Next-to-minimal weight of toric codes defined over hypersimplices

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 [Formula: see text], 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 [Formula: see text]. 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 [Formula: see text] 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 [Formula: see text] such that [Formula: see text] or [Formula: see text].