Indicator functions, v-numbers and Gorenstein rings in the theory of projective Reed-Muller-type codes
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914827971067904 |
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| author | González-Sarabia, Manuel Muñoz-George, Humberto Ordaz, Jorge A. Sáenz-de-Cabezón, Eduardo Villarreal, Rafael H. |
| author_facet | González-Sarabia, Manuel Muñoz-George, Humberto Ordaz, Jorge A. Sáenz-de-Cabezón, Eduardo Villarreal, Rafael H. |
| contents | For projective Reed--Muller-type codes we give a global duality criterion in terms of the v-number and the Hilbert function of a vanishing ideal. As an application, we provide a global duality theorem for projective Reed--Muller-type codes over Gorenstein vanishing ideals, generalizing the known case where the vanishing ideal is a complete intersection. We classify self dual Reed-Muller-type codes over Gorenstein ideals using the regularity and a parity check matrix. For projective evaluation codes, we give a duality theorem inspired by that of affine evaluation codes. We show how to compute the regularity index of the $r$-th generalized Hamming weight function in terms of the standard indicator functions of the set of evaluation points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_13728 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Indicator functions, v-numbers and Gorenstein rings in the theory of projective Reed-Muller-type codes González-Sarabia, Manuel Muñoz-George, Humberto Ordaz, Jorge A. Sáenz-de-Cabezón, Eduardo Villarreal, Rafael H. Commutative Algebra Information Theory Algebraic Geometry 13P25, 14G50, 94B27, 11T71 For projective Reed--Muller-type codes we give a global duality criterion in terms of the v-number and the Hilbert function of a vanishing ideal. As an application, we provide a global duality theorem for projective Reed--Muller-type codes over Gorenstein vanishing ideals, generalizing the known case where the vanishing ideal is a complete intersection. We classify self dual Reed-Muller-type codes over Gorenstein ideals using the regularity and a parity check matrix. For projective evaluation codes, we give a duality theorem inspired by that of affine evaluation codes. We show how to compute the regularity index of the $r$-th generalized Hamming weight function in terms of the standard indicator functions of the set of evaluation points. |
| title | Indicator functions, v-numbers and Gorenstein rings in the theory of projective Reed-Muller-type codes |
| topic | Commutative Algebra Information Theory Algebraic Geometry 13P25, 14G50, 94B27, 11T71 |
| url | https://arxiv.org/abs/2308.13728 |