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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.
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https://hal.inria.fr/hal-01409298
Contributor : Irene Márquez Corbella <>
Submitted on : Monday, December 5, 2016 - 7:50:31 PM
Last modification on : Thursday, March 5, 2020 - 4:55:14 PM
Long-term archiving on: : Monday, March 20, 2017 - 9:24:52 PM

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  • HAL Id : hal-01409298, version 1

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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. ⟨hal-01409298⟩

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