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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Becker T., Weispfenning V.: Gröbner Bases: A Computational Approach to Commutative Algebra. Springer, New York (1993).
Beelen P., Høholdt T., Piñero F., Justesen J.: On the Dimension of graph codes with Reed-Solomon component codes. In: ISIT, pp. 1227–1231 (2013).
Cox D., Little J., O’Shea D.: Ideals, Varieties and Algorithms. Springer, New York (2007).
Fitzgerald J., Lax R.: Decoding affine variety codes using Gröbner bases. DCC 13, 147–158 (1998). doi:10.1023/A:1008274212057.
Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. Online available at http://www.codetables.de (2007). Accessed on 22 Nov 2013.
Høholdt T., Justesen J.: The Minimum Distance of Graph Codes. Lecture Notes in Computer Science, vol. 6639, pp. 201–212. Springer, Berlin (2011).
Roth R.M.: Introduction to Coding Theory. Cambridge University Press, Cambridge (2006).
Sipser M., Spielman D.: Expander codes. IEEE Trans. Inf. Theory 42(6), 1710–1722 (1996). doi:10.1109/18.556667.
Stichtenoth H.: On the dimension of subfield subcodes. IEEE Trans. Inf. Theory 36(1), 90–93 (1990).
Tanner R.: A recursive approach to low complexity codes. IEEE Trans. Inf. Theory 27(5), 533–547 (1981). doi:10.1109/TIT.1981.1056404.
Zemor G.: On expander codes. IEEE Trans. Inf. Theory 47(2), 835–837 (2001). doi:10.1109/18.910593.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Computer Algebra in Coding Theory and Cryptography”.
Rights and permissions
About this article
Cite this article
Høholdt, T., Piñero, F. & Zeng, P. Optimal codes as Tanner codes with cyclic component codes. Des. Codes Cryptogr. 76, 37–47 (2015). https://doi.org/10.1007/s10623-014-9962-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-014-9962-4