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| Hauptverfasser: | , , , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2505.14472 |
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| _version_ | 1866910957679149056 |
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| author | Moreno, Eduardo Camps Fidalgo-Díaz, Adrián López, Hiram H. Martínez-Peñas, Umberto Ruano, Diego San-José, Rodrigo |
| author_facet | Moreno, Eduardo Camps Fidalgo-Díaz, Adrián López, Hiram H. Martínez-Peñas, Umberto Ruano, Diego San-José, Rodrigo |
| contents | Multivariate multiplicity codes have been recently explored because of their importance for list decoding and local decoding. Given a multivariate multiplicity code, in this paper, we compute its dimension using Gröbner basis tools, its dual in terms of indicator functions, and explicitly describe a parity-check matrix. In contrast with Reed--Muller, Reed--Solomon, univariate multiplicity, and other evaluation codes, the dual of a multivariate multiplicity code is not equivalent or isometric to a multiplicity code (i.e., this code family is not closed under duality). We use our explicit description to provide a lower bound on the minimum distance for the dual of a multiplicity code. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_14472 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Duals of multiplicity codes Moreno, Eduardo Camps Fidalgo-Díaz, Adrián López, Hiram H. Martínez-Peñas, Umberto Ruano, Diego San-José, Rodrigo Information Theory Multivariate multiplicity codes have been recently explored because of their importance for list decoding and local decoding. Given a multivariate multiplicity code, in this paper, we compute its dimension using Gröbner basis tools, its dual in terms of indicator functions, and explicitly describe a parity-check matrix. In contrast with Reed--Muller, Reed--Solomon, univariate multiplicity, and other evaluation codes, the dual of a multivariate multiplicity code is not equivalent or isometric to a multiplicity code (i.e., this code family is not closed under duality). We use our explicit description to provide a lower bound on the minimum distance for the dual of a multiplicity code. |
| title | Duals of multiplicity codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2505.14472 |