Skip to main content

Advertisement

Springer Nature Link
Log in

Lower bounds on minimal distance of evaluation codes

  • Published:
Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

In (J Pure Appl Algebra 196:91–99, 2005), the authors point out that the methods they use to find a lower bound for the minimal distance of complete intersection evaluation codes should apply to the case of (arithmetically) Gorenstein evaluation codes. In this note we show this is the case and we study other lower bounds on the minimal distance coming from the syzygies.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. Ballico E., Fontanari C.: The Horace method for error-correcting codes. Appl. Algebra Eng. Commun. Comput. 17(2), 135–139 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Buchsbaum D., Eisenbud D.: Algebra structures for finite free resolutions and some structure theorems for ideals of codimension 3. Am. J. Math. 99, 447–485 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cox D., Little J., O’Shea D.: Using Algebraic Geometry. Springer, New York (1998)

    MATH  Google Scholar 

  4. Davis E., Geramita A., Orecchia F.: Gorenstein algebras and the Cayley–Bacharach Theorem. Proc. Am. Math. Soc. 93, 593–597 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dodunekov, S., Simonis, J.: Codes and projective multisets. Electron. J. Comb. 5, No.1, Research paper R37, 23 p. (1998)

  6. Eisenbud D.: The Geometry of Syzygies. Springer, New York (2005)

    MATH  Google Scholar 

  7. Eisenbud D., Green M., Harris J.: Cayley–Bacharach theorems and conjectures. Bull. Am. Math. Soc. 33, 295–324 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Eisenbud D., Popescu S.: The projective geometry of the Gale transform. J. Algebra 230, 127–173 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Geramita A., Orecchia F.: On the Cohen–Macaulay type of s−lines in \({\mathbb{A}^{n+1}}\) . J. Algebra 70, 116–140 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gold L., Little J., Schenck H.: Cayley–Bacharach and evaluation codes on complete intersections. J. Pure Appl. Algebra 196, 91–99 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hansen, J.: Points in uniform position and maximum distance separable codes. In: Zero-Dimensional Schemes (Ravello, 1992), pp. 205–211. de Gruyter, Berlin (1994)

  12. Hansen J.: Linkage and codes on complete intersections. Appl. Algebra Eng. Comm. Comput. 14, 175–185 (2003)

    Article  MATH  Google Scholar 

  13. Huneke, C.: Hyman Bass and Ubiquity: Gorenstein Rings. arXiv:math.AC/0209199v1, 16 Sep (2002)

  14. Tsfasman M., Vladut S., Nogin D.: Algebraic Geometric Codes: Basic Notions. American Mathematical Society, Providence (2007)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, 45221-0025, USA

    Ştefan O. Tohǎneanu

Authors
  1. Ştefan O. Tohǎneanu
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Ştefan O. Tohǎneanu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tohǎneanu, Ş.O. Lower bounds on minimal distance of evaluation codes. AAECC 20, 351–360 (2009). https://doi.org/10.1007/s00200-009-0102-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00200-009-0102-8

Keywords

Mathematics Subject Classification (2000)