Abstract 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 $$(\mathbb {F}_q^*)^s$$ ( 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 $$3 \le d \le \frac{s - 2}{2}$$ 3 ≤ d ≤ s - 2 2 or $$\frac{s + 2}{2} \le d < s$$ 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 $$3 \le d < s$$ 3 ≤ d < s (respectively, $$3 \le d \le \frac{s - 2}{2}$$ 3 ≤ d ≤ s - 2 2 or $$\frac{s + 2}{2} \le d < s$$ s + 2 2 ≤ d < s ).

Projective Reed-Muller codes were introduced by Lachaud in 1988 and their dimension and minimum distance were determined by Serre and Sorensen in 1991. In coding theory, one is also interested in the higher Hamming weights, to study the code performance. Yet, not many values of the higher Hamming weights are known for these codes, not even the second lowest weight (also known as the next-to-minimal weight) is completely determined. In this paper, we determine all the values of the next-to-minimal weight for the binary projective Reed-Muller codes, which we show to be equal to the next-to-minimal weight of Reed-Muller codes in most, but not all, cases.

Høholdt, van Lint, and Pellikaan used order functions to construct codes by means of Linear Algebra and Semigroup Theory only. However, Geometric Goppa codes that can be represented by this method are mainly those based on just one point. In this correspondence, we introduce the concept of near order function with the aim of generalizing this approach in such a way that a wider family of Geometric Goppa codes can be studied on a more elementary setting.