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Designs, Codes and Cryptography

, Volume 87, Issue 4, pp 795–806 | Cite as

Linear codes from Denniston maximal arcs

  • Daniele BartoliEmail author
  • Massimo Giulietti
  • Maria Montanucci
Article
  • 90 Downloads
Part of the following topical collections:
  1. Special Issue: Finite Geometries

Abstract

In this paper we construct functional codes from Denniston maximal arcs. For \(q=2^{4\ell +2}\) we obtain linear codes with parameters \([(\sqrt{q}-1)(q+1),5,d]_q\) where \(\lim _{q \rightarrow +\infty } d=(\sqrt{q}-1)q-\sqrt{q}\). We also find for \(q=16,32\) a number of linear codes which appear to have larger minimum distance with respect to the known codes with same length and dimension.

Keywords

Denniston maximal arcs Functional codes Maximal arcs 

Mathematics Subject Classification

94B05 51E21 

Notes

Acknowledgements

The first two authors were supported in part by Ministry for Education, University and Research of Italy (MIUR) (Project PRIN 2012 “Geometrie di Galois e strutture di incidenza”) and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM). The first author carried out this research within the project “Progetto Codici correttori di errori”, supported by Fondo Ricerca di Base, 2015, of Università degli Studi di Perugia. The second author carried out this research within the project “Progetto Geometrie di Galois, Curve Algebriche su campi finiti e loro Applicazioni”, supported by Fondo Ricerca di Base, 2015, of Università degli Studi di Perugia.

References

  1. 1.
    Ball S., Blokhuis A., Mazzocca F.: Maximal arcs in Desarguesian planes of odd order do not exist. Combinatorica 17, 31–41 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barlotti A.: Sui \((k, n)\)-archi di un piano lineare finito. Boll. Un. Mat. Ital. 11, 553–556 (1956).MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bartoli D., Storme L.: On the functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric. Adv. Math. Commun. 8(3), 271–280 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bartoli D., De Boeck M., Fanali S., Storme L.: On the functional codes defined by quadrics and Hermitian varieties. Des. Codes Cryptogr. 71, 21–46 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Calderbank A.R., Kantor W.M.: The geometry of two-weight codes. Bull. Lond. Math. Soc. 18, 97–122 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Carrasco R.A., Johnston M.: Non-Binary Error Control Coding for Wireless Communication and Data Storage. Wiley, Chichester (2009).Google Scholar
  8. 8.
    De Clerck F., Van Maldeghem H.: Some classes of rank \(2\) geometries, Chapter 10. In: Buekenhout F. (ed.) Handbook of Incidence Geometry, Buildings and Foundations, pp. 433–475. Elsevier, Amsterdam (1995).CrossRefGoogle Scholar
  9. 9.
    Delsarte P.: Weights of linear codes and strongly regular normed spaces. Discret. Math. 3, 47–64 (1972).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Denniston R.H.F.: Some maximal arcs in finite projective planes. J. Combin. Theory 6, 317–319 (1969).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Edoukou F.A.B., Hallez A., Rodier F., Storme L.: On the small weight codewords of the functional codes \(C_{herm}(\rm X)\), \({\rm X}\) a non-singular Hermitian variety. Des. Codes Cryptogr. 56, 219–233 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Edoukou F.A.B., Hallez A., Rodier F., Storme L.: A study of intersections of quadrics having applications on the small weight codewords of the functional codes \(C\_2(Q)\), \(Q\) a non-singular quadric. J. Pure Appl. Algebra 214, 1729–1739 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Edoukou F.A.B., Ling S., Xing C.: Structure of functional codes defined on non-degenerate Hermitian varieties. J. Combin. Theory A 118, 2436–2444 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hallez A., Storme L.: Functional codes arising from quadric intersections with Hermitian varieties. Finite Fields Appl. 16, 27–35 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hamilton N.: Degree \(8\) maximal arcs in \(PG(2,2^h)\), \(h\) odd. J. Combin. Theory Ser. A 100, 265–276 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hamilton N., Mathon R.: On the spectrum of non-Denniston maximal arcs in \(PG(2,2^h)\). Eur. J. Combin. 25(3), 415–421 (2004).CrossRefzbMATHGoogle Scholar
  17. 17.
    Hirschfeld J.W.P.: Projective Geometries Over Finite Fields, 2nd edn. Oxford University Press, Oxford (1998).zbMATHGoogle Scholar
  18. 18.
    Johnston M., Carrasco R.A.: Performance of Hermitian codes using combined error and erasure decoding. IEE Proc. Commun. 153(1), 21–30 (2006).CrossRefGoogle Scholar
  19. 19.
    Jibril M., Tomlinson M., Zaki Ahmed M., Tjhai C.: Performance comparison between Hermitian codes and shortened non-binary BCH codes. In: IEEE International Conference on Microwaves, Communications, Antennas and Electronics Systems (2009).Google Scholar
  20. 20.
    Mathon R.: New maximal arcs in desarguesian planes. J. Combin. Theory Ser. A 97, 353–368 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mint Database, http://mint.sbg.ac.at.
  22. 22.
    Stichtenoth H.: A note on Hermitian codes over \(GF(q^2)\). IEEE Trans. Inf. Theory 34(5), 1345–1348 (1988).CrossRefzbMATHGoogle Scholar
  23. 23.
    Stichtenoth H.: Algebraic Function Fields and Codes. Graduate Texts inMathematics, vol. 254, 2nd edn. Springer, Berlin (2009).Google Scholar
  24. 24.
    Thas J.A.: Some results concerning \(\{(q+1)(n-1); n\}\)-arcs and \(\{(q+1)(n-1)+1; n\}\)-arcs in finite projective planes of order \(q\). J. Combin. Theory Ser. A 19, 228–232 (1974).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Thas J.A.: Construction of maximal arcs and partial geometries. Geom. Dedic. 3, 61–64 (1974).MathSciNetzbMATHGoogle Scholar
  26. 26.
    Thas J.A.: Construction of maximal arcs and dual ovals in translation planes. Eur. J. Combin. 1, 189–192 (1980).MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of PerugiaPerugiaItaly
  2. 2.Dipartimento di Matematica Informatica ed EconomiaUniversità degli Studi della BasilicataPotenzaItaly

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