Betti Numbers and Generalized Hamming Weights

Abstract : We can associate to each linear code C defined over a finite field the matroid M[H] of its parity check matrix H. For any matroid M one can define its generalized Hamming weights which are the same as those of the code C. In [2] the authors show that the generalized Hamming weights of a matroid are determined by the N-graded Betti numbers of the Stanley-Reisner ring of the simplicial complex whose faces are the independent set of M. In this talk we go a step further. Our practical results indicate that the generalized Hamming weights of a linear code C can be obtained from the monomial ideal associated with a test-set for C. Moreover, recall that in [3] we use the Gröbner representation of a linear code C to provide a test-set for C. Our results are still a work in progress, but its applications to Coding Theory and Cryptography are of great value.
Type de document :
Communication dans un congrès
22nd Conference on Applications of Computer Algebra (ACA 2016), Aug 2016, Kassel, Germany. 2016, 〈http://www.mathematik.uni-kassel.de/ACA2016/index.php〉
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Contributeur : Irene Márquez Corbella <>
Soumis le : lundi 5 décembre 2016 - 19:50:31
Dernière modification le : jeudi 26 avril 2018 - 10:28:25
Document(s) archivé(s) le : lundi 20 mars 2017 - 21:24:52

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CACTC16-6.pdf
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Irene Márquez-Corbella, Edgar Martínez-Moro. Betti Numbers and Generalized Hamming Weights. 22nd Conference on Applications of Computer Algebra (ACA 2016), Aug 2016, Kassel, Germany. 2016, 〈http://www.mathematik.uni-kassel.de/ACA2016/index.php〉. 〈hal-01409298〉

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