Skip to main content
SpringerLink
Log in

Maximal caps in AG (6, 3)

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

We show that there are no complete 44-caps in AG(5, 3). We then use this result to prove that the maximal size for a cap in AG(6, 3) is equal to 112, and that the 112-caps in AG(6, 3) are unique up to affine equivalence.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Barlotti, A.: Un’ estensione del teorema di Segre-Kustaanheimo. Boll. Un. Mat. Ital. 10, 96–98 (1955)

    MATH  MathSciNet  Google Scholar 

  • Bierbrauer, J., Edel, Y.: Bounds on affine caps. J. Combin. Des. 10, 111–115 (2002)

    Article  MathSciNet  Google Scholar 

  • Bose, R.C.: Mathematical theory of the symmetrical factorial design. Sankhyā 8, 107–166 (1947)

    MATH  Google Scholar 

  • Davis, B.L., Maclagan, D.: The card game set. Math. Intell. 25, 33–40 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Edel, Y., Bierbrauer, J.: 41 is the largest size of a cap in PG(4,4). Des. Codes Cryptogr. 16(2), 151–160 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Edel, Y., Ferret, S., Landjev, I., Storme, L.: The classification of the largest caps in AG(5, 3). J. Combin. Theory Ser. A 99, 95–110 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Hill, R.: On the largest size of cap in S 5,3. Atti Accad. Naz. Lincei Rend. 54, 378–384 (1973)

    Google Scholar 

  • Hill, R.: Caps and codes. Discrete Math. 22, 111–137 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  • Hill, R.: On Pellegrino’s 20-caps in S4,3. Combinatorial Geometries and their Applications (Rome 1981). Ann. Discrete Math. 18, 443–448 (1983)

    Google Scholar 

  • Hirschfeld J.W.P., Storme L.: The packing problem in statistics, coding theory and finite projective spaces: update 2001. In: Blokhuis A., Hirschfeld J.W.P., Jungnickel D., Thas J.A. (eds) Developments in Mathematics, vol. 3, Kluwer Academic Publishers. Finite Geometries, Proceedings of the Fourth Isle of Thorns Conference, Chelwood Gate, July 16–21. pp. 201–246 (2000).

  • Panella, G.: Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito. Boll. Un. Mat. Ital. 10, 507–513 (1955)

    MATH  MathSciNet  Google Scholar 

  • Pellegrino, G.: Sul massimo ordine delle calotte in S 4,3. Le Mathematiche 25, 149–157 (1971)

    MATH  MathSciNet  Google Scholar 

  • Qvist B.: Some remarks concerning curves of the second degree in a finite plane. Ann. Acad. Sci. Fenn. Ser. A 134, (1952).

  • Segre, B.: Sulle ovali nei piani lineari finiti. Atti Accad. Naz. Lincei Rend. 17, 1–2 (1954)

    MathSciNet  Google Scholar 

  • Segre, B.: Ovals in a finite projective plane. Can. J. Math. 7, 414–416 (1955)

    MATH  MathSciNet  Google Scholar 

  • Segre, B.: Le geometrie di Galois. Ann. Mat. Pura Appl. 48, 1–97 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  • Tits, J.: Ovoides et groupes de Suzuki. Arch. Math. 13, 187–198 (1962)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Princeton University, Princeton, NJ, 08544, USA

    Aaron Potechin

Authors
  1. Aaron Potechin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aaron Potechin.

Additional information

Communicated by J.D. Key.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Potechin, A. Maximal caps in AG (6, 3). Des. Codes Cryptogr. 46, 243–259 (2008). https://doi.org/10.1007/s10623-007-9132-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-007-9132-z

Keywords

AMS Classifications

Search

Navigation