Abstract
We study a family of primary affine variety codes defined from the Klein quartic. The duals of these codes have previously been treated in [12, Example 3.2]. Among the codes that we construct almost all have parameters as good as the best known codes according to [9] and in the remaining few cases the parameters are almost as good. To establish the code parameters we apply the footprint bound [7, 10] from Gröbner basis theory and for this purpose we develop a new method where we inspired by Buchbergers algorithm perform a series of symbolic computations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Cox, D.A., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, vol. 4. Springer, New York (2015)
Feng, G.L., Rao, T.R.N.: Decoding algebraic-geometric codes up to the designed minimum distance. IEEE Trans. Inform. Theory 39(1), 37–45 (1993)
Feng, G.L., Rao, T.R.N.: A simple approach for construction of algebraic-geometric codes from affine plane curves. IEEE Trans. Inform. Theory 40(4), 1003–1012 (1994)
Feng, G.L., Rao, T.R.N.: Improved geometric Goppa codes part I: basic theory. IEEE Trans. Inform. Theory 41(6), 1678–1693 (1995)
Fitzgerald, J., Lax, R.F.: Decoding affine variety codes using Gröbner bases. Des. Codes Crypt. 13(2), 147–158 (1998)
Geil, O.: Evaluation codes from an affine variety code perspective. In: Martínez-Moro, E., Munuera, C., Ruano, D. (eds.) Advances in Algebraic Geometry Codes. Coding Theory and Cryptology, vol. 5, pp. 153–180. World Scientific, Singapore (2008)
Geil, O., Høholdt, T.: Footprints or generalized Bezout’s theorem. IEEE Trans. Inform. Theory 46(2), 635–641 (2000)
Geil, O., Martin, S.: An improvement of the Feng-Rao bound for primary codes. Des. Codes Crypt. 76(1), 49–79 (2015)
Grassl, M.: Bounds on the minimum distance of linear codes and quantum codes (2007). http://www.codetables.de. Accessed 20 Apr 2017
Høholdt, T.: On (or in) Dick Blahut‘s’ footprint’. Codes, Curves and Signals, pp. 3–9 (1998)
Høholdt, T., van Lint, J.H., Pellikaan, R.: Algebraic geometry codes. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory, vol. 1, pp. 871–961. Elsevier, Amsterdam (1998)
Kolluru, M.S., Feng, G.L., Rao, T.R.N.: Construction of improved geometric Goppa codes from Klein curves and Klein-like curves. Appl. Algebra Engrg. Comm. Comput. 10(6), 433–464 (2000)
Salazar, G., Dunn, D., Graham, S.B.: An improvement of the Feng-Rao bound on minimum distance. Finite Fields Appl. 12, 313–335 (2006)
Acknowledgments
The authors gratefully acknowledge the support from The Danish Council for Independent Research (Grant No. DFF–4002-00367). They are also grateful to Department of Mathematical Sciences, Aalborg University for supporting a one-month visiting professor position for the second listed author. The research of Ferruh Özbudak has been funded by METU Coordinatorship of Scientific Research Projects via grant for projects BAP-01-01-2016-008 and BAP-07-05-2017-007.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Geil, O., Özbudak, F. (2017). Bounding the Minimum Distance of Affine Variety Codes Using Symbolic Computations of Footprints. In: Barbero, Á., Skachek, V., Ytrehus, Ø. (eds) Coding Theory and Applications. ICMCTA 2017. Lecture Notes in Computer Science(), vol 10495. Springer, Cham. https://doi.org/10.1007/978-3-319-66278-7_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-66278-7_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-66277-0
Online ISBN: 978-3-319-66278-7
eBook Packages: Computer ScienceComputer Science (R0)