On the generalized Hamming weights of hyperbolic codes
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| author | Camps-Moreno, Eduardo García-Marco, Ignacio López, Hiram H. Márquez-Corbella, Irene Martínez-Moro, Edgar Sarmiento, Eliseo |
| author_facet | Camps-Moreno, Eduardo García-Marco, Ignacio López, Hiram H. Márquez-Corbella, Irene Martínez-Moro, Edgar Sarmiento, Eliseo |
| contents | A hyperbolic code is an evaluation code that improves a Reed-Muller because the dimension increases while the minimum distance is not penalized. We give the necessary and sufficient conditions, based on the basic parameters of the Reed-Muller, to determine whether a Reed-Muller coincides with a hyperbolic code. Given a hyperbolic code, we find the largest Reed-Muller containing the hyperbolic code and the smallest Reed-Muller in the hyperbolic code. We then prove that similarly to Reed-Muller and Cartesian codes, the $r$-th generalized Hamming weight and the $r$-th footprint of the hyperbolic code coincide. Unlike Reed-Muller and Cartesian, determining the $r$-th footprint of a hyperbolic code is still an open problem. We give upper and lower bounds for the $r$-th footprint of a hyperbolic code that, sometimes, are sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_12594 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the generalized Hamming weights of hyperbolic codes Camps-Moreno, Eduardo García-Marco, Ignacio López, Hiram H. Márquez-Corbella, Irene Martínez-Moro, Edgar Sarmiento, Eliseo Information Theory Commutative Algebra 94B05, 13P25, 14G50, 11T71 A hyperbolic code is an evaluation code that improves a Reed-Muller because the dimension increases while the minimum distance is not penalized. We give the necessary and sufficient conditions, based on the basic parameters of the Reed-Muller, to determine whether a Reed-Muller coincides with a hyperbolic code. Given a hyperbolic code, we find the largest Reed-Muller containing the hyperbolic code and the smallest Reed-Muller in the hyperbolic code. We then prove that similarly to Reed-Muller and Cartesian codes, the $r$-th generalized Hamming weight and the $r$-th footprint of the hyperbolic code coincide. Unlike Reed-Muller and Cartesian, determining the $r$-th footprint of a hyperbolic code is still an open problem. We give upper and lower bounds for the $r$-th footprint of a hyperbolic code that, sometimes, are sharp. |
| title | On the generalized Hamming weights of hyperbolic codes |
| topic | Information Theory Commutative Algebra 94B05, 13P25, 14G50, 11T71 |
| url | https://arxiv.org/abs/2107.12594 |