On the number of minimal and next-to-minimal weight codewords of toric codes over hypersimplices

Toric codes are a type of evaluation code introduced by J.P. Hansen in 2000. They are produced by evaluating (a vector space composed of) polynomials at the points of (Fq^*)ˢ ( F q ∗ ) 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 homogeneous monomially square-free polynomials of degree d . The dimension and minimum distance of toric codes related to hypersimplices have been determined by Jaramillo et al. in (Des Codes Cryptogr 89:269–300, 2021). The next-to-minimal weight in the case $$d = 1$$ d = 1 has been determined by Jaramillo-Velez et al. in (São Paulo J Math Sci 17:188– 207, 2023), and has been determined in the cases where 3 ≤ d ≤ s - 2/2 3 ≤ d ≤ s - 2 2 or s + 2/2 ≤ d < s s + 2 2 ≤ d < s , by Carvalho and Patanker in 2024. In this work we characterize and determine the number of minimal (respectively, next-to-minimal) weight codewords when 3 ≤ d < s 3 ≤ d < s (respectively, 3 ≤ d ≤ s - 2/2 3 ≤ d ≤ s - 2 2 or s + 2/2 ≤ d < s s + 2 2 ≤ d < s ).