Designs, Codes and Cryptography

, Volume 76, Issue 1, pp 37–47 | Cite as

Optimal codes as Tanner codes with cyclic component codes

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Abstract

In this article we study a class of graph codes with cyclic code component codes as affine variety codes. Within this class of Tanner codes we find some optimal binary codes. We use a particular subgraph of the point-line incidence plane of \(\mathbf {A}(2,q)\) as the Tanner graph, and we are able to describe the codes succinctly using Gröbner bases.

Keywords

Tanner codes Graph codes Graph based codes Expander codes Affine variety codes Gröbner bases 

Mathematics Subject Classification

11T71 94B15 94B25 94B27 

Notes

Acknowledgments

The authors gratefully acknowledge the generous 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 third author would also like to acknowledge the support of the National Natural Science Foundation of China under Grants Nos. 61321064 and 61103222 and the Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20110076120016.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkKongens LyngbyDenmark
  2. 2.Shanghai Key Laboratory of Trustworthy ComputingEast China Normal UniversityShanghaiPeople’s Republic of China

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