Skip to main content
Log in

Abstract

In this paper we introduce a new family of codes, called projective nested cartesian codes. They are obtained by the evaluation of homogeneous polynomials of a fixed degree on a certain subset of \(\mathbb {P}^n(\mathbb {F}_q)\), and they may be seen as a generalization of the so-called projective Reed–Muller codes. We calculate the length and the dimension of such codes, an upper bound for the minimum distance and the exact minimum distance in a special case (which includes the projective Reed–Muller codes). At the end we show some relations between the parameters of these codes and those of the affine cartesian codes.

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

Access this article

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

  • Ballet, S., Rolland, R.: On low weight codewords of generalized affine and projective Reed-Muller codes. Des. Codes Cryptogr. 73(2), 271–297 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Mathematical Institute, University of Innsbruck, Austria. Ph.D. Thesis. An English translation appeared in J. Symbolic Comput. 41 (2006) 475-511 (1965)

  • Carvalho, C.: On the second Hamming weight of some Reed–Muller type codes. Finite Fields Appl. 24, 88–94 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Carvalho, C., Neumann, V.G.L.: Projective Reed–Muller type codes on rational normal scrolls. Finite Fields Appl. 37, 85–107 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Couvreur, A., Duursma, I.: Evaluation codes from smooth quadric surfaces and twisted Segre varieties. Des. Codes Cryptogr. 66, 291–303 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Cox, D., Little, J., O’Shea, D.: Ideals, varieties and algorithms, 3rd ed. Springer, New York (2007)

  • Duursma, I., Rentería, C., Tapia-Recillas, H.: Reed–Muller codes on complete intersections. Appl. Algebra Eng. Commun. Comput. 11, 455–462 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Fitzgerald, J., Lax, R.F.: Decoding affine variety codes using Göbner bases. Des. Codes Cryptogr. 13(2), 147–158 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Geil, O., Thomsen, C.: Weighted Reed–Muller codes revisited. Des. Codes Cryptogr. 66(1–3), 195–220 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • González-Sarabia, M., Rentería, C., Tapia-Recillas, H.: Reed–Muller-type codes over the Segre variety. Finite Fields Appl. 8, 511–518 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Lachaud, G.: The parameters of projective Reed–Muller codes. Discrete Math. 81(2), 217–221 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • López, H.H., Rentería-Márquez, C., Villarreal, R.H.: Affine cartesian codes. Des. Codes Cryptogr. 71(1), 5–19 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Rentería, C., Tapia-Recillas, H.: Reed–Muller codes: an ideal theory approach. Commun. Algebra 25(2), 401–413 (1997)

  • Sørensen, A.: Projective Reed–Muller codes. IEEE Trans. Inf. Theory 37(6), 1567–1576 (1991)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Cícero Carvalho.

Additional information

The Cícero Carvalho and V. G. Lopez Neumann are partially supported by CNPq and by FAPEMIG. The Hiram H. López was partially supported by CONACyT and Universidad Autónoma de Aguascalientes.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carvalho, C., Neumann, V.G.L. & López, H.H. Projective Nested Cartesian Codes. Bull Braz Math Soc, New Series 48, 283–302 (2017). https://doi.org/10.1007/s00574-016-0010-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00574-016-0010-z

Keywords

Mathematics Subject Classification

Navigation