On the regularity index of the minimum distance function in projective nested Cartesian codes
This work addresses a theoretical problem in algebraic coding theory, specifically for researchers studying projective nested Cartesian codes, and appears incremental as it extends known results to a more general setting.
The paper tackles the problem of determining the regularity index of the minimum distance function for projective nested Cartesian codes, providing a formula for it and characterizing when the underlying set is Cayley-Bacharach.
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.