., Pambianco F.: Transitive \(A_6\)-invariant \(k\)-arcs in \({{\rm PG}}(2, q)\). Des. Codes Cryptogr. 68, 73–79 (2013).MathSciNetCrossRefGoogle Scholar
  7. 7.
    Indaco L., Korchmáros G.: \(42\)-arcs in \({{\rm PG}}(2, q)\) left invariant by \({{\rm PSL}}(2,7)\). Des. Codes Cryptogr. 64, 33–46 (2012).MathSciNetCrossRefGoogle Scholar
  8. 8.
    Jackson W.-A., Martin K.M., O’Keefe C.M.: Geometrical contributions to secret sharing theory. J. Geom. 79(1–2), 102–133 (2004).MathSciNetCrossRefGoogle Scholar
  9. 9.
    Korchmáros G., Pace N.: Infinite family of large complete arcs in \({{\rm PG}}(2, q^n)\) with \(q\) odd and \(n>1\) odd. Des. Codes Cryptogr. 55, 285–296 (2010).MathSciNetCrossRefGoogle Scholar
  10. 10.
    Korchmáros G., Lanzone V., Sonnino A.: Projective \(k\)-arcs and \(2\)-level secret-sharing schemes. Des. Codes Cryptogr. 64(1–2), 3–15 (2012).MathSciNetCrossRefGoogle Scholar
  11. 11.
    Olson T., Swartz E.: Transitive \({{\rm PSL}}(2,11)\)-invariant \(k\)-arcs in \({{\rm PG}}(4,q)\) (including supporting GAP code). arXiv:1804.09707.
  12. 12.
    Pace N.: On small complete arcs and transitive \(A_5\)-invariant arcs in the projective plane \({{\rm PG}}(2, q)\). J. Comb. Des. 22(10), 425–434 (2014).CrossRefGoogle Scholar
  13. 13.
    Pace N., Sonnino A.: On linear codes admitting large automorphism groups. Des. Codes Cryptogr. 83(1), 115–143 (2017).MathSciNetCrossRefGoogle Scholar
  14. 14.
    Simmons G.J., Jackson W.-A., Martin K.M.: The geometry of shared secret schemes. Bull. Inst. Comb. Appl. 1, 71–87 (1991).MathSciNetzbMATHGoogle Scholar
  15. 15.
    Sonnino A.: Transitive \({{\rm PSL}}(2,7)\)-invariant \(42\)-arcs in \(3\)-dimensional projective spaces. Des. Codes Cryptogr. 72, 455–463 (2014).MathSciNetCrossRefGoogle Scholar
  16. 16.
    Thas J.A.: M.D.S. codes and arcs in projective spaces: a survey. Mathematiche (Catania) 47(2), 315–328 (1993).MathSciNetzbMATHGoogle Scholar
  17. 17.
    The GAP Group. GAP—Groups, Algorithms, and Programming, Version 4.8.9 (2017). https://www.gap-system.org.
  18. 18.
    Tolhuizen L.M.G.M., van Gils W.J.: A large automorphism group decreases the computations in the construction of an optimal encoder/decoder pair for a linear block code. IEEE Trans. Inf. Theory 34(2), 333–338 (1988).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsCollege of William & MaryWilliamsburgUSA

Personalised recommendations