×

zbMATH — the first resource for mathematics

Trial set and Gröbner bases for binary codes. (English) Zbl 1384.94123
Kotsireas, Ilias S. (ed.) et al., Applications of computer algebra, Kalamata, Greece, July 20–23, 2015. Cham: Springer (ISBN 978-3-319-56930-7/hbk; 978-3-319-56932-1/ebook). Springer Proceedings in Mathematics & Statistics 198, 21-27 (2017).
Summary: In this work, we show the connections between trial sets and Gröbner bases for binary codes, which give characterizations of trial sets in the context of Gröbner bases and algorithmic ways for computing them. In this sense, minimal trial sets will be characterized as trial sets associated with minimal Gröbner bases of the ideal associated to a code.
For the entire collection see [Zbl 1379.13001].
MSC:
94B05 Linear codes, general
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 1. Adams,W.W., Loustaunau, Ph.: An introduction to Gröobner bases. In: Graduate Studies in Mathematics, vol. 3. American Mathematical Society, Providence (1994)
[2] 2. Barg, A.: Complexity issues in coding theory. In: Handbook of Coding Theory, vol. I, pp. 649-754. North-Holland, Amsterdam (1998) · Zbl 0929.94028
[3] 3. Borges-Quintana, M., Borges-Trenard, M.A., Martínez-Moro, E.: On a Gröbner bases structure associated to linear codes. J. Discret. Math. Sci. Cryptogr. 10 (2), 151-191 (2007) · Zbl 1172.94632
[4] 4. Borges-Quintana, M., Borges-Trenard, M.A., Martínez-Moro, E.: A Gröbner representation for linear codes. In: Advances in Coding Theory and Cryptography, Ser. Coding Theory Cryptololgy, vol. 3, pp. 17-32. World Sci. Publ., Hackensack (2007) · Zbl 1141.94011
[5] 5. Borges-Quintana, M., Borges-Trenard, M.A., Fitzpatrick, P., Martínez-Moro, E.: Gröbner bases and combinatorics for binary codes. Appl. Algebra Eng. Comm. Comput. 19 (5), 393-411 (2008) · Zbl 1192.94130
[6] 6. Helleseth, T., Kløve, T., Vladimir, I.L.: Error-correction capability of binary linear codes. IEEE Trans. Inf. Theory 51 (4), 1408-1423 (2005) · Zbl 1293.94111
[7] 7. Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003) · Zbl 1099.94030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.