The dual of an evaluation code
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866910319144599552 |
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| author | López, Hiram H. Soprunov, Ivan Villarreal, Rafael H. |
| author_facet | López, Hiram H. Soprunov, Ivan Villarreal, Rafael H. |
| contents | The aim of this work is to study the dual and the algebraic dual of an evaluation code using standard monomials and indicator functions. We show that the dual of an evaluation code is the evaluation code of the algebraic dual. We develop an algorithm for computing a basis for the algebraic dual. Let $C_1$ and $C_2$ be linear codes spanned by standard monomials. We give a combinatorial condition for the monomial equivalence of $C_1$ and the dual $C_2^\perp$. Moreover, we give an explicit description of a generator matrix of $C_2^\perp$ in terms of that of $C_1$ and coefficients of indicator functions. For Reed--Muller-type codes we give a duality criterion in terms of the v-number and the Hilbert function of a vanishing ideal. As an application, we provide an explicit duality for Reed--Muller-type codes corresponding to Gorenstein ideals. In addition, when the evaluation code is monomial and the set of evaluation points is a degenerate affine space, we classify when the dual is a monomial code. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_10016 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The dual of an evaluation code López, Hiram H. Soprunov, Ivan Villarreal, Rafael H. Commutative Algebra Information Theory Algebraic Geometry Combinatorics 13P25, 14G50, 94B27, 11T71 The aim of this work is to study the dual and the algebraic dual of an evaluation code using standard monomials and indicator functions. We show that the dual of an evaluation code is the evaluation code of the algebraic dual. We develop an algorithm for computing a basis for the algebraic dual. Let $C_1$ and $C_2$ be linear codes spanned by standard monomials. We give a combinatorial condition for the monomial equivalence of $C_1$ and the dual $C_2^\perp$. Moreover, we give an explicit description of a generator matrix of $C_2^\perp$ in terms of that of $C_1$ and coefficients of indicator functions. For Reed--Muller-type codes we give a duality criterion in terms of the v-number and the Hilbert function of a vanishing ideal. As an application, we provide an explicit duality for Reed--Muller-type codes corresponding to Gorenstein ideals. In addition, when the evaluation code is monomial and the set of evaluation points is a degenerate affine space, we classify when the dual is a monomial code. |
| title | The dual of an evaluation code |
| topic | Commutative Algebra Information Theory Algebraic Geometry Combinatorics 13P25, 14G50, 94B27, 11T71 |
| url | https://arxiv.org/abs/2012.10016 |