Skip to main content
SpringerLink
Log in
Book cover

Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics pp 525–535Cite as

The Poincaré Polynomial of a Linear Code

  • Chapter
  • First Online:

  • 835 Accesses

Abstract

We introduce the Poincaré polynomial of a linear q-ary code and its relation to the corresponding weight enumerator. The question of whether the Poincaré polynomial is a complete invariant is answered affirmatively for q = 2, 3 and negatively for q ≥ 4. Finally we determine this polynomial for MDS codes and, by means of a recursive formula, for binary Reed-Muller codes.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley, Reading (1969)

    MATH  Google Scholar 

  2. Blackmore, T., Norton, G.H.: Matrix-product codes over \(\mathbb {F}_q\). Appl. Algebra Eng. Commun. Comput. 12, 477–500 (2001)

    Google Scholar 

  3. Brylawski, T.H., Lucas, D.: Uniquely representable combinatorial geometries. In: Teorie Combinatorie. Proceedings of the 1973 International Colloquium, pp. 83–104. Accademia Nazionale del Lincei, Rome (1976)

    Google Scholar 

  4. Campillo, A., Delgado, F., Gusein-Zade S.: Poincaré series of a rational surface singularity. Invent. Math. 155 45–53 (2004)

    Article  Google Scholar 

  5. Campillo, A., Delgado, F., Gusein-Zade S.: Poincaré series of curves on rational surface singularities. Comment. Math. Helvetici 80, 95–102 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Campillo, A., Delgado, F., Gusein-Zade, S.: Multiindex filtrations and motivic Poincaré series. Monatshefte. Math. 150, 193–209 (2007)

    Article  MathSciNet  Google Scholar 

  7. Campillo, A., Delgado, F., Gusein-Zade S., Hernando, F.: Poincaré series of collections of plane valuations. Int. J. Math. 21, 1461–1473 (2010)

    Article  Google Scholar 

  8. Campillo, A., Delgado, F., Gusein-Zade, S.: Equivariant Poincaré series of filtrations. Rev. Mat. Complut. 26, 241–251 (2013)

    Article  MathSciNet  Google Scholar 

  9. Campillo, A., Delgado, F., Gusein-Zade, S.: An equivariant Poincaré series of filtrations and monodromy of zeta functions. Rev. Mat. Complut. 28, 449–467 (2015)

    Article  MathSciNet  Google Scholar 

  10. Delgado, F., Moyano-Fernández, J.J.: On the relation between the generalized Poincaré series and the Stöhr zeta function. Proc. Am. Math. Soc. 137, 51–59 (2009)

    MATH  Google Scholar 

  11. Dür, A.: The automorphism groups of Reed-Solomon codes. J. Comb. Theory Ser. A 44(1), 69–82 (1987)

    Article  MathSciNet  Google Scholar 

  12. Galindo, C., Monserrat, F.: The Poincaré series of multiplier ideals of a simple complete ideal in a local ring of a smooth surface. Adv. Math. 225, 1046–1068 (2010)

    Article  MathSciNet  Google Scholar 

  13. Hernando, F., Ruano, D.: New linear codes from matrix-product codes with polynomial units. Adv. Math. Commun. 4, 363–367 (2010)

    Article  MathSciNet  Google Scholar 

  14. Hernando, F., Ruano, D.: Decoding of matrix-product codes. J. Algebra Appl. 12, 1250185 (2013)

    Article  MathSciNet  Google Scholar 

  15. Hernando, F., Lally, K., Ruano, D.: Construction and decoding of matrix-product codes from nested codes. Appl. Algebra Eng. Commun. Comput. 20, 497–507 (2009)

    Article  MathSciNet  Google Scholar 

  16. Hernando, F., Høholdt, T., Ruano, D.: List decoding of matrix-product codes from nested codes: an application to quasi-cyclic codes. Adv. Math. Commun. 6, 259–272 (2012)

    Article  MathSciNet  Google Scholar 

  17. Jurrius, R., Pellikaan, R.: Codes, arrangements and matroids. In: Martinez-Moro, E. (ed.) Algebraic Geometry Modeling in Information Theory. Series on Coding Theory and Cryptology, vol. 8, pp. 219–325. World Scientific, Hackensack (2013)

    Chapter  Google Scholar 

  18. Kahn, J.: On the uniqueness of matroid representations over GF(4). Bull. Lond. Math. Soc. 20, 5–10 (1988)

    Article  MathSciNet  Google Scholar 

  19. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North Holland Publishing, Amsterdam/New York (1977)

    MATH  Google Scholar 

  20. Pellikaan, R., Wu, X.-W., Bulygin, S., Jurrius, R.: Codes, Cryptology and Curves with Computer Algebra. Cambridge University Press, Cambridge (2017)

    Book  Google Scholar 

  21. Roman, S.: Coding and Information Theory. Springer, New York (1992)

    MATH  Google Scholar 

  22. Sokal, A.D.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In: Chapman, R. (ed.) Surveys in Combinatorics. London Mathematical Society Lecture Note Series, vol. 327, pp. 173–226. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  23. Tsfasman, M.A., Vlǎdut, S.G.: Algebraic-Geometric Codes. Kluwer Academic Publishers, Dordrecht (1991)

    Google Scholar 

  24. Tsfasman, M.A., Vlǎdut, S.G., Nogin, D.: Algebraic Geometric Codes: Basic Notions. Mathematical Surveys and Monographs, vol. 139. American Mathematical Society, Providence (2007)

    Google Scholar 

Download references

Acknowledgements

We like to thank Rudi Pendavingh for the given information about unique representable matroids.

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones de Castellón, Universitat Jaume I, Castelló, Spain

    Carlos Galindo & Fernando Hernando

  2. Instituto Universitario de Matemática Pura y Aplicada (IUMPA), Universidad Politécnica de Valencia, Valencia, Spain

    Francisco Monserrat

  3. Discrete Mathematics, Technische Universiteit Eindhoven, Eindhoven, The Netherlands

    Ruud Pellikaan

Authors
  1. Carlos Galindo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fernando Hernando
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Francisco Monserrat
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ruud Pellikaan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carlos Galindo .

Editor information

Editors and Affiliations

  1. Fachbereich Mathematik, Universität Kaiserslautern, Kaiserslautern, Germany

    Gert-Martin Greuel

  2. Departamento de Álgebra, Instituto de Matemáticas (IMUS), Universidad de Sevilla, Sevilla, Spain

    Luis Narváez Macarro

  3. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain

    Sebastià Xambó-Descamps

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Galindo, C., Hernando, F., Monserrat, F., Pellikaan, R. (2018). The Poincaré Polynomial of a Linear Code. In: Greuel, GM., Narváez Macarro, L., Xambó-Descamps, S. (eds) Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics. Springer, Cham. https://doi.org/10.1007/978-3-319-96827-8_23

Download citation

Publish with us

Policies and ethics

Search

Navigation