Lower bounds on minimal distance of evaluation codes



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.


Gorenstein ring Complete intersection Syzygies Evaluation codes 

Mathematics Subject Classification (2000)

Primary: 16E65 13D02 Secondary: 94B27 


  1. 1.
    Ballico E., Fontanari C.: The Horace method for error-correcting codes. Appl. Algebra Eng. Commun. Comput. 17(2), 135–139 (2006)MATHCrossRefMathSciNetGoogle Scholar
  2. 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)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cox D., Little J., O’Shea D.: Using Algebraic Geometry. Springer, New York (1998)MATHGoogle Scholar
  4. 4.
    Davis E., Geramita A., Orecchia F.: Gorenstein algebras and the Cayley–Bacharach Theorem. Proc. Am. Math. Soc. 93, 593–597 (1985)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dodunekov, S., Simonis, J.: Codes and projective multisets. Electron. J. Comb. 5, No.1, Research paper R37, 23 p. (1998)Google Scholar
  6. 6.
    Eisenbud D.: The Geometry of Syzygies. Springer, New York (2005)MATHGoogle Scholar
  7. 7.
    Eisenbud D., Green M., Harris J.: Cayley–Bacharach theorems and conjectures. Bull. Am. Math. Soc. 33, 295–324 (1996)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Eisenbud D., Popescu S.: The projective geometry of the Gale transform. J. Algebra 230, 127–173 (2000)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Geramita A., Orecchia F.: On the Cohen–Macaulay type of s−lines in \({\mathbb{A}^{n+1}}\) . J. Algebra 70, 116–140 (1981)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gold L., Little J., Schenck H.: Cayley–Bacharach and evaluation codes on complete intersections. J. Pure Appl. Algebra 196, 91–99 (2005)MATHCrossRefMathSciNetGoogle Scholar
  11. 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)Google Scholar
  12. 12.
    Hansen J.: Linkage and codes on complete intersections. Appl. Algebra Eng. Comm. Comput. 14, 175–185 (2003)MATHCrossRefGoogle Scholar
  13. 13.
    Huneke, C.: Hyman Bass and Ubiquity: Gorenstein Rings. arXiv:math.AC/0209199v1, 16 Sep (2002)Google Scholar
  14. 14.
    Tsfasman M., Vladut S., Nogin D.: Algebraic Geometric Codes: Basic Notions. American Mathematical Society, Providence (2007)MATHGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA

Personalised recommendations