Designs, Codes and Cryptography

, Volume 76, Issue 1, pp 49–79 | Cite as

An improvement of the Feng–Rao bound for primary codes

  • Olav Geil
  • Stefano Martin


We present a new bound for the minimum distance of a general primary linear code. For affine variety codes defined from generalised \(C_{ab}\) polynomials the new bound often improves dramatically on the Feng–Rao bound for primary codes (Andersen and Geil, Finite Fields Appl 14(1):92–123, 2008; Geil et al., Lecture Notes in Computer Science 3857: 295–306, 2006). The method does not only work for the minimum distance but can be applied to any generalised Hamming weight.


Affine variety code \(C_{ab}\) curve Feng–Rao bound Generalised Hamming weight Minimum distance One-way well-behaving pair 

Mathematics Subject Classification

94B65 94B27 94B05 



Part of this research was done while the second listed author was visiting East China Normal University. We are grateful to Professor Hao Chen for his hospitality. The authors also gratefully acknowledge the support from the Danish National Research Foundation and the National Science Foundation of China (Grant No. 11061130539) for the Danish-Chinese Center for Applications of Algebraic Geometry in Coding Theory and Cryptography. The authors would like to thank Diego Ruano, Peter Beelen and Ryutaroh Matsumoto for pleasant discussions. Finally we are grateful to the anonymous reviewers for their careful reading and useful suggestions.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematical SciencesAalborg UniversityAalborgDenmark
  2. 2.Engineering Software InstituteEast China Normal UniversityShanghaiChina

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