Narrow Results

Toric codes are a type of evaluation codes introduced by J. P. Hansen in 2000. They are produced by evaluating (a vector space composed by) polynomials at the points of ${({𝔽}_q^{\ast })}^s$, the monomials of these polynomials being related to a certain polytope. Toric codes related to hypersimplices are the result of the evaluation of a vector space of square-free homogeneous polynomials of degree $d$. The dimension and minimum distance of toric codes related to hypersimplices have been determined by Jaramillo et al. in 2021. The next-to-minimal weight in the case $d=1$ has been determined by Jaramillo–Velez et al. in 2023. In this work, we use tools from Gröbner basis theory to determine the next-to-minimal weight of these codes for $d$ such that $3\le d\le \frac{s-2}{2}$ or $\frac{s+2}{2}\le d<s$.

For the family of graded lattice ideals of dimension 1, we establish a complete intersection criterion in algebraic and geometric terms. In positive characteristic, it is shown that all ideals of this family are binomial set-theoretic complete intersections. In characteristic zero, we show that an arbitrary lattice ideal which is a binomial set-theoretic complete intersection is a complete intersection.

A hyperbolic code is an evaluation code that improves a Reed–Muller code because the dimension increases while the minimum distance is not penalized. We give necessary and sufficient conditions, based on the basic parameters of the Reed–Muller code, to determine whether a Reed–Muller code coincides with a hyperbolic code. Given a hyperbolic code ${𝒞}$, we find the largest Reed–Muller code contained in ${𝒞}$ and the smallest Reed–Muller code containing ${𝒞}$. We then prove that similar to Reed–Muller and affine Cartesian codes, the $r$th generalized Hamming weight and the $r$th footprint of the hyperbolic code coincide. Unlike for Reed–Muller and affine Cartesian codes, determining the $r$th footprint of a hyperbolic code is still an open problem. We give upper and lower bounds for the $r$th footprint of a hyperbolic code that, sometimes, are sharp.