On the regularity index of the minimum distance function in projective nested Cartesian codes
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915949543686144 |
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| author | Carvalho, Cicero Pinto, Maria Vaz Villarreal, Rafael H. |
| author_facet | Carvalho, Cicero Pinto, Maria Vaz Villarreal, Rafael H. |
| contents | Let $X$ be a projective nested product of fields and let $δ_X(d)$ be the minimum distance in degree $d\geq 1$ of the projective nested Cartesian code $C_X(d)$. The regularity index ${\rm reg}(δ_X)$ of the minimum distance function $δ_X$ is the minimum integer $d_0\geq 0$ such that $δ_X(d)=1$ for $d\geq d_0$. We give a formula for ${\rm reg}(δ_X)$ by determining an indicator function of least degree for each point of $X$ and using the fact that ${\rm reg}(δ_X)$ is the ${\rm v}$-number of the vanishing ideal $I_X$ of $X$. Then we give an arithmetical criterion that characterizes when $X$ is Cayley--Bacharach. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_20729 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the regularity index of the minimum distance function in projective nested Cartesian codes Carvalho, Cicero Pinto, Maria Vaz Villarreal, Rafael H. Commutative Algebra Information Theory Algebraic Geometry 13P25, 14G50, 94B27 Let $X$ be a projective nested product of fields and let $δ_X(d)$ be the minimum distance in degree $d\geq 1$ of the projective nested Cartesian code $C_X(d)$. The regularity index ${\rm reg}(δ_X)$ of the minimum distance function $δ_X$ is the minimum integer $d_0\geq 0$ such that $δ_X(d)=1$ for $d\geq d_0$. We give a formula for ${\rm reg}(δ_X)$ by determining an indicator function of least degree for each point of $X$ and using the fact that ${\rm reg}(δ_X)$ is the ${\rm v}$-number of the vanishing ideal $I_X$ of $X$. Then we give an arithmetical criterion that characterizes when $X$ is Cayley--Bacharach. |
| title | On the regularity index of the minimum distance function in projective nested Cartesian codes |
| topic | Commutative Algebra Information Theory Algebraic Geometry 13P25, 14G50, 94B27 |
| url | https://arxiv.org/abs/2604.20729 |